eel 4140 lab manual

LABORATORY MANUAL

DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING

UNIVERSITY OF CENTRAL FLORIDA

EEL 4140 ANALOG FILTERS DESIGN

Revised September 2005

TABLE OF CONTENTS

SAFETY RULES AND OPERATING PROCEDURES

LABORATORY SAFETY INFORMATION

EXPERIMENT # 1 Effect of Op Amp Frequency Dependence on Finite Gain Amplifiers and Bandwidth Extension Techniques using Composite Op Amps (CNOA)

Study Guide A Composite Operational Amplifiers: Generation and Finite-Gain Applications

Study Guide B Inverting Integrator and Active Filter Applications of Composite Operational Amplifiers

EXPERIMENT #2 Real Zero and Pole Synthesis

EXPERIMENT # 3 Sallen-Key Filters

EXPERIMENT # 4 State-Variable Biquads

EXPERIMENT # 5 Single Op Amp Band-Pass Filters

EXPERIMENT # 6 Two Op Amps Current Generalized Immittance

Structure (CGIC) Based Biquad Study Guide C Biquads II: The current Generalized Immittance (CGIC)

Structure

EXPERIMENT # 7 High-Order Low-Pass Filter Design

Study Guide C Biquads II: The current Generalized Immittance (CGIC) Structure

EXPERIMENT # 8 Butterworth Filter Approximation APPENDIX

LIST OF COMPONENTS

RESISTOR COLOR CODE TUTORIAL

ii

Safety Rules and Operating Procedures

1. Note the location of the Emergency Disconnect (red button near the door) to shut off power in an emergency. Note the location of the nearest telephone (map on bulletin board). 2. Students are allowed in the laboratory only when the instructor is present. 3. Open drinks and food are not allowed near the lab benches. 4. Report any broken equipment or defective parts to the lab instructor. Do not open, remove the cover, or attempt to repair any equipment. 5. When the lab exercise is over, all instruments, except computers, must be turned off. Return substitution boxes to the designated location. Your lab grade will be affected if your laboratory station is not tidy when you leave. 6. University property must not be taken from the laboratory. 7. Do not move instruments from one lab station to another lab station. 8. Do not tamper with or remove security straps, locks, or other security devices. Do not disable or attempt to defeat the security camera. 9. ANYONE VIOLATING ANY RULES OR REGULATIONS MAY BE DENIED ACCESS TO THESE FACILITIES. I have read and understand these rules and procedures. I agree to abide by these rules and procedures at all times while using these facilities. I understand that failure to follow these rules and procedures will result in my immediate dismissal from the laboratory and additional disciplinary action may be taken. ________________________________________ ________________ Signature Date Lab #

iii

Laboratory Safety Information Introduction The danger of injury or death from electrical shock, fire, or explosion is present while conducting experiments in this laboratory. To work safely, it is important that you understand the prudent practices necessary to minimize the risks and what to do if there is an accident. Electrical Shock Avoid contact with conductors in energized electrical circuits. Electrocution has been reported at dc voltages as low as 42 volts. 100ma of current passing through the chest is usually fatal. Muscle contractions can prevent the person from moving away while being electrocuted. Do not touch someone who is being shocked while still in contact with the electrical conductor or you may also be electrocuted. Instead, press the Emergency Disconnect (red button located near the door to the laboratory). This shuts off all power, except the lights. Make sure your hands are dry. The resistance of dry, unbroken skin is relatively high and thus reduces the risk of shock. Skin that is broken, wet, or damp with sweat has a low resistance. When working with an energized circuit, work with only your right hand, keeping your left hand away from all conductive material. This reduces the likelihood of an accident that results in current passing through your heart. Be cautious of rings, watches, and necklaces. Skin beneath a ring or watch is damp, lowering the skin resistance. Shoes covering the feet are much safer than sandals. If the victim isn’t breathing, find someone certified in CPR. Be quick! Some of the staff in the Department Office are certified in CPR. If the victim is unconscious or needs an ambulance, contact the Department Office for help or call 911. If able, the victim should go to the Student Health Services for examination and treatment. Fire Transistors and other components can become extremely hot and cause severe burns if touched. If resistors or other components on your proto-board catch fire, turn off the power supply and notify the instructor. If electronic instruments catch fire, press the Emergency Disconnect (red button). These small electrical fires extinguish quickly after the power is shut off. Avoid using fire extinguishers on electronic instruments. Explosions When using electrolytic capacitors, be careful to observe proper polarity and do not exceed the voltage rating. Electrolytic capacitors can explode and cause injury. A first aid kit is located on the wall near the door. Proceed to Student Health Services, if needed.

iv

EEL 4140 ANALOG FILTERS LABORATORY 1

Effect of Op Amp Frequency Dependence on Finite Gain Amplifiers and Bandwidth

Extension Techniques using Composite Op Amps (CNOA) I. Objective

To understand the effect of finite gain bandwidth product of practical Op Amps in finite gain applications, and to study the concept of Composite Op Amps (CNOA).

II. Introduction

The simplified model of a practical Op Amp is shown in Fig.1. In this model, and

are the input and output resistance, respectively. The open-loop gain

IR OR

)( ωjAOL can be written

as:

0

0

1)(

ωωω

j

AjAOL

+= (1)

Where is the DC open-loop gain, and 0A 0ω is the dominant-pole frequency.

IROR

a

b

o+

−

+

−

aV

bV

oV

[ ]baOL VVjA −)( ω

Fig.1 The simplified model of a practical Op-amp

The relationship between )( ωjAOL and the frequency ω is shown in Fig.2. In this figure,

the gain-bandwidth product Gω , or the unity-gain bandwidth, is defined as:

00ωω AG ≈ (2)

( )dB )( ωjAOL

( )ωlog( )0log ω ( )Gωlog

( )dB 0A

( )dB 0

Fig.2 The magnitude response of the open-loop gain

The positive finite gain amplifier, as shown in Fig. 3, is used to illustrate the bandwidth shrinkage by the voltage gain K . In this amplifier, the input and output resistances of the Op Amp are assumed to be infinite and zero, respectively. The transfer function of this amplifier can be derived as:

⎟⎠⎞

⎜⎝⎛ +

+×

+=

=

KA

jKA

AjVjVjT

i

o

00

0

0

11

1

1

)()()(

ω

ω

ωωω

(3)

Normally, it is true that

KA >>0 (4)

Therefore, Equation (3) can be simplified as:

KAj

KjT

00

1

1)(

ω

ωω

+×≈ (5)

and the cutoff frequency, or 3dB frequency, of this amplifier, is given by:

⎟⎠

⎞⎜⎝

⎛=KA

cutoff0

0ωω (6)

From Equation (6), it is clear that the cutoff frequency is inversely proportional to the gain of the amplifier, K . Therefore, the bandwidth of a single Op Amp amplifier realization

shrinks by a factor of K1 .

+

-RR =1

( )RKR 12 −=

)( ωjVi

)( ωjVo

Fig 3 The positive finite gain amplifier

One method to increase the bandwidth is to use the N-stage amplifier, each stage having the

gain of N K to realize an overall gain K . Consequently, the bandwidth of each stage shrinks

by N K

1 . In addition, cascading N stages introduces another shrink factor of 121

−N [1].

In total, the bandwidth of the N-stage amplifier shrinks by a factor of N

N

K12

1

− .

Composite Op Amp (CONA)

The CNOA is constructed using N Op Amps and 2(N-1) resistors, resulting in (N-1) resistor ratios. These resistor ratios can be used advantageously to reduce the deviation of the overall active realization and to guarantee the system stability. The CNOA is versatile since it has three external terminals that correspond to those of the regular Op Amps. CNOA has other important

properties, such as stability with one- and two-pole Op Amps model, low sensitivity to component and Op Amps mismatch, and wide dynamic range.

As shown in References [2, 3] attached, the bandwidth of the amplifier constructed by

CNOA shrinks only by a factor ofN K

1 . Compared with the shrinkage factor of K1 for the

single Op Amp amplifier, and N

N

K12

1

− for the N-stage amplifier, the CNOA considerably

extends the useful bandwidth.

In this experiment, we only discuss C2OA, composed of two Op Amps and 2 resistors. For the general case of CNOA, see References [2, 3].

Four different C2OA structures, referred as C2OA-1, C2OA-2, C2OA-3, and C2OA-4, are found to meet the good performance criteria, and are shown in Fig. 4. Here, α is the resistor ratio. The open-loop gain of the Op Amps used in the modeling of the C2OAs (assuming a single-pole model) is

2or 1 )(,0

,0, =+

= ijA

jAi

iioi ωω

ωω (7)

where , and ioA , i,0ω are the DC open-loop gain and the cutoff frequency of the ith Op Amp,

respectively.

The output voltage of the C2OAs is given by:

4,3,2,1 ),()( =−= mjAVjAVV bmbamaom ωω (8)

where for C2OA-1

( )( )( )

( )( )αω

αωωαω

αωω+++

−++

++=

1)(1)()(

1)(1)(1)(

1

21

1

121 jA

jAjAVjA

jAjAVV bao (9)

for C2OA-2

( )( )

( )( )αω

αωωαωαωω

+++

−+++

=1)(

1)()(1)(

1)()(

2

21

2

212 jA

jAjAVjA

jAjAVV bao (10)

for C2OA-3

( )( )( )α

ωωα

ωω++

−+

=1

)(1)(1

)()( 12213

jAjAVjAjAVV bao (11)

and for C2OA-4

( )( )

( )[ ]( )α

αωωα

αωω+

++−

++

=1

1)()(1

)()( 12124

jAjAVjAjAVV bao (12)

+

-

R Rα

A1

A2

a

b

o

+

-

oa

b

C2OA1

+

-

+

-

R

Rα

A1

A2

a

b

o

+

-

oa

b

C2OA2

+

-

(a) (b)

+

-R

RαA1

A2

a

b

o

+

-

o

a

bC2OA3

+

-

+

-

R

RαA1

A2

a

b

o

+

-

o

a

b

C2OA4

+

-

(c) (d)

Fig 4. The Composite Operational Amplifiers (C2OAs).

(a) C2OA-1. (b) C2OA-2. (c) C2OA-3. (d) C2OA-4.

From Equations (9), (10), (11), and (12), the transfer functions of the circuits using C2OA’s can be derived. The applications of the four proposed C2OA’s in the positive and negative finite-gain amplifiers are summarized in Table 1. From this table, it is clear that the 3-dB

bandwidth of the finite amplifier shrinks by a factor of K1 .

Design Procedure for Negative Finite-Gain Amplifiers Employing C2OA-1

1. Design a negative finite-gain amplifier employing C2OA-1. The gain of this amplifier k is 10, and the quality factor Qp of C2OA is 1.

2. Assume the two Op Amps in the C2OA-1 are identical. That is

21 ωω = (13)

and

21 AA = (14)

The quality factor can be simplified as:

kQP

++

=1

1 α (15)

3. Calculate the resistor ratio as:

32.2 11

=−+= kQPα (15)

4. Choose the resistor value in the C2OA-1 as: R

Ω= kR 10 (16)

5. Calculate the resistor value as:

Ω= kR 2.23α (17)

III. Design

1. Positive Finite Gain Amplifiers a. Design a single-stage positive finite gain amplifier with an overall gain of 100 using LM471

Op Amps. Repeat for an overall gain of 25.

b. Design a two-stage positive finite gain amplifier. Each stage has a gain of 10 to realize the overall gain of 100. Use LM471 Op Amps.

c. Design a positive finite gain amplifier using C2OA-1 ( 707.0=pQ ) with close-loop gain of

100. Use LM471 Op Amps.

2. Negative Finite Gain Amplifiers a. Design a single-stage negative finite gain amplifier with an overall gain of 100 using LM471

Op Amps. Repeat for an overall gain of 25.

b. Design a two-stage negative finite gain amplifier. Each stage has a gain of 10 to realize the overall gain of 100. Use LM471 Op Amps.

c. Design a negative finite gain amplifier by using C2OA-1 ( 707.0=pQ ) with an overall gain of

100. Use LM471 Op Amps.

IV. Computer Simulations

1. Positive Finite Gain Amplifiers a. Simulate the single-stage positive finite gain amplifier with overall gain of 100, and plot the

magnitude responses of the amplifier. Repeat for an overall gain of 25. Record and compare these two cutoff frequencies. Relate the bandwidth shrinkage to the amplifier close-loop gain.

b. Simulate the two-stage positive finite gain amplifier with close-loop gain of 100. Plot the magnitude response. Determine the cutoff frequency.

c. Simulate the positive finite gain amplifier with finite gain of 100 using C2OA-1. Plot the magnitude response. Determine the cutoff frequency.

Compare these cutoff frequencies of the three realizations of finite overall gain of 100, and show the useful bandwidth improvement.

2. Negative Finite Gain Amplifiers a. Simulate the single-stage negative finite gain amplifier with closed-loop gain of 100, and plot

the magnitude response of the amplifiers. Repeat for an overall gain of 25. Record and compare these two cutoff frequencies. From the formulas and the graphs compare the bandwidth shrinkage to the amplifier closed-loop gain.

b. Simulate the two-stage negative finite gain amplifier with overall gain of 100. Plot the magnitude response. Determine the cutoff frequency.

c. Simulate the negative finite gain amplifier with close-loop gain of 100 using C2OA-1. Plot the magnitude response. Determine the cutoff frequency.

Compare these cutoff frequencies of the three realizations of finite overall gain of 100, and comment on the useful bandwidth in each realization.

V. Experiments This lab is a computer simulation lab. No actual experiment.

References [1]. J. Millman, and C. Halkias, “Integrated Electronics; Analog and Digital Circuits and Systems,” Mcgraw-Hill, Inc. 1972, pp.386

[2]. Wasfy B. Mikhael, and Sherif Mickael, “Composite Operational Amplifiers: Generation and Finite-Gain Applications,” IEEE Trans. on Circuits and Systems, Vol. 34, No. 5, pp. 449-460, May 1987.

[3]. Sherif Mickael, and Wasfy B. Mikhael, “Inverting Integrator and Active Filter Applications of Composite Operational Amplifiers,” IEEE Trans. on Circuits and Systems, Vol. 34, No. 5, pp. 461-470, May 1987.

Table 1 Negative and Positive Finite Gains i

oV

V Using the C2OA’s

C2OA-i Negative Finite Gain Transfer Function

Positive Finite Gain Transfer Function

The 3-dB bandwidth

pω

The quality factor pQ

C2OA-1 2

2

1

1

ppp

i sQs

T

ωω++

2

21

1

1

ppp

i sQs

s

T

ωω

ω

++

+

k+121ωω

1

2

11

ωωα

k+

+

C2OA-2

2

2

1

1

ppp

i sQs

T

ωω++

2

2

1

1

ppp

i sQs

T

ωω++

k+121ωω

2

1

11

ωωα

k++

C2OA-3

2

21

1

1

ppp

i sQs

s

T

ωω

ω

++

+

2

2

1

1

ppp

i sQs

T

ωω++

( )( )αωω++ 1121

k

( )( )2

111ω

ωα++ k

C2OA-4 ( )

2

21

1

11

ppp

i

ws

Qs

s

T

ωω

ωα

++

++

2

21

1

1

ppp

i sQs

s

T

ωω

ωα

++

+

( )( )α

ωω++ 1121

k

( )( ) 2

1

11

ωαω

++ k

+

-

R

kR

iVoV

C2OA

+

-

R

kR

iVoV

C2OA

ii

o TkVV

=−= i

i

o TkVV

=+= )1( iT is the ideal

transfer function

Study Guide A


Composite Operational Amplifiers: Generation and Finite-Gain Applications

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-34, NO. 5, MAY 1987 449

Composite Operational Amplifiers: Generation * and Finite-Gain Applications

WASFY B. MIKHAEL, FELLOW, IEEE, AND SHERIF MICHAEL, MEMBER, IEEE

Abstract-A practical and effective general approach is presented for extending the useful operating frequencies and improving the performance of linear active networks realized using operational amplifiers (OA’s). This is achieved by replacing each OA in the active network by a composite operational amplifier (CNOA) constructed using N OA’s.’ The technique of generating the CNOA’s for any given N is proposed. The realizations employing the CNOA are examined according to a stringent performance criterion satisfying such important properties as extended bandwidth, stability with one- and two-pole OA models, low sensitivity to the components and OA mismatch, and wide dynamic range. Several families of CNOA’s, for N = 2, 3, and 4, are shown to satisfy ,the suggested performance criterion. In this contribution, the CNOA’s applications in inverting, noninverting, and differential finite-gain amplifiers are given and shown theoretically and experimentally to compare favorably with the state-of-the-art realizations using the same number of OA’s. Applications of the CNOA in inverting integrator and active filter realizations are presented in a companion contribution (321.

I. INTRODUCTION

L INEAR ACTIVE circuits, namely, positive, negative, and differential finite-gain amplifiers, integrators, and

active filters are usually realized with operational amplifiers (OA’s) as the active elements. These active elements have frequency-dependent gains which restrict the operating frequencies of the linear active circuits. The operating frequencies are defined to be those frequencies for which the deviation of the actually obtained transfer function T’(s) of an active realization from its ideal value q(s) (due to the OA’s finite gain and frequency dependence) falls within a predetermined acceptable range. The frequency limitations due to the passive components are not addressed here. In practice, the passive components restrict operation in a higher frequency range relative to that of the OA.

For practical reasons, extending the useful bandwidth (BW) of the most commonly used linear active circuits has received the attention of many researchers in this field. This has resulted in many contributions, each dealing with the solution of this frequency limitation in specific applications [l]-[14]. Generally, three approaches were considered to minimize the dependence of the realization on the active

Manuscript received October 29,1985; revised August 26, 1986. W. B. hfikhael is with the Electrical Engineering Department, West

Virginia University, Morgantown, WV 26506. S. Michael is with the Department of Electrical and

Engineering, Naval Postgraduate School, Monterey, CA 93943. Computer

IEEE Log Number 861?469.

:@p;+“+ ~qfjj&!+!F~ , 1) N”ll.fOl

(4 (b) A--t-m Fig. 1. (a) An operational amplifier (VCVS) and (b).its nullor represen-

tation.

device parameters and consequently its variations [15], [16]. In the first approach, for a given fixed number of OA’s, the passive configuration in which the OA’s are embedded (called the companion network) is carefully designed. In the second approach, an increased number of OA’s are used to realize a given 7).(s). This results in increased degrees of freedom in choosing the companion network: In the third approach, each OA in a given configuration is simply replaced by an OA that has improved characteristics, such as wider gain-bandwidth product (GBWP).

Recently, the authors suggested an approach using so- called composite operational amplifiers (CNOA’s) that achieved a considerable performance improvement and bandwidth extension of almost all linear active networks used in signal processing and amplification for audio and video communications as well as instrumentation [17]-[20]. This has been verified by other researchers [21], [22]. In addition, the CNOA concept has proved to be useful in nonlinear and high-speed, high-accuracy applications, such as fast A/D and D/A conversion, digital communications, and switched capacitor filtering [23]-[26], [34]. The objective of this paper is to present a comprehensive treatment of the CNOA method and its application in linear active signal processing. In this technique, each OA is replaced by a composite operational amplifier (CNOA) [17]-[20] without modifying the topology of the companion network.

In Section II, the procedure for generating the CNOA’s is presented. First, a general technique for the generation of a number of C20A’s (N = 2) using imllator-norator pairing [27]-[30] is described. Four of the C20A’s are found to meet useful performance criteria; these are re- tained for use in design. To further illustrate the generality of the proposed technique, the generation of the CNOA’s

0098-4094/87/0500-0449$01.00 01987 IEEE

450 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-34, NO. 5, MAY 1987

(4 0) .

7h (4 (H-I)R

Fig. 2. (a)-(d) Four different networks fbr generating the composite operational amplifiers using two single OA’s (C20A’s). (e) -H and (f) +H finite-gain amplifier realizations used in Fig. 4(a)-(d). (g) The composite operational amplifier (C20A-i) symbol.

for N > 2 is described. Sample results of C30A’s and C40A’s, which meet the above performance criteria, are presented.

Use of the proposed CNOArs in inverting and noninverting finite-gain applications is given in Section III. -It is shown theoretically and experimentally that appreciable performance improvements are realized over the present state-of-the-art designs which utilize the same number of OA’S.

II. GENERATION OF COMPOSITE OPERATIONAL AMPLIFIERS (CNOA’s) USING N SINGLE OA’s

A. Generation of the C2OA’s TN = 2) An operational amplifier, shown in Fig. l(a) is a

voltage-controlled voltage source (VCVS). In the ideal case, the input impedance Z, -+ cc. This corresponds to the model shown in Fig. l(b), which uses nullator and norator singular elements [27]-[30]. The ideal OA is replaced by a nullor which is described by

[ :]= [: :I*[ -q. (1)

The matrix in (1) is called the nullor chain transmission matrix of an ideal OA. In any physical circuit that contains N OA’s, replacing each OA by a nullor results in a nullor equivalent network. The nullors can further be split into nullators and norators to yield a nullator-norator equivalent network.

Similarly, a nullator-norator equivalent network containing N nullators and N norators yields N! nullor equivalent networks, since nullators and norators can be paired into nullors in an arbitrary manner.

Although the nullator (or norator) alone is not an ad- missible element for modeling a physical network, the nullor, like the infinite-gain controlled source, can be used for this purpose. The equivalence established is valid whether A + cc or A + - cc, and so in practice a nullor can be replaced by a high-gain differential controlled source in two ways. In general, a nullor equivalent network containing N nullors corresponds to 2” physical networks. Each of these N! nullor networks yields a physical realization which has a different dependence on the nonideal active elements. This subject is well documented in the literature [27]-[30].

The procedure to generate. the C20A’s is as follows. In the first step, a redundant amplifier of finite gain + H

MIKHAEL AND MICHAEL: COMPOSITE OPERATIONAL AMPLIFIERS 451

.Fig. 3.

I//’ (4

The composite operational amplifiers (C20A’s).

(Fig. 2(e) and (f)) is combined with a single OA such that the chain matrix of the resulting two-amplifier network (assuming ideal amplifiers) corresponds to that of a nullor, as given in (1). That is, although each network contains two VCVS’s, the overall two-port network realizes one VCVS. Six possible topologies can be obtained for each of the four networks shown in Fig. 2(a)-(d). Two topologies are obtained, one for + H and the other for - H, for each position of the three-way switch, leading to six topologies per network. It is easy to show that 17 of the 24 topologies realize true nullors; i.e., no special stipulation on network elements or signals is required. Eight possible OA realizations can be obtained from each of these 17 topologies (nullor networks). This results in 136 composite operational amplifiers (C20A’s), each constructed using two single OA’s. The resulting C20A’s, shown in Fig. 2(g), are examined according to the following performance criteria.

i) Let A,(s) and Ah(s) denote the noninverting and inverting open-loop gains of each of the 136 C20A’s examined. The denominator polynomial coefficients of A,(s) and Ah(s) should show no change in sign. This satisfies the necessary (but not sufficient) conditions for stability. Also, none of the numerator or denominator coefficients of A,(s) and Ah(s) should be realized through differences. This eliminates the need for single OA’s with matched GBWP’s and results in low sensitivity of the C20A with respect to its components.

ii) The external three-terminal performance of the C20A should resemble as closely as possible that of the single OA.

iii) No right-half s-plane (RHS) zeros due to the single OA pole should be allowed in the closed-loop gains of the C20A’s (for minimum phase shifts).

i ’ I/ /’ (4

(a) C20A-1. (b) C20A-2. (c) C20A-3. (d) C20A-4

iv) The resulting input-output relationship T,(s) in the applications considered should have extended frequency operation with minimum gain and phase deviation from the ideal q(s). The improvement should be sufficient to justify the increased number of OA’s.

Four C20A’s referred to as C20A-1, C20A-2, C20A-3 and C20A-4, of the 136 examined are found to meet these performance criteria, and are shown in Fig. 3.

It is interesting to note that a special case of C20A-3 can be derived from the transistor Darlington pair [31], where the norators are both at ac ground in the Darlington pair enabling an OA realization.

The open-loop gain of the single OA’s used in the modeling of the C20A’s (assuming a single-pole model) is

AoitiLi wi Ai=-== i=lor2

WLi+S s + WLi ’ (4

where Aoi, wLi and wi are the dc open-loop gain, the 3-dB bandwidth, and the GBWP of the ith single OA, respectively.

It can be easily shown that the open-loop input-output relationships for the C20A-1 to C20A-4 are given by

V,, = K’,Aai(s)- GAbi (i=l..-4)

where for C20A-1

v = v Ad+ Ad(l+ a) _ v A,A,(l+ a) 01 a A,+(l+a) bAI+(l+ci) (3)

for C20A-2

Yd = v, AAl + a> A,+(l+a)

_ v AAl + 4 bA2+(1+a) (4

452 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS. VOL. CAS-34, NO. 5, MAY 1987

a + 0

I. C30A- I ’ ‘\ b

I ‘\ a-

cl& ‘\

I -\

I ‘.

‘\

Fig. 4. The composite operational amplifiers (C30A’s). (a) C3OA-1. (b) C30A-2. (c) C30A-3. (d) C30A-4. (e) C30A-5. (f) C30A-6.

for C20A-3

and for C20A-4

4(4+d -v T/od=K (l+a)

A,[A,+(l+a)l b

(1+4 (6)

where QI is a resistor ratio, as illustrated in Fig. 3. Assuming identical OA’s, i.e.,

A,l=A,z=A, and 01=w2=wi

it is interesting to examine the open-loop gains given by (3)-(6) in the single-ended inverting application, i.e., when v, = 0.

For C20A-1 and C20A-2, the open-loop dc gain Aocl is given by

A A&+ a>

Ocl = 1+ (1 -I- a)/A, = A,(1 + a) for (l+ a) <<A,.

(74

From (2) and (7a), the composite amplifier has a single-pole rolloff from q/A, to w,/(l+ or), where the second pole occurs. As (Y increases, the dc gain increases while the frequency of the second pole decreases.

Also, from (5) and (6), both the C20A-3 and the C20A-4 have an open-loop dc gain given by

(7b)

From (2) and (7b), AoC2 has double poles (12-dB/octave) at wi/A,, and as (Y increases the dc gain decreases without affecting. the location of the second pole.

Only the C20A-2 has identical expressions for the positive and negative open-loop gains A, and A,. Thus, common-mode rejection ratio (CMRR) problems should not be encountered using C20A-2, even for relatively large common-mode signal applications. From (3), (53, and (6), the CMRR of the C20A-1 and C20A-3 is (A,, + l/2), while that of the C20A-4 is (A,, + (Y + l/2). For single- ended applications (small common-mode signal), no prob-

MIKHAEL MICHAEL: COMPOSITE OPERATIONAL AMPLIFIERS 453

Fig. 5. The composite operational amplifiers (C40A’s). (a) C40A-1. (b) C40A-2. (c) C40A-3. (4 C40A-4 (e) C40A-5.

lems are anticipated in using the C20A-1, C20A-3, or C20A-4, as verified experimentally later.

It is easy to show that the voltage swing at the first OA(A,) output, which is an internal node in each of C20A-1 to C20A-4, is always less than the output voltage V,. Hence, the dynamic range is determined by the voltage swing of the output voltage V,. Consequently, no dynamic-range reduction of V, or harmonic distortion problems should arise.

B. Generation of CNOA’s (N > 2) Following an analogous approach, CNOA’s for N > 2

can be generated for extending the operating frequencies at the expense of additional amplifiers. The CNOA’s can be obtained in two different ways. The first approach starts from the basic single OA with additional redundant amplifiers. Then, nullator-norator pairing is used as described in Section II-A.

In the second approach, which is used here, the C20A’s are used as single OA replacements in the C20A structure.

Although this second approach is not exhaustive, it will be shown to yield excellent results. Hence, C30A’s are obtained by starting with one of the proposed C20A’s and replacing one of its single OA’s by any ,of the C20A’s in Fig. 3. Thirty-two possible combinations of C30A’s can be obtained using the four proposed C20A’s.

Similarly, C40A’s are generated here by replacing each of the single OA’s in a C20A with any of the C20A’s or by replacing one of the single OA’s in C30A’s with a C20A. This results in many possible combinations of C40A’s. The process can be continued (by using C20A’s, C30A’s, and C40A’s) to obtain CNOA’s for any number N. N should be limited in practice; the increased complex- ity is expected to give rise to practical problems in spite of the advantages of an extended operating range.

Samples of C30A and C40A novel designs, which meet the performance criteria described in Section II-A, are shown in Figs. 4 and 5. The open-loop expressions of these C30A’s and C40A’s, as well as others, can be found elsewhere [33].


TABLE I NEGATIVE AND POSITIVE FINITE GAINS V,/ v] USING THE C20A’s

CZOA-i Negative Finite Gain Trans. Function Positive Finite Gain Trans. Function (1.3) (Ta) wP QP

1 (1Ww) WlW2 C20A-1 T,

1 + (ShPQP) + (S%;) Ti 1 + (S/upQ;) + (S%;) I-

(W w2 - - l+k i- ‘fl+k WI

1 1 w1w.T C20A-2 T,

1 + (S/opQp) + (S*/$) Ti

1 + WwpQp) + (S'/$) F

(14 WI --

l+k l- f-K wy

* (l+S/u) 1

CZOA-3 Ti 1 + (ShPQP) + (S*/w;)

Ti 1 + (S/wpQp) + (S%;)

f--x-& yyaz

Cl+(l+~wwl (l+Ww1) C20A-4 Ti

1 + (S/wPQP) + (S%;) Ti 1 + (VwPQP) + (S'/$) ji$$ igz

vow *va

kR kR

!!.c= vi

- k=Ti F= (ltk) = Ti 1 Ti(idea1 Transfer Function)

*aR1 e kR (for maximum up).

III. REALIZATION OF POSITIVE, NEGATIVE, AND LR

DIFFERENTIAL FINITE-GAIN AMPLIFIERS V2

A. Finite Gain Amplifiers Using the Proposed C2OA’s The application of the four proposed C20A’s in positive

and negative finite-gain amplification is given in Table I. For the differential finite-gain realization, often referred to

VI

as the instrumentation amplifier shown in Fig 6, C20A-2 is used. The network in Fig. 6, when each single OA is modeled by (2), can be shown to have the input-output relationship given by

1 1 + s/wPQP + s ‘/o, I

v,

1 + T2 1 + s/oPQP + s ‘,‘a,” 1 v, (8)

where q1 = x(1 + k)/(l+ x) q2=-k

Qp= w1

J 020+ k)

(1+ Lx).

For the differential gain application given above and the finite-gain applications in Table III, the actual input-output relationship T, has the form

T,=q.; (9)

Fig. 6. Application of the C20A-2 as a differential finite-gain amplifier.

where q = the transfer function realized assuming ideal OA’s and

N=l+a~=l+~ *z

(a is zero ( w, + cc) in some cases)

D=1+b,s+b2s2=1+(s/WpQp)+(s2/~;).

Thus, N/D indicates the amplitude and phase deviation of T, from q. Also, b, and b, determine the stability of T,, while a, b,, and b,, and consequently oZ, wP, and QP, are functions of the circuit parameters or, w2, and (Y. None of the a and b coefficients is realized through differences, which guarantees the low sensitivity of T,, o,, o,,, and Q.P to the circuit parameters. On the other hand, the b coefficients are always positive (assuming a single-pole ,OA model), which is necessary for the stability of the transfer function. From Table I, a mismatch of k 5 percent in w1 and w2 results in a +Spercent change in wP and a *2.5-percent change in QP. Hence, single OA’s with mis- matched gain-bandwidth products within practical ranges can be used without appreciably affecting the stability or the sensitivity of the finite-gain realizations.


TABLE II VALUESOF~ FORMAXIMALLYFLATANDFOR Q,, =l FINITE-GAINREALIZATIO~S~SING C20A’s ANDTHE

CORRESPONDINGBANDWIDTHAND STABILITYCONDITIONS

CZOA-i 1%

1

PP -OF! Stability Condition

for (I used

c2oA-1 m 1 -2

Satisfied

fm

(ind$ment Satisfied

1 C20A-3 ( 0 1 OPmin = m(. & ( Unsatisfied (

I I I I I I (l+k) ! 9 Unsatisfied

m

C20A-4

2(l+k) 1

-z- 9 Unsatisfied

&-(ltk)

I I I I I I

1) Effect of the Single OA’s Second Pole on the Stability of the C2OA’s Finite-Gain Realizations: In the following, the stability properties of the positive and negative finite- gain amplifier realizations using a two-pole, open-loop model of the single OA’s is studied. If the dc gains of the first and second OA’s (A, and A,, respectively) are assumed to be equal, this greatly simplifies the analysis without affeCting the reliability.of the conclusions. This is due to the absence of gain difference terms in all the gain expressions obtained, as seen from (3)-(6) (8), and Table I. Let I

A=A,=A,

where 114 is given by

.f= (1+;)( --&+k), (10) 1

and wh >> wL. By invoking the Routh-Hurwitz stability criterion [16],

the necessary ‘and sufficient condition for stability using the C20A-i’ .or C20A-2 [33] is found to be

0+9<(1+W (11) while for the C20A-3 the condition [33] is found to be

(l&)>/m. (14 Finally, for the C20A-4 the condition [33] is given by

(1+X) >4(1$k). 03) Upon examining (ll), (12), and (13), one finds that e

imposed hy the stability conditions (not necessarily BW conditions): is physically realizable for all k. In practice, (Y should be chosen in the stable range for a given .k’ that results in” the best realizable value of QP and ‘o,.“Table II gives the values of (Y required to yield QP = l/a and Q, = 1 for the realizations in Table I. The usefui BW’s of the different finite-gain amplifiers can be obtained by

dB b

C20A-I Proposed design 32 - for op= ,707

a:4 30 , , r , , , , -f kh.

IO 20 30 40 50 60 70 80 90 100 110 120 130 140 I50

(4

0

-10

-20

-30

-40

-50

-60

-70

-80

(b) (b) Fi 7. Fi 7.

8 - 8 - Theoretical responses of the negative finite-gain amplifiers using Theoretical responses of the negative finite-gain amplifiers using

20A 1 and the existing two-OA realizations (assuming OA GBWP = 1 20A 1 and the existing two-OA realizations (assuming OA GBWP = 1 MHz). (a) Frequency response of negative finite-gain amplifiers. (b) MHz). (a) Frequency response of negative finite-gain amplifiers. (b) Phase response of negative finite-gain amplifiers. Phase response of negative finite-gain amplifiers.

comparing the up’s in Table II. As oP increases for a fixed Q,, both amplitude and phase deviations of T, from 7; at a given frequency o (w < wP) decrease. It can be easily shown that the differential finite-gain amplifier in Fig. 6 has similar excellent bandwith and stability properties as those obtained for C2OA-2 in’pbsitive and negative finite- gain applications. In conclusi?n, it is clear that the C2OA-I and C2OA-2 are the most attractive configurations in finite- gain applications from SW aid stability considerations. It should be noted that some -special cases of finite-gain amplifiers using C20A’s have been reported in the literature [l]-[3], [5], [6], [8] and Qted for their improved performance.

2) Comparisons of the Proposed C2OA’s Finite-Gain Reizlizations with Others: The SW of a finite-gain amplifier realized using a single OA shrinks approximately by a multiplying factor lik relative, to its unity gain 3-dB BW (wi). Also, the optimum n@paily flat 3-dB BW using a cascade of two (single-OA realization) finite-gain amplifiers is obtained when each -am+fier has a gain fi to realize an overall gain k. .The resulting BW shrinks by m/G = 0.66/G relative’to ‘wl’[31]. The C20A-1 and C20A-2 circuit BW’s can be designed to shrink by only a factor of =l/fi for QP = 0.707 (maximally flat) and greater than l/G for QP = 1 (k’> 1); see Table II. In addition, the C20A’s require only two accurate gain-

456 IEEE TRANSACTIONS ON CIRCUITS AND SY~TEMS,VOL. CAS-34, NO. &MAY 1987

20 LOG Vo/Vi

I I k

+2 dB

-6-

-10 a= 6

c 20 40 60 80 100 120 140 160 180 200 220

,f kHr

0 = 0.707

5 %

5 % --_

-6 I .f 20 30 40 50 60 70 80 90 100 II0 120 130 IhO

kHz

(4 (b)

20 LOG Vo/Vin

I I k

dB

k’ 100

-4-

10 20 30 40 50 60 70 80 so 100

20LOG b/Vi

I

I I k Qp = 0.707

d0 k = 100 a =6

(4 (4 Fig. 8. Experimental results using C2OA-1 in negative gain applications. (a) (Qp = 0.707) Maximally flat closed-loop

gain = - 25, - 50, - 100 (LM747 OP AMPS). (b) Effect of compensation resistor-ratio variation by T 5 percent (LM747 OP AMPS). (c) Effect of active compensation on extending the bandwidth (LM747 OP AMPS). (d) Effect of power supply variation from T 9 V to T 15 V on the closed-loop gain for k = 100 (LM 747 OP AMPS).

determining components, compared with four in the cascade realization.

To further illustrate the usefulness of the C20A’s, one of the common applications considered in this paper is chosen, namely, negative finite-gain amplification. The performance of the C20A-1 in this application is illustrated and compared in Fig. 7 with some of the most recently published negative finite-gain realizations which utilize a similar number of OA’s. The results shown in Fig. 7 are for nominal gains > 1 for practical reasons since an increase in k decreases the useful bandwidth, so that extending the range of operating frequencies becomes more important. The proposed realizations are seen to be far superior in both amplitude and phase responses relative to those reported in [7] and [9]. Upon examining those of [2] and [8], one may erroneously conclude that, in spite of their inferior amplitude characteristics, they have better phase response. In fact, the realizations of [2] and [8], in contrast with the proposed ones, can be easily shown in theory to be unstable for all useful values of closed-loop gains, due to the second Ok pole. This has been verified experimentally as well. Indeed, the results in Fig. 7 show clearly the excellent gain and phase performance of the proposed realizations.

3) Experimental Results Using C2OA’s in Finite-Gain Applications: Experimental results of negative finite-gain amplifier realizations are given in Fig. 8(a) and (b) using the C20A-1 of Fig. 3(a). LM747’s with a GBWP ranging from 1 to 1.5 MHz were used to implement the C2OA’s in this section as well as in the experiments throughout this work. The stability and low sensitivity to the power supply and to the active compensation resistor variations are examined as shown in Fig. 8(c) and (d). Comparing the results in Fig. 8 with those obtainable using single-amplifier realizations illustrates the considerable’ improvement in the useful BW without sacrificing any of the single-OA desirable features, namely, its low sensitivity to circuit elements and power supply variations, stability, and versa- tility.

B. Finite-Gain Applications Using C3OA’s and C4OA’s Wide-band positive, negative, and differential composite

amplifiers can be designed using the CNOA structures proposed in Section II-B and shown in Figs. 4 and 5.

Positive and negative finite-gain expressions for C30A-1 through C30A-6 are given in Table III, while those for C40A-1 through C40A-4 can be found elsewhere [33]. In the finite-gain expressions of these C30A’s and C40A’s,


TABLE III NEGATIVE AND POSITWE FINITE GAINS V,/ y USING THE C30A’s

C30A-i

C30A-1

C30A-2

C30A-3

C30A-4

kc -k(l+ i3;-) Negative Finite

1 t (1%) $+ (E) &+ (l+k) & Gain Trans. Func.

"i (l+k)>a(lw)

"- (ltk)(l+ ' s 52 L Positive Finite -+-, T+a TlFi w,w:, Gain Trans. Func.

"i l+(l+# ++ ($1 St (l+k) ?- 4 w2w3 V - K (W./w,) Neqative Finite .o _ q - ,+ CL+ l+k w* ol(l+B) 1 s‘+ f&g+ (l+kJ$g

!!L (l+K)(l+S/w2)

"i 1+ [+ + s 1 S + #$ & + (I+@&

Gain Trans. Func.

Positive Finite Gain Trans. Func.

9

(l'tk)> a(l+B)

Finite Gain Transfer Function

V -K AZ= Negative Finite

"i z 3

l+(-$)$+(&&+(l+U& Gain Trans. Func.

(l+k)>(l+a)(l+B) L

V (l+K)( 1 + & g+ & ) Positive Finite o= "i l+(s)+

L J + ( -$ )$q+ W+&-

/ Gain Trans. Func.

I -K '1 Neaative Finite1

l+k SL 5 +(K)w

19 + (l+k+--

Gain Trans. Func

w1w2w3 1 (l+k)>(l+a)(l+6) Positive Finite Gain Trans. Func.

I, h

Negative Finite Gain Configura- tion

no terms containing differences are encountered; thus, low coefficient sensitivities are obtained and reasonable OA mismatch is tolerated. Also, all the denominator coefficients are positive, which is necessary for stability. Apply- ing the same technique used in Sections II and III-A, it can be shown that the resistor ratios (Y, p, and i can be chosen to extend the BW and to’ satisfy the necessary stability conditions assuming single-pole OA models.

1) Comparisons of the Proposed C3OA’s and C4OA’s Finite-Gain Realizations with Others: The optimum maximally flat 3-dB BW using three (four) single-CiA finite-gain building blocks is obtained by cascading three (four) identical blocks, each with gain k113(k114) to realize an overall gain k. The overall BW shrinks by a multiplying factor 0.51/k’/3(0.435/k”4) relative to wi [31]. The BW of the new proposed C30A (C40A) circuits are found to shrink by only a factor 1/k’/3(1/k”4) (k B 1).

Maximally flat -response (Butterworth) as well as Chebyshev characteristics, using CNOA’s, can be achieved by controlling the resistor ratios OL, /3, and y while still satisfying the stability conditions. Computer plots of the C30A-1 and C40A-1 transfer functions in the positive and negative finite-gain configurations are given in Figs. 9 and 10 for gains of 100. From Figs. 9 and 10, it is seen that the 3-dB BW available from these C30A (C40A) finite- gain amplifiers implemented using l-MHz single OA’s corresponds to the BW attainable from a single OA with

zero-dB BW in excess of 25 (35) MHz! The performance of the C30A-1 is compared with the performance of recently published three-OA realizations, called the zero second derivative (ZSD) amplifiers, proposed in [lo]. A positive finite gain of 38.7 is chosen to permit direct comparison with the theoretical results previously published in [lo]. For practical reasons, both ZSD amplifiers are designed such that the stability condition, using Routh’s test on the third-order denominator coefficients, is exceeded by a margin of 10 percent. The best theoretical results using the ZSD are obtainable with the minimum stability margin to allow for maximum bandwidth, i.e., (rl - 1) = 1.1 pk. Fig. 11(a) and (b) shows the theoretical magnitude and phase characteristics of the ZSD amplifiers, as well as the C30A-1 and C40A-1 amplifiers (which satisfy the stability constraints), with positive finite gain of 38.7, for different compensation values. The figure depicts the extended frequency range of operation attainable with these C30A and C40A designs over the ZSD realization [lo], or a realization which uses three (four) cascaded single-OA finite-gain stages. It is interesting to note that the condition for stability of the amplifier shown in Table III using the C30A-1 is

where k = 38.7.

l+k a< 1+p

458

40.09

dB

3i.81

i3.54

i0.26

26.99

23.71

, dB 37.20 -1 34.27

31.35

28.43

25.50

- 14i.6

aI0

a= 1.8 - 71.8

p= I2

k= 100 0.0

- 71.8

:l43.6

-215.0

0 60 160 240 320 400 f kHr

IEEE‘TRANSACTIONS ON CIRCUITS Ah-SYSTEMS, VOL. CAS-34, NO. 5, MAY 1987

( MAXIMALLY FLAT 1

(a)

a = 2.6

(3 = Il.8

k = 100

0 60 160 240 320 4

(CHEBYCAEV)

143.6

9’ 71.8

-71.8

143.0

?I5 .o

0

:. @I Fig. 9. Computer plots (magnitude and phase) of the C30A-1 transfer

function for gain 100. (a) Computed frequency response (amplitude and phase) of the positive finite-gain amplifier (k =lOO) using C30A-1 (single OA GBWP = 1 MHz). (b) Computed frequency response (amphtude and phase) of the positive finite-gain amplifier (k =lOO) using C30A-1 (single OA GBWP = 1 MHz).

This is satisfied by a wide ,margin in. the ,.C30A-1 responses in Fig. 11(a) and (b),for both the maximally flat response and the Chebyshev ,response., All of: these finite- gain designs have the same attractive dynamic-range, stability, and low-sensitivity properties as the C20A designs in Section III-A. Also, it is interesting to note that some of the finite-gain designs presented here (C30A-2, C30A-4, C40A-2, C40A-3, and C40A-5) have identical N/D multiplying factors in the positive and negative gain applications, which makes them suitable in differential gain applications.

2) Experimental Results Using C3OA’s and ,C4OA’s in Finite-Gain Applications: Only sample experimental results using the C30A-1 and C40A-1 are given to illustrate the performance. Exhaustive test results are documented in [33]. Fig. 12 gives the experimental results using the C30A-1 in positive finite-gain applications of 38.7. The computer frequency response plots of the C40A-1 negative finite-gain realization in Fig. 10 closely agree with the experimental results of Fig. 13.

The stability and low sensitivity to power supply as well as to the active compensation resistor variations were

(I i 2 32.73 - P = 2

v = 3.9 k = 100

30.29 -

27 .85

200 300

MAXIMALLY FLAT

(4

kHz

36.11 a-2 p=3

34.16 . i = 3.9 k = 100

32.21 -

(CHEBYCHEV)

(b)

Fig. 10. Computer plots (magnitude and phase) of the;C40A-1 transfer function for, gain 100. (a) ‘Maximally flat computed frequency response (amplitude and phase) of the negative finite-gain .aihplifier (k = 100) using .C40A-1 (single C)A,GBWP = 1 MHz). (b) Chebyshev computed frequency response (amphtude and phase) of the negative finite-gain amplifier (k = 100) using C40A-1 (single OA GBWP = 1 MHz).

IV. CONCLUSIONS

A new approach is presented for extend& the useful operating frequency range in a wide variety *of linear active networks which utilize OA’s. The extended SW is. achieved by replacing each of the single OA’s in theactive realization by a composite OA (CNOA). The application of the CNOA’s in finite-gain amplifiers is also given.

A systematic procedure is given for the generation of the CNOA’s. Each CNOA is constructed using. N-single OA’s and 2(N - 1) active compensating low-spread and low- accuracy resistors, resulting in (N - 1) resistor ratios. The CNOA is versatile since it has three external terminals that correspond to those of a single OA. The suggested generation method gives rise to a large number of QJOA’s for a given N. For N = 2, 3, and 4, CNOA’s%are generated and examined according to a stringent perfor.mance criterion that considers stability, sensitivity, dynar$ range,: CMRR, BW, and the GBWP mismatch effect of single OA’s. Several of the CNOA’s, namely the C2OA:J, to C20A-4, C30A-1 to C30A-6, and C40A-1 to C40A-5, .meet the performance criterion and have been found to be very useful in practice. In these CNOA’s, simple resistor ratios can be used advantaeeouslv to reduce the deviatidh of the verified [33].


20 Log - I I

vo/ vi

d0 4 k

+2 - (experimental)

-2 - -3 dB __-------.-----. -_-

-4 -

-6 -

f kHz -IO, f kHz e

200 300 400 50 100 150 200 250 300 350 400 450 500

(4 Fig. 13. The effect of active compensation on extending the bandwidth of the C40A-1 (using LM747 OP AMPS).

IO 20

30 40

50 60

70

00

90 f ‘KHz

m 100 200

(b)

300 400

Fig. 11. Theoretical results for C30A-5, C40A-5, and [lo] for positive finite-gain applications (gain k = 38.7). 0: [lo] r =l.l, /3 =10m8 (stable with min. margin); 0 : [lo] r = 2, B = 0.2 (unstable): @ : C30A-5 (Y =1.9, p = 5.4; @ : C30A-5 OL =i, p = 6.4; @ : C40A-5 OL = 0.6, p = 2.4, y = 12.5; @ : cascade of three single-OA finite-gain stages, each of gain (38.7)1/3; 0 : c ascade of four single-OA finite-gain stages, each of gain (38.7)‘i4. (a) Theoretical amplitude responses of C30A-5, C40A-5, and [lo] for positive finite-gain applications. (b) Theoretrcal phase responses of C30A-5, C40A-5, and [lo] in positive finite-gain applications.

d0 L

33.7.

27.7.

100 200 300

f kHz e

Fig. 12. Experimental results of C30A-1 in positive finite-gain application, for a gain of 38.7, and the effect of variation of the compensating resistor ratios a and j3 (LM747 OP AMPS).

overall active realization’s response from the ideal while guaranteeing stability.

Finite-gain applications utilizing the proposed CNOA’s are shown both theoretically and experimentally to be stable and to exhibit wide dynamic range and low sensitivity. Comparisons with the the state-of-the-art realizations using similar numbers of OA’s in these applications show the appreciable improvement of the realizations obtained

using the proposed CNOA’s with respect to stability and useful BW.

Although the examples given which use CNOA’s are for high-gain applications, it is easy to show that the deviation in amplitude and phase from the ideal is much lower than other existing realizations, even for closed-loop gains as low as unity. Also, another attractive feature of the proposed technique is that, in an integrated implementation, the chip area and the power consumption of a CNOA are much less than N times those of a single OA. This is because in a CNOA there is one output OA only that drives an external load and that may be required to have power handling capability.

In addition, it is worthwhile to mention that the method used to generate the CNOA’s is actually a composite dependent source generation technique, i.e., it is applicable to any of the four types of dependent sources: voltage (and current) controlled voltage (or current) sources. Novel composite dependent sources with considerable performance improvements are expected to result when the procedure described is applied to the other dependent sources. Moreover, employing elements other than resistors for active compensation and using different types of OA’s in the same CNOA (e.g., a high-accuracy OA for the input OA and a high-speed one for the output OA) are prom- ising and challenging topics for further research and are presently under investigation.

It is to be noted that the CNOA’s, when implemented using ideal OA’s, represent ideal nullors independent of the absolute values of the compensating resistors. In certain configurations using CNOA’s implemented using frequency-dependent OA’s, the ratios of one or more of the compensating resistors to other resistors outside the CNOA appear in the high-order parasitic terms of T, of the overall active realization. Although such terms are small, they can be made negligible by the proper choice of the impedance level of the appropriate compensating resistors. This is the approach taken here for simplicity while still yielding excellent results. Other researchers may find the impedance level as an added degree of freedom that may be used advantageously.

460 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS,VOL. CAS-34,N0. 5,MAY 1987

PI PI

[31

141

[51

[61

171

PI

191

UOI

WI

WI

1131

u41

P51

WI

u71

WI

P91

WI

1211

P21

1231

1241

1251

WI

1271

P81

REFERENCES

A. M. Soliman, “Instrumentation amplifiers with improved bandwidth,” IEEE Circuits Syst. Magazine, vol. 3, no. 1, pp. 7-9, 1981. M. A. Reddy, R. Ravishankar, B. Ramamurthy, and K. R. Rao, “A high-quality double-integrator building-block for active-ladder filters,” IEEE Trans. Circuits Sysr., vol. CAS-28, pp. 1174-1177, Dec. 1981. A. Budak, G. Wullink, and R. L. Geiger, “Active filters with zero transfer function sensitivity with respect to the time constant of operational amplifiers,” IEEE Trans. Circuits Syst., vol. CAS-27, pp. 849-854, Oct. 1980. B. B. Bhattacharyya, W. B. Mikhael, and A. Antoniou, “Design of RC-active networks by using generalized immittance converters,” in Proc. IEEE Int. Symp. Circuit Theory, Apr. 1973, pp. 290-294. K. R. Rao, M. A. Reddy, S. Ravichandran, B. Ramamurthy, and R. R. Sankar, “An active-compensated double-integrator filter without matched operational amplifiers,” IEEE Proc., vol. 68, pp. 534-538, Apr. 1980. A. M. Soliman, “A eneralized active compensated noninverting VCVS with reduced p fl ase error and wide bandwidth,” IEEE Proc., vol. 67, pp. 963-965, June 1979. S. Natarajan and B. B. Bhattacharyya, “Design and some applications of extended bandwidth finite gain amplifiers,” J. Frank/in Inrt., vol. 305, no. 6, pp. 320-341, June 1978. R. L. Geiger and A. Budak, “Active filters with zero amplifier sensitivity,” IEEE Trans. Circuits Syst., vol. CAS-26, pp. 277-288, Apr. 1979. A. M. Soliman and M. Ismail, “Active compensation of op-amps,” IEEE Trans. Circuits Syst., vol. CAS-26, pp. 112-117, Feb. 1979. R. L. Geiger and A. Budak, “Design of active filters independent of first- and second-order operational amplifier time constant effects,” IEEE Trans. Circuits Syst,, vol. CAS-28, pp. 749-757, Aug. 1981. A. M. Soliman, “Classification and generation of active compensated non-inverting VCVS building blocks,” Znt. J. Circuit Theory and Ap lications, vol. 8, pp. 395-405, 1980. G. Wilson, 4 ompensation of some operational-amplifier based RC-active networks,” IEEE Trans. Circuits Syst., vol. CAS-23, pp. 443-446, July 1976. A. Nedungadi, “A simple inverting-noninverting voltage amplifier,” IEEE Proc., vol. 68, pp. 414-415, Mar. 1980. R. Nandi and A. K. Bandyopadhyay, “A high-input impedance inverting/noninverting active gain block,” IEEE Proc., vol. 67, pp. 690-691, Apr. 1979. A. Sedra and P. Brackett, Filter Theory and Design: Active and Passive. Beaverton, OR: Matrix, 1978. M. Ghausi and K. Laker, Modern Filter Design Active-RC and Switched Ca acitor. Englewood Cliffs, NJ: Prentice-Hall, 1981. W. B. Mi ki ael and S. Michael, “Active filter design for high frequency operation,” in Midwest Symp. Circuits Syst. (Al- buquerque, NM), June 1981, pp. 573-576. W. B. Mikhael and S. Michael, “A systematic general approach for the generation of composite OA’s with some useful applications in linear active networks,” in Proc. 25th Midwest Symp. Circuits Syst., (Houghton, MI), Aug. 1982, pp. 454-463. W. B. Mikhael and S. Michael, “Generation of actively compensated operational amplifiers and their use in extending the operating frequencies of linear active networks,” in IEEE Znt. Symp. Circuits Syst. (Newport Beach, CA), May 1983, pp. 1290-1293. S. Michael and W. B. Mikhael, “High frequency filtering and inductance simulation using new composite generalized immittance converters,” in IEEE Inf. Symp. Circuits Syst. (Kyoto, Japan), June 1985, pp. 299-300. R. Schaumann, “Two-amplifier active-RC biquads with minimized dependence on op-amp parameters,” IEEE Trans. Circuits Syst., vol. CAS-30, pp. 797-803, Nov. 1983. W. F. Stephenson, “Composite amplifier structures for use in active RC biquads,” IEEE Trans. Circuits Syst., vol. CAS-31, pp. 420-423, Apr. 1984. T. Fleming, “Monolithic sample/hold combines speed and precision,” Harris Application Note #538, Jan. 1983. M. Ismail, S. R. Zarabadi, and G. Myers, “Application of composite op-amps in nonlinear circuits,” in 27th Midwest Symp. Circuits Syst. (Morgantown, WV), June 1984, pp. 44-47. S. Michael and W. B. Mikhael, “High-speed high-accuracy integrated operational amplifiers,” in 27th Midwesf Symp. Circuits Syst. (Morgantown, WV), June 1984, pp. 792-795. CLC103, “Fast settling wideband operational amplifiers,” Comlin- ear Corp., Loveland,, CO, Nov. 1984. A. Antomou, “Realization of gyrators using operational amplifiers, and their use in RC-active-network synthesis,” IEEE Proc., vol. 116, pp. 1838-1850, Nov. 1969. A. C. Davies, “The significance of nullators, norators and nullors in active-network theory,” Radio Electron. Eng., vol. 34, pp. 259-267. 1967.

1291

[301

1311

~321

[331

1341

J. Braun, “Equivalent N.I.C. networks with nullators and norators,” IEEE Trans. Circuit Theory, vol. CT-12, pp. 411-412, 1965. B. D. Tellegen, “On nullators and norators,” IEEE Trans. Circuit Theory, vol. CT-13, pp. 466-469, 1966. M. S. Ghausi, Electronic Devices and Circuits: Discrete and In- tegrated. New York: Holt, Rinehart and Winston, 1985. S. Michael and W. B. Mikhael, “Inverting integrators and active filter applications of composite operational amplifiers,” pp. 461-470, this issue. S. Michael, “Composite operational amplifiers and their applications in active networks,” Ph.D. dissertation, West Virginia Univer- sity, Morgantown, July 1983. T. J. Groom, “Precision high speed op. amp. parallel transconduc- tance implementation HA2.548,” Harris Semiconductor, Melbourne, FL.

rIc ’

Wasfy B. Mikhael (S’70-h4’73-SM’83-F’86) was born in Manfalout, Egypt, on November 3,1944. He received the BSc. degree (honors) in electronics and communications from Assiut University, Assiut, Egypt, the M.Sc.E.E. degree from the University of Calgary, Calgary, Alberta, Canada, and the D.Eng. degree from Sir George Williams University, Montreal, Quebec, Canada, in 1965, 1970, and 1973, respectively.

From 1965 to 1968, he was an Engineer with the Telecommunications Organization, Cairo,

Egypt. From 1970 to 1973, he taught in the Computer Science Depart- ment at Sir George Williams University, and in 1973 he taught in the Mathematics Department, Dawson College, Montreal, Quebec, Canada. Since May of 1973, he has been a Member of the Scientific Staff at Bell-Northern Research, Ottawa, Ontario, Canada, as well as an adjunct Associate Professor in Electrical Engineering at Sir George Williams University (now known as Concordia University). In August 1978, he joined the faculty of West Virginia University, where he is now a Professor. of Electrical Engineering. His research interests are active networks, switched-capacitor circuits, and adaptive signal processing. He has several patents and publications in the area of communication networks and active filters.

Dr. Mikhael was the recipient of the Bell Northern Research Outstand- ing Contribution Patent Award in 1978, the Outstanding Researcher Award from the College of Engineering, West Virginia University, in 1982 and 1983, and the Halliburton Best Researcher Award in 1984. He served as Chairman of the 1984 Midwest Symposium on Circuits and Systems. Dr. Mikhael is presently Associate Editor of the Letters Section of the IEEE TRANSACTIONSON CIRCUITSAND SYSTEMS.

Sherif Michael (S’78-M’83) was born in Alexandria, Egypt. He received the B.Sc. degree in electrical engineering (electronics and communications) from Cairo University, Cairo, Egypt, in 1974. He received the MSc. degree in industrial engineering and the Ph.D. degree in electrical engineering from West Virginia Uni- versity, Morgantown, WV, in 1980’and 1983, respectively.

He received technical training at Philips Industries, Eindhoven, Holland. He served as a

First Lieutenant in an Engineering Corps specializing in water well drilling, where he conducted, as a Field Engineer, electronic measure- ments and supervised drilling operations in cooperation with Schlum- berger Co. Dr. Michael worked as a research and teaching fellow at West Virginia University. As a Research Engineer with the National Transpor- tation Research Center, Morgantown, WV, he worked on designing and implementing a new digital communication system for the Morgantown Personal Transit System (MPRT). Since 1983, he has been an Assistant Professor with the Department of Electrical and Computer Engineering at the Naval Postgraduate School, Monterey, CA. His present research interests are in the area of analog integrated circuits and active networks design, radiation hardening, solar cells and space power applications.

Dr. Michael is a member of Eta Kappa Nu, Alpha Pi Mu, and Tau Beta Pi and is a registered Professional Engineer.

Study Guide B


Inverting Integrator and Active Filter Applications of Composite Operational Amplifiers

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-34, NO. 5, MAY 1987 461

Inverting Integrator and Active Filter Applications of Composite

Operational Amplifiers SHERIF MICHAEL, MEMBER, IEEE, AND WASFY B. MIKHAEL, FELLOW, IEEE

Abstract -A new approach for extending the useful operating frequency range of linear active networks realized using operational amplifiers (OA’s) has been reported [l]. The extension in bandwidth (BW) is achieved by replacing each of the single OA’s in the active realization by a suitable composite OA (CNOA) that has been constructed using N OA’s.

The use of the CNOA’s to realiie inverting integrators and active filters is presented here. The considerable performance improvement of these realizations is demonstrated both theoretically and experimentally. Their comparison with state-of-the-art designs is also given.

I. INTRODUCTION

I N RECENT YEARS, a great deal of attention has been directed toward designing high-performance integrators

and active RC filters. Several contributions using a variety of techniques have been reported to extend the useful. operating frequency range of these networks and to reduce their sensitivities with respect to the active elements, namely, the operational amplifiers (OA’s) [3]-[24]. It is well known that the poles and zeros actually realized are displaced from their nominal positions because of the frequency-dependent characteristics of the operational amplifier gains. This results in both phase and magnitude errors in the response, especially if the active networks are designed to operate at high frequencies and/or with high Q ‘s.

In [l], the authors propose a technique for extending the operating frequency range of linear active networks by using composite OA’s. A general procedure is described where N OA’s are combined to form a new active device that resembles externally an OA and is referred to as a composite operational amplifier CNOA. The technique generates a very large number of CNOA’s for a given N. The effect of using suitable CNOA’s as one-to-one replacements of the single OA’s in finite-gain realizations is found to result in an extended operating frequency range relative to that of existing realizations that use a similar number of OA’S.

Manuscript received October 29, 1985; revised August 26, 1986. S. Michael is with the Department of Electrical and Computer

Engineering, Naval Postgraduate School, Monterey, CA. W. B. Mikhael is with the Department of Electrical Engineering, West

Virginia University, Morgantown, WV 26508. IEEE Log Number 8613470.

In this paper, the use of the CNOA families in the realization of inverting integrators and active filters is investigated. In Section II, it is shown, both theoretically and by computer simulations, that the members of these new C20A and C30A families extend the operating frequency range of inverting integrator realizations considerably beyond the presently available state-of-the-art designs. In addition, the proposed technique results in practi- cally useful designs, as is demonstrated by experiment later in Section III.

Appreciable improvements in the performance of different types of modern active filters using the CNOA’s are shown in Section III. Here, the active filters are considered to belong to two categories. The first category consists of filters which are realized with functional building blocks (finite-gain amplifiers and integrators), while the second category consists of filters in which the OA’s are embedded in the passive network, so that functional building blocks cannot be identified from the filter structure. Active filters in the first category are shown, theoretically in Section III-A and experimentally in Section III-B, to provide significant improvement at high operating frequencies when the improved inverting integrators proposed here and finite-gain amplifiers proposed in [l] are used. An application is given in Section III-B by employing the C20A’s in the two-iniegrator-loop filters as an example. The extension in BW is achieved while maintaining low sensitivity to passive and active elements, wide dynamic range, and stable operation.

In Section III-C, the theory for extending the active filters’ operating frequencies in the second category is presented. The application of the theory, supported by experiments, is given in Section III-D, where a suitable C30A is used in one of the well-known multiple-feedback (MFB) structures. The resulting filter maintains the practical useful features such as stability, low sensitivity to the compensating elements, and signal handling capacity. Also, comparison with one of the state-of-the-art designs is given which shows considerable improvements in bandwidth and stability of the proposed technique. It is worthwhile to note that Schaumann [l, ref. [21]] successfully demon-

009%4094/87/0500-0461$01.00 01987 IEEE

462

TABLE I INVERTINGINTEGRATORS V,/ I: USINGTHE C20A’s

IEEE TRANSACTIONS ON CIRCUITSAND SYST!2MS,VOL.CAS-34,N0. 5,MAY1987

CZOA-i Actual Negative Integrator Transfer Function (T,)

CZOA-1

& T. . c

1 1 l+S/wpQp+wwpZ

I* 'W,(l+a)

l+T,UJ,(l+al

C20A-2

CZOA-3 Ti * [ 1

CZOA-4 1 + (,+a) s/w,

Ti . '1 + S/wpQp + SLl~p~l

v Ideal Transfer Function Ti = F= $ = &-

i ,

(Where Tt is the integrator time constant = RC = 11 wt )

Fig. 1. The composite

I-- YW2 -lx

"i

I I’ i _’ i ,. /

operational amplifiers C20A’s [l].

MICHAEL AND MIKHAEL: APPLlCATIONSOFCOMPOSlTE OPERATIONAL AMPLIFIERS 463

strated the drastic active sensitivity improvement of the Deliyannis filter, which belongs to category 2, by using C20A-4.

II. APPLICATIONOFTHEPROPOSEDCNOA'SIN INVERTING INTEGRATORS

A. Inverthg Integrators Using C.JOA’s

The amplifiers C20A-1 through C2OA-4 were previously proposed by, the authors and are repeated in Fig. 1 for convenience. The actually realized transfer functions T, using the C2OA-1 to C20A-4 in inverting integrator applications, assuming a single-pole OA model, are given in Table I. The Ta’s have the same general form found in [l]:

where T. is the transfer function realized assuming ideal OA’s and

N=l+a.s=l+(s/o,) ( c-i is zero (0, + w) in some cases)

Thus, N/D determines the amplitude and phase deviation of T, from T.. Also, b, and b, determine the stability of T,. Here, a, b,, and b, and consequently wZ, wp, and Qp (as defined in Table I) are functions of the circuit parameters which are the single-OA GBWP’s ot, 02, and the C20A compensation ratio (Y (as well as TV, the integrator time constant). None of the a and b coefficients are realized through differences, thus guaranteeing the low sensitivity of T,, wZ, (jp, and Qp with respect to the circuit parameters. On the other hand, the b coefficients are always positive (assuming a single-pole OA model), which

x guarantees the stability of the transfer function. Now let us assume a two-pole open-loop gain Ai of the

ith single OA (i = 1,2). l/Ai is given by

where

oL, 3-dB bandwidth 2 wi/A,,, oi gain-bandwidth product (GBWP) of the single-pole

model of the ith OA, Aoi dc gain of the ith OA, w,,, second pole frequency (w,, Z+ wL,).

It is easy to show [l, ref. [33]] that, for identical single-OA models (wh, = o,, = oh and oi= w2 = wi), the necessary and sufficient stability conditions (the parasitic poles are in the left half of the s plane) for the C20A-1 and C20A-2 integrator realizations are given by

TABLE11 VALUESOF~EC~OA'~RESISTORCOMPENSA~ONRATIO &FOR SELECTEDVALUESOFQ ,ANDT~CORRESPONDINGSTABXLITY

CONDITIONSFORTHE NVERTINGINTEGRATOFSUSINGTHE P

CZOA-i a

0

CZOA-1

Unrealizable

0

CZOA-2 Unrealizable

0

CZOA-3

CZOA-4

r

Unrealizable

0

1 .

:

1

1

z?

1

1 A

1

1 =- A

I 0.666

ity for (I used

= wj 3w.

Wh ' +

= qfi Wh > 3"'

wi is the GBWP of each single OA and. w,, is tile single-OA second pole frequency.

For the C20A-3, the condition is given by

(4)

Also, for the C20A-4, the stability condition is given by

It is clear from (3), (4), and (5) that, for a given (Y, there is a minimum value of wh/oi which satisfies the stability condition. In C20A-1 and C20A-2 integrators, the condition a = 0 results in the minimum value for wh, since wh increases as (Y increases. The C20A-2 integrator requires wh/wi to be greater than 3/2 for stable operation when (Y equals zero and wh/wi to be greater than 2 when a tends toward infinity. The C20A-4 integrator has an advantage, since for wh = oi, a value of OL exists (a = l/3) for which the integrator is stable with excellent frequency response. In general, as wh increases, the stability improves. From physical considerations, if o,, -+ cc, all the integrators become stable, as can be seen from Table I, since the two-pole OA model reduces to a single-pole model (Ai = w/(si + oL )). The stability conditions for particular values of (Y are summarized in Table II. From graphical data sheets of internally compensated OA’s such as’ 741’s and 747’s, w,, is seen to slightly exceed wi [2]. This results in stable realizations down to 0-dB closed-loop gain. For

464 . IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-34, NO. 5, MAY 1987

externally compensated OA’s such as the 702 and 709, (Y can be chosen in the range that guarantees stability while yielding the most desirable T,(s).

A theoretical comparison of the C20A-4 negative integrator with state-of-the-art negative integrator realizations [3]-[5] is given in Fig. 2 for ot/oi = 0.05 (w, =l/rt, where rr ,is the integrator time constant). The theoretical results in Fig. 2 shows that the percentage deviation in magnitude and phase from the ideal for the proposed C20A-4 negative integrator, with (Y = l/3, very nearly matches the excellent performance of the integrator in [3]. (The C20A-4 integrator with (Y = 0 is identical to the integrator in [3].) In practice, the integrator in [3] suffers from stability problems while the proposed one does not, as verified experimentally by the authors in active filters (two-integrator-loop biquad-Section III-B) when. internally compensated OA’s with a phase margin of less than 60” at 0-dB closed-loop gain (wh/wi < 3/2) are used. Thus, the importance of the controlling parameter (Y is apparent since it guarantees stable operation using commercial internally compensated OA’s without sacrificing performance.

B. Inverting Integrators Using C3OA’s

In this section, three high-frequency integrators which use actively compensated multiple OA’s (CNOA’s) for N = 3 are introduced. The different structures of these C30A’s, together with their actually realized open-loop transfer functions, are given in [l] and [l, ref. [33]]. C30A-2, C30A-5, and C30A-6, which are employed here, are given in Fig. 3(a) for convenience.

Using the C30A-2, C30A-5, and C30A-6 in the inverting integrator circuit yields the following integrator transfer functions.

Using the C30A-2,

% deviation from ideal

IHI t CBOA-4

6 %

7 % 6 %

5 %

4 %

3 %

2 %

I %

% deviofion from ideal

04 Fig. 2. Comparison of the C20A-4 negative integrator for (Q, =

0.707, Qp = 0.835) with existing negative integrators proposed in [3]-[5]. (a) Percentaee deviation from ideal of the transfer function magnitude versus no&lized frequency for the proposed C20A-4 i&g&r and several other integrators (l/r,w, = 0.05). (b) Percentage deviation from ideal of the transfer function phase versus normalized frequency for the proposed C20A-4 integrator and several other integrators (l/r,wi = 0.05).

T,=q.

Using the C30A-5,

T,=T.

Using the C30A-5,

T, = q.

(6)

(7)

(8)

MICHAEL AND MIKHAEL: APPLICATIONS OF COMPOSITE OPERATIONAL AMPLIFIERS 465

4 %

3 %

2 %

I %

from ideal I

b

f ktiz

20 40 60 80 100 120 140 160 180 200 220 240 260 280

(b)

5 %

4 % .

20 40 60 60 ‘100 120 I40 I60 I60 200 220 240 260 260 f kHz

(4 Fig. 3. Comparison of C30A-6 negative integrator and the one pro-

posed in [3] (0,/w, = 0.1). (a) The composite operational amplifiers C30A-2, C30A-5, and C30A-6 [l]. (b) Percentage deviation from ideal of the transfer function magnitude versus normalized frequency for the proposed C30A-6 negative integrator and the one proposed in [3] (o,/w, = 0.1). (c) Percentage deviation from ideal of the transfer function phase versus normalized frequency for the proposed C30A-6 negative integrator and the one proposed in [3] (at/w, = 0.1).

where wi is the GBWP of the OA’s used. T. the ideal integrator transfer function is equal to - w,/s (T, --) 7J as the OA Ai’s -+ co). In (6)-(8), it can be shown that no difference terms appear in any of the numerator and denominator coefficients when OA’s with different gains are used. Thus, the low coefficient sensitivities are achieved and the necessary condition for stability is satisfied without the need for matched amplifiers. Assuming a single-pole OA model, the necessary and sufficient conditions for stability, i.e., for the roots of D to be strictly in the left half of the s plane, are as follows.

For the C30A-2 integrator,

I

+[;][1+ ,(1”;.)] >l. tsa) For the C30A-5 integrator,

[ l+;f+)][(~+@+~+;(l+P)]>l. (9b) For the C30A-6 integrator,

(1+0)(1+~) 51. I

It can be easily shown that these new integrators can be designed to satisfy the above stability conditions, while allowing a wide range of (Y and p variations. For illustra- tion, Fig. 3(b) and (c) shows sample theoretical results using the C30A-6 and those obtained using the integrator in [3] that employs two OA’s. The performance improvement in both magnitude and phase is obvious.

III. EXTENDING THE ACTIVE FILTER OPERATING FREQUENCY RANGE USING CNOA’s

Active filters have been designed using a wide variety of approaches [6]-[lo]. In this contribution, active filters are considered to belong to one of two categories. The first category includes active filters that are realized using functional building blocks, namely, inverting integrators and finite-gain amplifiers. Examples’ are the positive-gain Sallen and Key filter, the two-integrator-loop filter, and the SFG filter [6]. The second category contains those filters whose OA’s are embedded in the passive network, and functional building blocks cannot be isolated in the filter structure. Examples are the multiple-feedback (MFB) filter and the generalized immittance converter (GIC) filters WI, [91-

A. Improving the Performance of Active Filters in the First Category

It is easy to show that for filters in the first category, as . the behavior of the active functional building blocks approaches the ideal over a wider frequency range, extended operating frequencies are obtained. This can be explained as follows.


Let us assume an active filter that realizes a transfer function T, using functional active building blocks whose individual transfer functions are G,, G,, . . . , G,. Here, T, is the actually realized transfer function using nonideal (frequency-dependent) active building blocks. GpGz,--*, G,, are the actually realized (nonideal) transfer functions of the building blocks using frequency-dependent active elements (OA’s). In addition, let q, Gii,. . . , Gni be the corresponding ideal transfer functions when the active elements (OA’s) used are frequency independent; i.e., the gain-bandwidth product (GBWP,) of the OA’s is cc (7i = 0, where 7i =l/GBWP =l/bi). T, can be written as

Ta=f(G1,GZ;-,G,). (10)

Similarly, the jth gain Gj realized using the jth CNOA, whose OA ri’s are r,j to rNj, can be expressed as

Gj=g(r1j;r2j,...,r11[1) 01)

where j=l,2;.., n. Here, f (.) and g( .) denote functional dependence. For simplicity, let n = 1 and N = 2. The following argument can be shown to be valid for any integer values of n and N. Also, let G, = G and Gii = Gi.

The Taylor’s series expansion of T, about Gi is given by

Tl= %=a, + dG G=C, (G - Gi) 1 a’T,

(G-GJ*+ -. (12)

Equation (12) can be rewritten as

AT=T’(AG)+ ; T”(AG)2+ ... i I

= T’AG for AG very small

where (13)

AT=T,-T. and AG=G-Gi.

Similarly, from a Maclaurin series expansion of (11) in ri and r2 about their ideal values of ri = r2 = 0, it can be shown that


AG = Girl + G$r2 + i [G ;‘rf + G;‘rz + 2G$r1r2] + + . . .

(15)

It is to be noted that, even if Gi is real, AG is complex in general. Also, T ‘, T “, . . . are dependent on the topology

of the active filter, but independent of the nonidealness of G. Thus, from (13) and (15), to minimize AT, both the phase and gain deviations in G of the functional building block, corresponding to AG’s real and imaginary parts, must be appropriately minimized using the CNOA’s compensating resistor ratios in the expansion of G given by (15). The minimization can be carried out in many different ways. A simple but less exact approach is to minimize each ] AG] individually. This is found useful in practice and is applied in Section III-B. Another more exact but more involved approach is to substitute for AG from (15) in AT given by (13). Then, the compensating resistor ratios are chosen to minimize AT as desired. In both approaches, AT may be minimized at a critical frequency or over a band of frequencies.

At this point, two pitfalls are pointed out. The first may occur if one attempts to choose a structure for realizing G which results in zero lower order derivatives G;, G;, G;‘, . . . of G with respect to its parameters. Great care should be exercised in attempting this, since the higher order derivatives of G may be increased in a manner that offsets the reduction in AG obtained by nulling the lower order derivative terms, particularly as the frequency increases. Note that, for a given number of OA’s, a realization that is characterized by zero low-order sensitivities may not be optimum; there may exist a structure that fails to satisfy the zero low-order sensitivity property but results in a smaller AT. This can be easily seen by comparing the performance of the C20A-4 with the C20A-1 in finite-gain applications [l].

The second pitfall may result if one controls the phase angle of G by moving some of the parasitic poles of G to the right half of the s plane to cancel the phase shift due to the other poles and zeros. This is intolerable in finite-gain applications and very undesirable even when the active building block is embedded in filter structures, since it leads to local instabilities. This has been examined by researchers and referred to as stability during activation [25]-[27].

To conclude this subsection, it follows from (13) that if the active building blocks in a filter are replaced by the proposed building blocks which have a smaller AG, the filter’s performance can be improved.

B. Improving the Performance of an Active Filter in the First Category (A Multiple-Amplifier Biquad) Using C2OA’s

In this subsection, it is shown how the operating frequencies of the first category of filters can be extended through the use of the functional building blocks that are presented in [l] and in Section II. These blocks allow the minimization of AG and the necessary tradeoff between gain and phase deviations to achieve stable high-frequency operation with low sensitivity to the active compensation elements. As mentioned above, several techniques are possible for error minimization and bandwidth extension in this category of filters. A less global and more straightforward approach is employed in this subsection for demon- strating where the bandwidth of each block is individually extended.

MICHAEL AND MIKHAEL: APPLICATIONS OF COMPOSITE OPERATIONAL AMPLIFIERS 467

Cl CP

‘i

(4

CZOA-2) C20Ad’rl

I oo- MEASURED “SING C2OA’l

50.

OQf 20 40 60 80 100 120

350.

300

250 -

200.

150.

I oo-

50

-% 100 / SINGLE OA

')Pf 4 Lop;

30 SlNGLE OA

30

2s

20

15

IO

5

(4 (4

Fig. 4. Experimental results of the two-integrator-loop BP filter using the proposed C20A-2 and C20A-4 and the theoretical results obtainable with regular OA’s. (a) Bandpass filter (two-integrator-loop) using CZOA-2 and C20A-4. (b) Percentage variation of Qp, (filter) as a function of Q

“d for bandpass filter (wpf = 50 krad/s). (LM747 OP AMP). (c) Percentage

variation of Qp/ as a function of Qp, for ban pass filter (o , = 30.8 krad/s) (LM747 OP AMP). (d) Percentage variation of Q p, as a function of fp, for bandpass filter (LM747 OP ARIP).

A biquadratic active filter, which uses the functional building blocks, is designed and tested. The filter that is chosen is the well-known state-variable filter [lo], and is shown in Fig. 4(a). It uses two inverting integrators and a differential finite-gain amplifier, which are constructed using the C20A-4 integrators proposed in Section II and the C20A-2 differential amplifier [l]. The biquad’s transfer function 7;(s) (at the bandpass output) is given by

l+R,/R, s _. _ .-

WhP = 1+&/R, W,

7 s l+R,/R, RJR, ' (16) r&I - I

.J I

R,C, l+R,/R, ' R,R,C,C,!

The elements are chosen as

C,=C,=C R,=R,=R,=R,=R,=R

and aPr (the complex pole-pair resonant frequency) = 1/RC.

Referring to [lo], these design values correspond to

X=2Q*,-1+4 3

(17)

where QPr is the complex pole-pair selectivity factor, and

&=l. (18)

From [l], for maximally flat response (Q, = l/a) of the

differential amplifier, 1y = 9 - 1 = 0, independent of X and QP,. Also, from Sect& II and Table II, a! is set equal to unity (maximally flat) in the C20A-4’s integrators. In Fig. 4(b)-(d), the experimental results for the filter using the C20A’s are ‘compared to those utilizing the single OA’s. In addition to the appreciable improvement in the useful operating frequency range demonstrated in Fig. 4(b)-(d), excellent theoretical sensitivity, stability, and dynamic range are also verified experimentally. Total harmonic distortion (THD) of much less than 1 percent at any of the filter outputs is measured over a wide range of frequencies and signal levels; e.g., the THD was found to be less than 50 dB below the fundamental at fPf, for an output voltage swing of 12 V peak to peak, where v- power supply =-- 12 V, r,f = 16 kHz, and QP, =-IO. It is very interesting to note that when (Y is set to zero (which results in a previously reported integrator [3], [ll], [12]), the filter


0 "0

Fig. 5. MFB active BP filter using C30A-1.

became unstable in practice, as predicted from the stability analysis in Section II.

C. Improving the Performance of Active Filters in the Second Category

In the second category, T, can be expressed directly as a function of the t's of the CNOA’s as follows:

T,=f(t,,t,,--,t,) 09) where t,, t,, . . e, t, are reciprocals of the open-loop gains of the first, second,. . . , n th composite amplifier, respectively. For a composite amplifier constructed using N OA’s, tj can be expressed as

tj=g(71j,T2j,"',7NI) (20)

where tj is the reciprocal of the open-loop gain of the jth CNOA and 7ij is the reciprocal of the GBWP of the i th OA used in constructing the jth CNOA. For simplicity, let n =1 and N = 2. Also, let t, = t, ~ii = 7i, and 721 = r2. Hence,

T,=f(t) (21)

and

t = &I, 72). (22)

Thus, the Maclaurin series expansion of (21) about its ideal value of t = 0 is given by

T,= T,ltzO+ aT, at (t=o+)+( f)Jgq=o*(‘,‘+ *-..


AT23 1 a2T,

at t=O *t+Yi at,2 .(t)2+

t-o

=T’t+ ; zy(t)‘+ **-. i I

Also, t is given by

at at t=-

aT1

T1 + - Jr2

72 + 71 - 7* = 0 7, = T2= 0

. .

(23)

. (24)

(25)

(26)

From (24) and (26), to minimize AT, t, which is complex in general, has to be minimized appropriately using the CNOA’s compensating resistor ratios (OL’S, R’s, etc). As stated before, in Section III-A, error minimization can be done in many different ways depending on the cost function and the optimization criterion. In the following subsection, a way of improving the performance of a filter in the second category by employing CNOA’s is given.

D. Improving the Performance of an Active Filter in the Second Category (A Single-timplifier Biquad Using C3OA)

For a given filter in this category, it is feasible to derive closed-form expressions, by combining (24) and (26), to minimize the error appropriately. In this subsection, a simpler but equally useful approach is adopted to demon- strate the usefulness of the proposed technique. In the following results, the CNOA’s compensating resistor ratios (a’s and R’s) are chosen using the search method in a straightforward computer program that minimizes the filter’s response deviation from the ideal and satisfies stability.

A multiple-feedback (MFB) biquadratic bandpass active filter is considered here (Fig. 5), as an example to illustrate the improved filter performance using one of the proposed C30A’s [l], namely, the C30A-1. For ideal OA’s, i.e., Ai + cc (i = 1,2,3), the filter transfer function Ti is given by

$L - R,CsH

V. (27) I 1+2RCs+R R C2s2 1 1 3

where H = R2/(Ri + R2), R, = Ri//R2, uPr = l/(Cm), and Qp, = (1/2)-/m. Assuming, for simplicity, identical frequency-dependent OA’s and R’ X-

R,, the actually realized transfer function T, is given by

T,s ( -sCR3H(1+;/ l+s[zCR,+~

1

+wi(l+P) 1 [ +s2 C’R,R,

(2CR, + CR,) 2 1 +‘ A <‘+

CR1

o;.(l+P) -+

0, f&l + a) 1 C2R,R, C2R,R2 + (2CR, + CR,) 1

0,(1+/3) + wi w;(l+ a) +-

w; I

+S4 C2R,R3

w?(1+ a) +

(28)

Again, it can be shown that the dynamic range of this filter is identical to that obtained if a single OA replaces the C30A. Also, not only is OA gain matching not required, but also difference terms do not appear in any of the

MICHAEL AND MIKI-IAEL: APPLICATIONS OF COMPOSITE OPERATIONAL AMPLIFIERS 469

dB

46

QPpxpsrimsntol = 9.7

fptexperimental = 37.2

IDEAL MFE FILTER (ideal OA’s)

fpfhHz c

1 42 46

Fig. 6. Experimental amplitude response of the MFB bandpass filter using C30A-1, and its variations due to the compensating element variations.

(+$dooI

% t

10 _ C3OAl a=o,p=m

9- II31 9= 0.2, P = I

8-

7.

6-

J-

4.

10 20 30 40 50 60 70 90 90 100

(a>

AAx 100

I I QPf

%

IO 20 30 40 50 60 70 80 90

fp, kH*

-%- (b)

Fig. 7. Experimental results of the MFB filter designed using the proposed C30A-1 and the design proposed in [13]. (a) Percentage varia- trons in center frequency of the MFB filter for Q,,/ = 5, 10, 20. (b) Percentage variations in Q,,f for Q,,, 7 5, 10, 20. * Trapezoidal oscilla- tions were encountered. ---Limiting diodes were necessary.

transfer function T, coefficients, even when different OA’s are used.

A BP filter with j”, = 37.9 kHz and Qp, = 10 is designed and tested. The design values are C = 1 nF, Rj = 250 Cd, R, = 210 a, R, = 1.3 k& and R, = 84 kQ. The OA’s used are LM747’s with a GBWP ~1 MHz. Applying the

Routh-Hurwitz stability criterion, it can be easily shown numerically that the necessary and sufficient conditions are well satisfied with a wide margin for the component values (Y and p. Ideal and experimental results are shown in Fig. 6. The experimental results in Fig. 6 illustrate the appreciable improvement over a single-OA realization in the MFB structure under consideration at this QP, and aP, W[101.

Performance comparisons are made with the recently reported high-frequency filter design given in 1131, which requires the same number of OA’s as C30A-1, namely three. ’

The experimental results using the proposed C30A-1 design and the design in [13] of the MFB BP filter in Fig. 5 are given in Fig. 7. No error minimization is attempted at each operating aPr and Q,,. Rather, the compensation remains fixed for all the results in Fig. 7. In the C30A-1 design IX = 0, p = cc (compensating resistors R' and /3R are open, while aR' and R resistors are short).

In the design in [ 131, 0 = 0.2, p = 1. Fig. 7(a) shows the percentage deviation in &, as a

function of fPf for Qpr = 5, 10, and 20. Fig. 7(b) shows the percentage deviation in Qpr as a function of jp, for Q,,, = 5, 10, and 20. The comparisons in Fig. 7 show clearly the performance improvements in the proposed filter design.

IV. CONCLUSIONS

A general technique for the synthesis of composite OA’s (CNOA’s) and for using them in extending the BW of finite-gain amplifiers has been presented in [l].

In this paper, applications of the CNOA’s in inverting integrators and active filters are investigated. Theoretical and experimental results of utilizing the proposed CNOA’s (1v = 2,3) in inverting integrator applications show that the BW is extended considerably beyond present state- of-the-art techniques, while other important properties such as stability and low sensitivity are maintained.

To analyze the effectiveness of this approach, active filters are considered to belong to one of two categories. The first category consists of filters realized using func- ‘tional building blocks (finite-gain amplifiers and inverting integrators), while in the second category the OA’s are embedded in the passive network so that functional building blocks cannot be isolated from the filter structure.

Filter examples are considered from both categories and are constructed using the proposed CNOA’s and the improved functional building blocks. The circuits thus attained are shown to perform as well as or better than the best available state-of-the-art designs with respect to ,extension of BW, stability, sensitivity, and deviation from the ideal response.

In conclusion, an integrated technique for alleviating the problem of BW limitation due to OA frequency dependence has been presented which extends the useful BW of linear active networks using OA’s without impairing other important properties such as stability, sensitivity, tolerance of OA mismatch, and dynamic range. It is worthwhile

470 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS,VOL. CAS-34,N0. &MAY 1987

mentioning that the technique described here and in [l] is general and is applicable to a wide range of linear active realizations using OA’s, as well as other controlled sources.

PI PI

131

141

151

161

171

PI

[91

[lOI

WI

WI

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[I41

WJI

1161

P71

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1191

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P21

1231

]241

]251

REFERENCES

W. B. Mikhael and S. Michael, “Composite operational amplifiers: Generation and finite-gain applications,” pp. 449-460, this issue. The Linear and Interface Circuit Data Book for Design Engineers, Texas Instruments, Inc. G. Bailey and R. Geiger, “A new integrator with reduced amplifier dependence for use in active RC-filter synthesis,” IEEE Int. Symp. Circuits Syst., 1980, pp. 87-90. A. M. Soliman and M. Ismail, “On the active compensation of noninverting integrators,” IEEE Proc., vol. 67, pp. 961-963, June 1979. P. 0. Brackett and A. S. Sedra, “Active compensation for high- frequency effects in op. amp.‘circuits with applications to active-RC filters,” IEEE Trans. Circuits Syst., vol. CAS-23, pp. 68-73, Feb. 1976. A. Sedra and P. Brackett, Filter Theory and Design: Active and Passive. Beaverton. OR: Matrix. 1978. S. Mitra, , 4nalysis and Synthesis bf Linear Active Networks. New York: Wiley, 1969. G. C. Temes and S. K. Mitra, Modern Filter Theoty and Design. New York: Wilev 1071 M. Ghausi an ;i’KT’L”&er, Modern Filter Design Active-RC and Switched CaDacitor. Enelewood Cliffs. NJ: Prentice-Hall. 1981. L. P. Huelsman and P.“E. Allen, Intioduction to the Theory and Design of Actibe Filters. New York: McGraw-Hill, p. 217. M. A. Reddy, R. Ravishankar, B. Ramamurthy, and K. R. Rao, “High-quality double-integrator building-block for active-ladder filters,” IEEE Trans. Circuits Syst., vol. CAS-28, pp. 1174-1177, Dec. 1981. E: R., R;&g A. Reddy, S. Ravichandran, B. Ramamurthy, and

“An active-compensated double-integrator filter without matched operational amplifiers,” IEEE Proc., vol. 68, pp. 534-538, Apr. 1980: R. L. Geiger and A. Budak, “Design of active filters independent of first- and second-order operational amplifier time constant effects,” IEEE Trans. Circuits Syst., vol. CAS-28, pp. 749-757, Aug. 1981. K. Martin and A. S. Sedra, “On the stability of the phase-lead integrator,” IEEE Trans. Circuits Syst., vol. CAS-24, pp. 321-324, June 1977. A. M. Soliman and M. Ismail, “A universal variable phase 3Lport VCVS and its application in two-integrator loop filters,” in IEEE Int. Symp. Circuits Syst., 1980, pp. 83-86. S. Ravichandran and K. R. Rao, “A novel active compensation scheme for active-RC filters,” IEEE Proc., vol. 68, pp. 743-744,, June 1980. S. Natarajan, “Active sensitivity minimization in SAB’s with active compensation and optimization,” IEEE Trans. Circuits Syst., vol. CAS-29, pp. 239-245, Apr. 1982. S. Natarajan, “Synthesis of actively compensated double-integrator filter without matched operational amplifiers,” IEEE Proc., vol. 68, pp. 1547-1548, Dec. 1980. A. K. Mitra and V. K. Aatre, “Low sensitivity high-frequency active R filters,” IEEE Trans. Circuits Syst., vol. CAS-23, pp. 670-676, Nov. 1976. M. A. Reddy, “An insensitive active-RC filter for high Q and high frequencies,” IEEE Trans. Circuits Syst., vol. CAS-23, pp. 429-433, July 1976. A. K. Mitra and V. K. Aatre, “A note on frequent limitations of active filters,” IEEE Trans. Circuits Syst., vo Y

and Q CAS-24,

pp. 215-218, A B

r. 1977. A. S. Sedra an J. L. Esoinoza. “Sensitivitv and freauencv limitations of biquadratic act&e filters,” IEEE T>ans. Circbits gyst., vol. CAS-22, pp. 122-130, Feb. 1975. J. J. Friend? A. Harris, and D. Hilberman, “Star: An active biquadratic filter section,” IEEE Trans. Circuits Syst., vol. CAS-22, pp: 115-121,1975. A. Budak and D. M. Petrela, “Frequency limitations of active filters using onerational amnlifiers.” IEEE Trans. Circuit Theorv, vol. CT-19,-p;. 322-328, Jul-y 1972:

,

A. Antoniou, “Realization of gyrators using operational amplifiers and their use in RC-active network synthesis,” Proc. Inst. Elee. Eng., vol. 116, pp. 1838-1850, Nov. 1969.

[26] A. Antoniou, “Stability properties of some gyrator circuits,” Elec- tron. Left., vol. 4, pp. 510-512, 1968.

[27] W. B. Mikhael and B. B. Bhattacharyya, “Stability properties of some RC-active realizations,” 288-289, June 1972.

Electron. Let?., vol. 8, no. 11, pp.

Wasfy B. Mikhael (S’70-M’73-SM’83-F’86) was

born in Manfalout, Egypt, on November 3,1944. He received the B.Sc. degree (honors) in electronics and communications from Assiut University, Assiut, Egypt, the M.Sc.E.E. degree from the

‘University of Calgary, Calgary, Alberta, Canada, and the D.Eng. degree from Sir George Williams University, Montreal, Quebec, Canada, in 1965, 1970, and 1973, respectively.

From 1965 to 1968, he was an Engineer with the Telecommunications Organization, Cairo,

Egypt. From 1970 to 1973, he taught in the Computer Science Depart- ment at Sir George Williams University, and in 1973 he taught in the Mathematics Department, Dawson College, Montreal, Quebec, Canada. Since May of 1973, he has been a Member of the Scientific Staff at Bell-Northern Research, Ottawa, Ontario, Canada, as well as an adjunct Associate Professor in Electrical Engineering at Sir George Williams University (now known as Concordia University). In August 1978, he joined the faculty of West Virginia University, where he is now a Professor of Electrical Engineering. His research interests are active networks, switched-capacitor circuits, and adaptive signal processing. He has several patents and publications in the area of communication networks and active filters.

Dr. Mikhael was the recipient of the Bell Northern Research Outstand- ing Contribution Patent Award in 1978, the Outstanding Researcher Award from the College of Engineering, West Virginia University, in 1982 and 1983, and the Halliburton Best Researcher Award in 1984. He served as Chairman of the 1984 Midwest Symposium on Circuits and Systems. Dr. Mikhael is presently Associate Editor of the Letters Section of the IEEE TRANSACTIONSON CIRCUITSAND SYSTEMS.

Sherif Michael (S’78-M’83) was born in Alexandria, Egypt. He received the B.Sc. degree in electrical engineering (electronics and communications) from Cairo University, Cairo, Egypt, in 1974. He received the M.Sc. degree in industrial engineering and the Ph.D. degree in electrical engineering from West Virginia Uni- versity, Morgantown, WV, in 1980 and 1983, respectively.

He received technical training at Philips Industries, Eindhoven, Holland. He served as-a

First Lieutenant in an Engineering Corps specializing in water well drilling, where he conducted, as a Field Engineer, electronic measure- ments and supervised drilling operations in cooperation with Schlum- berger Co. Dr. Michael worked as a research and teaching fellow at West Virginia University. As a Research Engineer with the National Transpor- tation Research Center, Morgantown, WV, he worked on designing and implementing a new digital communications system for the Morgantown Personal Transit System (MPRT). Since 1983, he has been an Assistant Professor with the Department of Electrical and Computer Engineering at the Naval Postgraduate School, Monterey, CA. His present research interests are in the area of analog integrated circuits and active networks design, radiation hardening, solar cells and space power applications.

Dr. Michael is a member of Eta Kappa Nu, Alpha Pi Mu, and Tau Beta Pi and is a registered Professional Engineer.


Real Zero and Pole Synthesis I. Objective

To study real zero and pole synthesis, and cascade design of first-order circuits.

II. Introduction The general form of the transfer functions can be written as the ratio of two polynomials as

follows:

( )01

11

011

1

......

bsbsbsasasasasT n

nn

mm

mm

++++

++++=

−−

−− (1)

where the numerator coefficients and denominator coefficients are

real numbers, and is the order of the filter.

maaa ,...,, 10 110 ,...,, −nbbb

n

The degree of the numerator polynomial must be less than or equal that of the denominator polynomial for causality reasons. That is

nm ≤ (2)

The numerator and denominator polynomials can be factored, and can be expressed

as:

)(sT

( ) ( )( ) ( )( )( ) ( )n

mmpspspszszszsasT

−−−−−−

=......

21

21 (3)

where the zeros, , and the poles, , can be either real number or

complex number.

mzzz ,...,, 21 nppp ,...,, 21

Complex zeros and poles, however, must occur in conjugate pairs. In this experiment, only

real zeros and poles are discussed. In this case, the transfer function is factored as: )(sT

( ) ( ) ( ) ( )sTsTsTsT n...21= (4)

where are bilinear transfer functions. ,...,n,issTi 21 ,)'( =

Once the transfer function is factored into multiple stages, the task is to synthesize each stage using first-order circuits.

Real Zero and Pole Synthesis

Synthesizing real poles and zeros is based on the cost of the energy storage elements. For resistor, capacitor and Op Amp circuits, this requirement consists of limiting size or value of capacitors. Capacitors remain the most expensive and sensitive components in the discrete, hybrid, and integrated circuit design. Cost implies not only dollars but also size, weight, parasitic sensitivity, tolerance allocation, and sensitivity to environment changes. One of the designer’s main functions is to choose reasonable capacitor values.

According to pole and zero locations, we choose suitable circuits to synthesize these poles and zeros. Then, we find a set of relations between zeros and poles, and unknown resistor and capacitor values. The number of the unknown values is generally more than the number of equations. This property does not represent a problem but rather an opportunity. Usually, the design procedure begins with a selection of reasonable capacitors. Subsequently, other components can be calculated.

Cascade Design of First-Order Circuits

First, we factor the transfer function in terms of bilinear transfer functions s

Equation (4). Second, we synthesize the using first-order circuits. Third, we connect

these circuits in the sense that each successive circuit does not load the previous circuit. In this

way, the transfer function of this cascading circuit is .

)(sT ssTi )'( a

ssTi )'(

)(sT

Design Procedure for Real Zero and Pole Synthesis

1. The transfer function is given by: )(sT

( )( )( )( )53

43

1010210105)(

+×++×+

=sssssT (5)

Equation (5) is expressed as the product of two bilinear functions, and . )(1 sT )(2 sT

( )( )3

3

1 102105)(

×+×+

=sssT (6)

and

( )( )5

4

2 1010)(

++

=sssT (7)

2. Choose proper circuits to realize poles and zeros. For both transfer functions and

, the same first-order circuit is used, as shown in Fig.1. The transfer function of the

circuit in Fig. 1 is given by:

1( )T s

2 ( )T s

( )( )

( )(( )

)( )

1 12

1 2 2

1 11

2 2 2

1( )

1

1

1

cir

sC RRT sR sC R

s C RCC s C R

+= −

+

+= −

+

(8)

3. Synthesize the transfer functions and . For both circuits, the capacitance

values are chosen as:

1( )T s 2 ( )T s

uFCCCC 01.02,22,11,21,1 ==== (9)

In the first circuit, which synthesizes , the resistor values are computed as: 1( )T s

Ω= kR 201,1 (10)

and

Ω= kR 501,2 (11)

In the second circuit, which synthesizing , the resistor values are computed as: 2 ( )T s

Ω= kR 102,1 (12)

and

Ω= kR 12,2 (13)

4. Cascade the designed two circuits to realize the transfer function . Since the output of

the first-order circuit is just the output of Op Amps, the output resistance of the first-order circuit can be considered as zero. Thus, these two circuits can be connected directly, as shown in Fig. 2.

)(sT

+

-

1R)(sVi

)(sVo

2R

1C 2C

Fig.1 The noninverting first order operational amplifier circuit

+

-

Ω= kR 201,1)(sVi

)(sVo

uFC 01.01,1 =

+

-

Ω= kR 12,2

uFC 01.01,2 = uFC 01.02,1 = uFC 01.02,2 =

Ω= kR 501,2 Ω= kR 102,1

Fig.2 The cascading design example, which synthesizes real poles and zeros. The resulting transfer

function is ( )( )( )( )53

43

1010210105)(

+×+

+×+=

sssssT

III. Design The transfer function of the band-pass filter is given by:

( ) ( )( )( )( )43

43

1010210210

+×+×++

=ss

sssT (14)

Decompose the transfer function in the factor format. Synthesize real zeros and poles using the circuit shown as in Fig.1. Follow the above procedure, and compute the resistance and capacitance values.

IV. Computer Simulations 1. Simulate the designed band-pass filter.

2. Plot the magnitude and phase responses of each first-order circuit in the frequency range from 30Hz to 40kHz.

3. Plot the magnitude and phase responses of the overall circuit in the frequency range from 30Hz to 40kHz.

Note that the magnitude response of the cascading circuit should equal the product of the magnitudes responses of two first-order circuits. In addition, the phase response of the cascading circuit should equal the summation of the phase responses of two first-order circuits

V. Experiments 1. Build this band-pass filter using two first-order circuits. Use LF 351 Op Amps with a split

power supply voltage of ±15V.

2. Measure the magnitude response of each first-order circuit in the frequency range from 30Hz to 40kHz.

3. Measure the magnitude responses of the overall circuit in the frequency range from 30Hz to 40kHz.

VI. Lab Report In the report, you need present the experiment results and compare them with the simulation results. Comment on deviations from expected results, if any, and the reasons for these deviations. Your report should include the following: 1. The design steps and results. 2. The simulation results. 3. The experiment results. 4. Compare the simulation results with the experiment results. Discuss any discrepancies, and

make comments.

References [1]. M. E. Van Valkenburg, Analog Filter Design, Oxford University Press, 1982.

[2]. Dr. Robert Janes Martin, “EEL 4140: Lab Manual for the Design of Analog Filters,” University of Central Florida, 1997.


Sallen-Key Filters I. Objective

To study design and implementation of Sallen-Key filters.

II. Introduction The term biquad is an edited form of the word biquadratic, which means the ratio of two

quadratic polynomials. Thomas C. Lee named this title several decades ago, and it is now in common usage. All realizable polynomials used in analog filters can be factored to second order forms and generally are of quadratic nature. Thus, the biquad is a useful and universal building block. There are several circuits that can implement the biquad transfer functions. Sallen-Key circuit is one of these circuits.

Sallen-Key Filters

Sallen and Key proposed a class of circuits, named as Sallen-Key filters, in 1955. The basic Sallen-Key structure is shown in Fig. 1. This circuit incorporates a single amplifier embedded in a passive RC network to generate any type of second-order transfer functions: low-pass, high-pass, band-pass, and notch. In Fig.1, the box labeled “second-order passive RC

network” contains resisters and two (or sometime three) capacitors. is the input voltage,

is the output voltage of an amplifier having gain

)(sVi

)(sVo K , and is the amplifier input

voltage. If we apply superposition to the circuit, we obtain

)(sV

)()()()()( sVsTsVsTsV oFBiFF += (1)

where the feedforward gain of the passive network, , is defined as: )(sTFF

0)()()()(

=

=sVi

FF

osVsVsT (2)

and the feedback gain of the passive network, , is defined as: )(sTFB

0)()()()(

=

=sVo

FB

isVsVsT (3)

K

Second-orderpassive RC network

)(sVi )(sVo

)(sV

Fig. 1 The basic Sallen-Key topology, where is the input voltage, is the output voltage of an amplifier having gain

)(sVi )(sVo

K , and is the amplifier input voltage. )(sV

Also, we have the amplifier relation as:

)()( sKVsVo = (4)

We obtain the relationship between the input voltage and the output voltage in

term of

)(sVi )(sVo

)(sTFF and )(sTFB as:

)()(1

)()( sVsKT

sKTsV iFB

FFo −

= (5)

Unless the RC network is degenerate, and have the same denominator

polynomial as:

)(sTFF )(sTFB

)(sD

012)( bsbssD ++= (6)

Therefore, we can express and as: )(sTFF FB )(sT

)()()(

sDsNsT FF

FF = (7)

and

)()()(

sDsNsT FB

FB = (8)

where )(sNFF and )(sNFB are polynomials with degree of at most two.

Using Equations (5), (7), and (8), we can derive the transfer function of the Sallen-Key filter as:

)()()(

)()()(

sKNsDsKN

sVsVsT

FB

FF

i

o

−=

= (9)

)(sKNFB can modify the coefficients of the denominator polynomial of . It is

evident that the poles of can be placed anywhere in the complex plane by appropriately

choosing

)(sT

)(sT

K . According to the different K values, we can classify two kinds of Sallen-Key filters. If the amplifier gain , this kind of circuit is called as the positive Sallen-Key filter. Otherwise, if the amplifier gain , the circuit is referred to as the negative Sallen-Key filter.

0>K0<K

Low-Pass Positive Sallen-Key Filter

+

-

1R)(sVi

)(sVo

2R

1C

2C

aR bR

Fig. 2 The low-pass positive Sallen-Key filter

The low-pass positive Sallen-Key filter is shown in Fig. 2. From this configuration, we can

compute the feedforward and feedback gains of the passive network as:

2121121

21

22

2

212111

1

)()()(

CCRRs

CRRRR

CRs

CCRR

sDsNsT FF

FF

+⎥⎦

⎤⎢⎣

⎡ +++

=

=

(10)

and

2121121

21

22

2

2211

1

)()()(

CCRRs

CRRRR

CRs

sCR

sDsNsT FB

FB

+⎥⎦

⎤⎢⎣

⎡ +++

=

=

(11)

Substituting Equations (10) and (11) into Equation (9), we can derive the transfer function

as: )(sT

2121121

21

22

2

2121

11)(

CCRRs

CRRRR

CRKs

CCRRK

sT+⎥

⎦

⎤⎢⎣

⎡ ++

−+

= (12)

where the amplifier gain K is given by:

a

ba

RRRK +

= (13)

From Equation (12), we can obtain the cutoff frequency as:

21210

1CCRR

=ω (14)

and the quality factor as:

( )12

21

1

2

22

11

121

21

22

2121

11

1

1

1

CRCR

RR

CRCRK

CRRRR

CRK

CCRRQ

⎟⎟⎠

⎞⎜⎜⎝

⎛++−

=

++

−=

(15)

Design Procedure for Low-Pass Positive Sallen-Key Filters

1. There are five adjustable parameters in the circuit as shown in Fig. 2: the four passive

component values ( ) and the active parameter (the amplifier gain 2121 ,,, CCRRa

ba

RRR

K+

= ).

Thus, we can arbitrary choose three values. Each choice leads to a filter with different

properties. Here, we choose RRR == 21 and CCC == 21 . The rationale for this choice relies

on two principal aims: mathematical convenience and low element spread. The first needs no elaboration. The second means that resistance and capacitance values should not be spread too widely.

2. Based on the above choice, we can rewrite the expression for 0ω and as: Q

RC1

0 =ω (16)

and

KQ

−=

31 (17)

Hence, given values of 0ω and Q , we can calculate K and . RC

3. We choose the proper capacitance value of , and then compute the resistance value of C R .

4. If we decide the value of , we can easily obtain the value of as: aR bR

ab RKR )1( −= (18)

Note that the DC gain is equal to K for the above design. If we want the DC gain to be unity,

for instance, a voltage divider can be used to replace . Although the element spread for the 1R

equal-R and equal-C filter is excellent, it turns out that the quality factor is strongly sensitive to variations in component values. Thus, it is of some interest to investigate an alternative design in practice.

Low-Pass Negative Sallen-Key Filter

+

-1R)(sVi

)(sVo

2R

1C 2C

oR

4R

3R

Fig. 3 The low-pass negative Sallen-Key filter

The low-pass negative Sallen-Key filter is shown in Fig. 3. We can use the same method to derive the transfer function of the low-pass negative Sallen-Key filter as:

2142414321322224121113

2

2121

11111111111)(

CCRRRRRRRRRRKs

CRCRCRCRCRs

CCRRK

sT×⎥

⎦

⎤⎢⎣

⎡++++

++⎥

⎦

⎤⎢⎣

⎡+++++

−

= (19)

where the amplifier gain K is given by:

4RRK o= (20)

The cutoff frequency is given by:

2142414321320

111111CCRRRRRRRRRR

K×⎟⎟⎠

⎞⎜⎜⎝

⎛++++

+=ω (21)

and the quality factor is expressed as:

2

1

421

2

321

4241432132

11111

11111

CC

RRCC

RRR

RRRRRRRRRRK

Q

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛++

⎟⎟⎠

⎞⎜⎜⎝

⎛++++

+

= (22)

Design Procedure for Low-Pass Negative Sallen-Key Filters

1. We choose the resistance and capacitance values as RRRRR ==== 4321 and

. Then, Equations (21) and (22) can be simplified as: CCC == 21

RCK 5

0+

=ω (23)

and

55+

=KQ (24)

2. Given the value, we obtain Q K as:

525 2 −= QK (25)

3. We solve the following equation for the value of . RC

0

5ω+

=KRC (26)

4. We choose the proper capacitance value of , and then compute the resistance value of C R .

5. We compute the resistance value of as: oR

4KRRo = (27)

III. Design Design example 1: the low-pass positive Sallen-Key filter

The filter specifications are given by srad / 1000*20 πω = and 5=Q . Follow the above

procedure to choose the appropriate values of capacitors and resistors. Design example 2: the low-pass negative Sallen-Key filter

The filter specifications are given by srad / 4000*20 πω = and 1=Q . Follow the above

procedure to choose the appropriate values of capacitors and resistors. IV. Computer Simulations 1. Simulate the above two designed low-pass filters.

2. Plot the magnitude and phase responses of these two filters in the frequency range 10Hz-30kHz.

V. Experiments 1. Build two designed filters, using LF 351 Op Amps with a split power supply voltage of ±15V.

2. Use the Channel One of digital oscilloscope to show the input voltage waveform, and channel Two to show the output voltage waveform. Record the input and output voltage waveforms at the frequencies, 10Hz, 100Hz, 1kHz, and 10kHz.

3. Measure the magnitude and phase responses of two designed filters. The frequency range is from 10Hz to 30kHz.

VI. Lab Report In the report, you need present the experimental results and compare them with the expected results. Discuss any discrepancies, make comments, and write conclusions. Your report should include the following: 1. The complete circuit design processes and results. 2. The computer simulation results: the magnitude and phase responses of both filters. 3. The experiment results: the magnitude and phase responses of both filters and the recorded graphs. 4. Summary and conclusions. References [1]. M. E. Van Valkenburg, Analog Filter Design, Oxford University Press, 1982, pp 171-179.


[3]. Artice M. Davis, “Sallen-Key Filters,” Chapter 6, in RC Active Filter Design Handbook,

Edited by F. W. Stephenson, John Wiley & Sons, Inc., 1985.


State-Variable Biquads I. Objective

To study design and implementation of state-variable biquads.

II. Introduction In the previous experiment, one realization of biquads, Sallen-Key filter, was discussed.

In the Sallen-Key filter, a single amplifier embedded in a passive RC network is used to generate a second-order transfer function. The structure of the Sallen-Key filter is relatively simple. However, it is subject to great sensitivity to the constituent components’ values.

In this experiment, another biquad, the state-variable structure, will be studied. A relatively high quality value can be achieved in this circuit. In addition, state-variable biquads provide flexibility, good performance, and low sensitivity. The implementation of the state-variable baud is based on the state-variable approach. State-variable methods of solving differential equations are employed in the development of the realization. In the implementation of these realizations, three basic active building blocks are generally used: the summer, the integrator, and the lossy integrator.

In this experiment, two kinds of state-variable biquads will be introduced: Tow-Thomas biquad and Kerwin-Huelsman-Newcomb (KHN) biquad.

Tow-Thomas Biquad

The structure of the Tow-Thomas biquad is shown as in Fig. 1. In this configuration, all positive terminals of the Op Amps are grounded. The first basic building block composed

of the Op Amp is a lossy integrator, while the second block is a summer amplifier, and

the third one is an integrator. The path composed of is feedback. , , and

constitute feed forward paths to obtain the transmission zeros.

1U

3R 4R 5R 6R

Analyzing the Tow-Thomas biquad, we can derive the second-order transfer function from this realization as:

21732

8

11

221537

6

7

6

1411

2

6

81

111

)(

CCRRRR

CRss

CCRRRR

RR

CRCRss

RRsT

++

+⎟⎟⎠

⎞⎜⎜⎝

⎛−+

−= (1)

From Equation (1), we can obtain the cutoff frequency as:

21732

80 CCRRR

R=ω (2)

and the quality factor as:

2732

181 CRRR

CRRQ = (3)

Since the numerator is of general second-order form, we can achieve any filter type by choosing proper resistor values. For example, we can realize the second–order band-pass filter by choosing

∞== 65 RR (4)

Where denotes the infinite value. ∞

In practice, Equation (4) implies that and are not present in the circuit. We

substitute Equation (4) into Equation (1). Therefore, the transfer function specified in Equation (1) is simplified as:

5 6R R

21732

8

11

2

174

8

1)(

CCRRRR

CRss

CRRRs

sT++

= (5)

In Equation (5), the voltage gain at the center frequency 0ω is given as:

74

81

RRRRH BP = (6)

)(1 sV

+

-

+

-

+

-

1R

1C

7R

3R

2C

2R

)(sVo

5R

4R

6R

8R

1U 2U 3U

Fig.1 The feed forward Tow-Thomas circuit

Design Procedure for band-pass Tow-Thomas filters

1. Given the filter specifications as ω , , and Q BPH .

2. Choose the resistance and capacitance values and C R such that

CCC == 21 (7)

and

RRRR === 873 (8)

3. Define a positive constant α such that

R

RR2

32

2

α

α

=

= (9)

4. Based on the above chosen parameter values, Equations (2), (3), and (6) can be simplified as following:

RCαω 1

0 = (10)

RRQα

1= (11)

4

1RRH BP = (12)

5. Compute the positive number α as:

RC0

1ω

α = (13)

6. Determine the resistance value of as: 2R

RR 22 α= (14)

7. Compute the resistances values of and as: 1R 4R

RQR α=1 (15)

and

BPHRR 1

4 = (16)

Kerwin-Huelsman-Newcomb Biquad

The Kerwin-Huelsman-Newcomb (KHN) biquad is shown in Fig.2. This circuit is

composed of a summer amplifier and two integrators. and form the feedback

paths. The three output terminals voltages, , , and , achieve high-pass,

band-pass, and low-pass filter, respectively. Moreover, these three transfer functions have the same poles.

4R 5R

)(2 sV )(3 sV )(4 sV

)(1 sV)(4 sV

+

-

+

-

+

-3R

6R 1C

1R

5R

2C

2R

)(2 sV

)(3 sV4R1U 2U 3U

Fig.2 The Kerwin-Huelsman-Newcomb (KHN) circuit

The high-pass transfer function is obtained by the ratio of and as: )(2 sV )(1 sV

2002

2

1

2)()(

ωω +⎟⎠⎞⎜

⎝⎛+

=sQs

sHsVsV HP (17)

Where HPH is given by:

4

35

6

1

1

RRRR

H HP+

+= (18)

the cutoff frequency 0ω is given by:

2121

5

620 CCRR

RR

=ω (19)

and the quality factor is given by: Q

116

225

5

6

3

4

1

1

CRRCRR

RR

RR

Q

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+= (20)

From the relationship between and , the band-pass transfer function is

given as:

)(1 sV )(3 sV

2002

0

1

3)()(

ωω

ω

+⎟⎠⎞⎜

⎝⎛+

⎟⎠⎞⎜

⎝⎛−

=sQs

sQH

sVsV BP

(21)

where BPH is the voltage gain at the frequency 0ω as:

3

4RRH BP = (22)

The low-pass filter is achieved by the relationship between and as: )(1 sV )(4 sV

2002

20

1

4)()(

ωωω

+⎟⎠⎞⎜

⎝⎛+

=sQs

HsVsV LP (23)

where LPH is the DC gain, expressed as:

4

36

5

1

1

RRRR

HLP+

+= (24)

Design Procedure for band-pass KHN filters

1. Given the design specifications: 0ω , , and Q BPH .

2. Choose the capacitance value such that C

CCC == 21 (25)

3. Let , , , and have the same resistance value as: 2R 3R 5R 6R

RRRRR ==== 6532 (26)

4. Define a positive constant number α such that

R

RR2

22

1

α

α

=

= (27)

5. Equations (19), (20), and (22) are simplified as:

RCαω 1

0 = (28)

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

3

412 R

RQ α (29)

12 −=αQHBP (30)

6. Compute the positive constant number α as:

12+

=BPHQα (31)

7. Compute the resistance value of R as:

CR

αω0

1= (32)

8. Compute the resistance values of and as: 1 4R R

RR 21 α= (33)

and

RQR ⎟⎠⎞

⎜⎝⎛ −= 124 α

(34)

III. Design Design example 1: the band-pass Tow-Thomas filter

Design a band-pass filter having the gain 3=BPH , the quality factor , and the cutoff

frequency

5=Q

srad / 1000*20 πω = , using the Tow-Thomas circuit. Choose the appropriate

values of capacitors and resistors, following the above procedure. Design example 2: the band-pass Kerwin-Huelsman-Newcomb (KHN) filter

Design a band pass filter having the gain 3=BPH , the quality factor , and the cutoff

frequency

10=Q

srad / 1000*20 πω = , using the KHN circuit. Choose the appropriate values of

capacitors and resistors, following the above procedure. IV. Computer Simulations 1. Simulate the two designed band-pass filters.

2. Plot the magnitude and phase responses in the frequency range from 10Hz to 20 kHz.

3. Compare these two magnitude response plots, and understand the mechanism of the

quality factorQ .

V. Experiments 1. Build above two designed biquad circuits, using LF 351 Op-amps with a split power supply voltage of ±15V.


3. Measure the magnitude response for two circuits. The frequency range is from 10Hz to 20kHz.

VI. Lab Report In the report, you need present the experiment results and compare them with the simulation results. Discuss any discrepancies, make comments, and write conclusions. Your report should include the following: 1. The complete circuit design processes and results. 2. The computer simulation results: the magnitude and phase responses for both circuits. 3. The experiment results: the magnitude responses for both filters and the recorded graphs. Note that two band-pass filters have the same center frequency and the same voltage gain, but the different quality factors. Compare the magnitude responses of these filters, and

understand the rule of the quality factor . pQ

4. Summary and conclusions. References [1]. M. E. Van Valkenburg, “Analog Filter Design”, Chapter 5, Oxford University Press, 1982.


[3]. E. Sanchez-Sinencio, “Biquad I: The State-Variable Structure,” Chapter 8, in RC Active Filter Design Handbook, Edited by F. W. Stephenson, John Wiley & Sons, Inc., 1985.


Single Op Amp Band-Pass Filters I. Objective

To design and implement single Op Amp band-pass biquad filters.

II. Introduction Band-pass filters find wide use in modems, radio receivers, electro-optical systems, and

communications systems. The figure of merit for a band-pass filter is the quality factor which is defined as the ratio of the center frequency to the 3dB bandwidth. The term quality factor was first coined for band-pass filters in the first public communication systems. In early radio, tuned radio frequency filters were used directly in front of the detector. These radios were very noisy and had poor selectivity by today’s standards. The sharper the transition from the pass band to the stop band, the better the rejection of the adjacent channels. This led to better separation of stations. A larger center frequency to bandwidth ratio was associated with higher quality factor.

The circuit used in this experiment is the multi-loop feedback filter. Its basic circuit configuration is shown in Fig. 1. This structure is similar to Sallen-Key filters. There are, however, two major differences. The first difference is that the active element is an Op Amp in the multi-loop feedback structure rather than a finite-gain amplifier. The second difference is that there are two feedback paths (rather than one) from the output of the amplifier to the RC network. This is the reason for the name, multi-loop feedback filter.

The multi-loop feedback circuit can achieve the low-pass, high-pass, or band-pass filter by choosing the different RC networks. In this experiment, only the band-pass filter is discussed. First, a basic band-pass structure without the positive feedback is introduced. Then, another band-pass realization increases the quality factor by incorporating the positive feedback. The design procedures for these two realizations are also given.

Passive RC network)(sVi

)(sVo+

-

Fig. 1 The basic multi-loop feedback filter configuration

Band-Pass Multiple-Loop Feedback Filters without the Positive Feedback

The band-pass multiple-loop feedback filter without the positive feedback is shown in Fig. 2. The transfer function is given by:

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛++

−

=

=

322121212122

2

211111

)()()(

RRCCRRCCs

RCRCs

CRs

sVsVsT

i

o

(1)

Notice that this band-pass transfer function is an inverting one. From Equation (1), we

obtain the quality factor and the cutoff frequency Q 0ω as:

12

23

22

13

1

31

CRCR

CRCR

RR

Q+

+= (2)

and

2132

1

3

0

1

CCRRRR

+=ω (3)

In addition, we can get the midband gain as:

1

21

2

1CC

RR

H BP+

= (4)

+

-1R)(sVi

)(sVo

2R

1C

2C

3R

Fig. 2 The band-pass multi-loop feedback filter without positive feedback

Design Procedure for Band-Pass Multiple-Loop Feedback Filters without the Positive Feedback

1. Given the design specifications, 0ω ,Q , and BPH .

2. In the band-pass multi-loop feedback filter, it is convenient to use the equal-value capacitor

design. Choose a suitable capacitance value, , for the capacitors and , as: C 1C 2C

CCC == 21 (5)


BPCHQR

01 ω= (6)


CQR0

22ω

= (7)


( )BPHQCQR−

= 20

3 2ω (8)

Note that in Equation (8), BPH must be less than in order that be finite and

positive.

22Q 2R

Band-Pass Multiple-Loop Feedback Filters with Positive Feedback

A modification of the band-pass filter shown in Fig.2 can be used to reduce the spread of

element values for high realizations. The modified circuit is shown in Fig. 3. The positive

feedback is provided by the voltage divider consisting of two resistors and . To simplify

equations describing the circuit, the effect of these two resistors is presented by defining a constant

Q

aR bR

K as:

b

a

RRK = (9)

+

-1R)(sVi

)(sVo

2R

1C

2C

bRaR

Fig. 3 The band-pass multi-loop feedback filter with positive feedback

The transfer function for this circuit is given by:

( )

2121211222

2

21111

1

)()()(

CCRRCRK

CRCRss

CRKs

sVsVsT

i

o

+⎟⎟⎠

⎞⎜⎜⎝

⎛−++

+−

=

=

(10)

From Equation (10), we obtain the quality factor and the cutoff frequency Q 0ω as:

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

1

2

1

2

1

2

2

1

2

11CC

RRK

CC

CC

RR

Q (11)

and

21210

1CCRR

=ω (12)

The midband gain is given as:

( )

111222

1111

1

CRK

CRCR

CRK

H BP−+

+

= (13)

From Equation (11), it is clear that the quality factor is improved by introducing positive feedback K . The design procedure for this filter is based on the solution of these equations, and is summarized in following.

Design Procedure for Band-Pass Multiple-Loop Feedback Filters with the Positive Feedback

1. Given the design specifications 0ω and . Q

2. As before, it is convenient to use an equal-value capacitor design, in which the capacitors

and have the same value. Choose a suitable capacitance value, , for the capacitors

as:

1C 2C C

CCC == 21 (14)

3. Determine the ratio parameter that would be required if there are no positive feedbacks

as:

0m

204

1Q

m = (15)

4. Select the desired resistor ratio m , which is less than one, and greater than . 0m

2

1RRm = (16)

Use to determine the amount of positive feedback m K as:

QmmK −= 2 (17)

5. Choose a convenient value for , and calculate as: bR aR

ba KRR = (18)


mCR

02

1ω

= (19)

7. Determine the value of using the relation 1R

21 mRR = (20)

8. The value of BPH , the gain at resonance, is determined by the relation.

KmKH BP −+

=2

1 (21)

III. Design Design example 1: the second-order band-pass multi-loop filter without positive feedback

Design a second-order band-pass filter having the gain 2=BPH , the quality factor 2=Q , and

the center frequency , using the multi-loop structure without positive feedback.

Choose the appropriate values of capacitors and resistors, following the above procedure.

srad /1040 =ω

Design example 2: the second-order band-pass multi-loop filter with positive feedback

Design a second-order band-pass filter having the quality factor and the center 10=Q

frequency , using the multi-loop structure with positive feedback. Choose the

appropriate values of capacitors and resistors, following the above procedure.

srad /1040 =ω

IV. Computer Simulations 1. Simulate the above two band-pass filters with the designed resistance and capacity values.

2. Plot the magnitude and phase responses, for the frequency range 10Hz-30kHz.

V. Experiments 1. Build above two designed band-pass filters, using LF 351 Op-amps with a split power supply voltage of ±15V.


3. Measure the magnitude responses of two filters. The frequency range is from 10Hz to 30kHz.

VI. Lab Report In the report, you need present the experiment results and compare them with the simulation results. Discuss any discrepancies, make comments, and write conclusions. Your report should include the following: 1. The complete circuit design processes. 2. The Computer simulation results: the magnitude and phase responses for both circuits. 3. The Experiment results: the magnitude responses for both filters and the recorded graphs. 4. Summary and conclusions. References [1]. M. E. Van Valkenburg, Analog Filter Design, Oxford University Press, 1982.


[3]. L. P. Huselsman, “Multiple-Loop Feedback Filters,” Chapter 7, in RC Active Filter Design Handbook, Edited by F. W. Stephenson, John Wiley & Sons, Inc., 1985.


Two Op Amps Current Generalized Immittance Structure (CGIC) Based Biquad

I. Objective

To study CGIC biquads, to use this structure to design second-order low-pass and band-pass filters with given specifications, and to functionally tune the above circuits to get the specified values of the cutoff frequency and the quality factor.

II. Introduction The general CGIC structure is shown in Fig. 1. The transfer functions between the input

and output terminals, , , and , assuming ideal Op Amps, are given below: )(2 sV )(3 sV )(4 sV

)(

1)(

)()()(

2

85

2

675

31

sDYYY

YYYshY

sVsVsT

i

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛++

=

=

(1)

)(

)(1

)()()(

74

76

4

85

1

42

sD

YshYYY

YYY

sVsVsT

+−⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

=

(2)

and

)()(

)()()(

75

23

sDYshY

sVsVsT

i+

=

= (3)

where

41

32)(YYYY

sh = (4)

( ) ( )8765 )()( YYshYYsD +++= (5)

The most commonly used second-order transfer functions can be easily generated from the above equations, as summarized in Table 1. By using minimum sensitivity constraints in circuit 1, 3, 7, 10, and 12, possible sets of element values have been obtained, as shown in Table 2. In this experiment, two kinds of filters will be discussed in detail: a low-pass filter using the number 1 circuit, and a band-pass filter using the number 7 circuit.

+

-

)(1 sV)(2 sV

+

-1Y

2Y

3Y

4Y

8Y6Y

7Y5Y

)(sVi

)(4 sV

)(3 sV

Fig. 1 The basic CGIC configuration

Design Procedure for Low-Pass CGIC Filters

1. Given the design specifications, 0ω and . Q

2. Circuit 1 in Table 2 realizes a low-pass filter. The resistor and capacitor are chosen as follows:

GGGGG ==== 8541 (7)

QGG =3 (8)

and

CCC == 32 (9)

where G is the conductance define as .1R

G =

3. Then, the transfer function is simplified as:

222

222 1

2

)(

CRRCQss

CRsT++

= (10)

From the definition of 0ω , we obtain

RC1

0 =ω (11)

4. Choose an appropriate capacitor value . Then, compute the resistor value C R as:

oCR

ω1

= (12)

5. Consequently,

RQR =3 (13)

+

-1R

)(sVi)(sVo

4R

3C

2C

3R

+

-

8R

5R

Fig. 2 The second-order low-pass CGIC filter

Design Procedure for Band-Pass CGIC Filters

1. Given the design specifications 0ω and . Q

2. Circuit 7 in Table 2 realizes a band-pass filter. The resistance and capacitance values are chosen as follows:

CCC == 83 (14)

GGGGG ==== 6421 (15)

QGG =7 (16)

and

08 =G (17)

G is the conductance define as RG 1= .

3. The transfer function is simplified as:

222

1 1

2

)(

CRRCQss

RCQs

sT++

= (18)

4. Choose an appropriate capacitor value . Then, compute the resistance value of C R as:

oCR

ω1

= (19)

5. Consequently,

RQR =7 (20)

+

-1R

)(sVo

4R

3C

8C

+

-

2R

6R )(sVi7R

Fig. 3 The second-order band-pass CGIC filter

III. Design Design example 1: the second-order Butterworth low-pass CGIC filter

Design a second-order low-pass filter with a quality factor 707.0=Q , and a cutoff frequency

. Choose the capacitance value as srad /10*2*2 30 πω = uFC 01.0= . Compute the

appropriate values resistors, following the above procedure. Design example 2: the second-order band-pass CGIC filter

Design a second-order band-pass filter having a quality factor 10=Q , and a cutoff frequency

. Choose the capacitor value as srad /10*3*2 30 πω = uFC 01.0= . Compute the

appropriate values resistors, following the above procedure. IV. Computer Simulations 1. Simulate both of the above design filter examples with the calculated resistance values.

2. Plot the magnitude and phase responses in the frequency range from 50Hz to 20kHz.

V. Experiments 1. Build above the CGIC filters designed in part III, using LF 351 Op Amps with a split power supply voltage of ±15V.

2. The circuits need to be functionally tuned to yield the specified values of the quality factor

Q and the cutoff frequency 0ω . A sinusoidal input is used during the tuning processes.

In the case of the low-pass filter, the circuit is first tuned for 0ω . Applying a sinusoidal input of

the frequency 0ω , then adjusting and monitoring the phase angle difference between the

output and input voltages. The circuit is tuned for

8R

0ω when the output voltage lags the input

voltage by 90 degrees.

To tune the circuit to attain the specified Q , the gain LPH (the ratio of the output voltage

between the input voltage) at a low frequency, much lower than 0ω (usually a few Hz but not

to 0Hz), is determined. Then, a sinusoidal input at the frequency 0ω is applied. is

adjusted until the gain of this filter become times

3R

Q LPH .

In case of the band-pass filter, the desired value of 0ω is realized by adjusting until the

phase angle between the output and the input voltage equals 0 degrees at the sinusoidal input of

the frequency

2R

0ω . can be attained by adjusting until the output voltage advances the

input voltage by 45 degrees when the frequency of the sinusoidal input is the lower cutoff

frequency

Q 7R

1ω .


4. Measure the magnitude response for each of two circuits after functional tuning is complete. The frequency range is from 10Hz to 30kHz.

VI. Lab Report In the report, present the experimental results and compare them with the simulation results. Comment on the deviations from the expected results, and the reasons for these deviations. Your report should include the following: 1. Derive the transfer function for the above two circuits in the form supplied in the table from the general form, using the specified element values.

2. Compute the quality factor and the cutoff frequency Q 0ω from the measured resistance

values after functional tuning is complete. Compare Q and 0ω with the desirable

specifications. 3. The computer simulation results: the magnitude and phase responses for both circuits. 4. The experiment results: the magnitude responses for both filters and the recorded graphs. 5. Summary and conclusions. References [1]. Wasfy B. Mikhael, “2 OA Current Generalized Immittance Structure (CGIC) Based Biquad,” Lab Manual, University of Central Florida.

[2]. Wasfy B. Mikhael, “Biquad II: The Current Generalized Immittance (CGIC) Structure,” Chapter 9, in RC Active Filter Design Handbook, Edited by F. W. Stephenson, John Wiley & Sons, Inc., 1985.

[3]. Wasfy B. Mikhael, “The Current Generalized Immittance (CGIC Biquad),” Chapter 82, in Circuits and Filter Design Handbook, Edited by F. W. Stephenson, CRC Press, 2003, pp 2495-2514.

Table 1 Element identification for realizing the most commonly transfer functions

Circuit Number

Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Transfer Function Remarks

1 G1 sC2 sC3+G3 G4 G5 0 0 G8 ( )541832832

28451

2 )(GGGGGsCGCCs

GGGGsT

+++

= Low-pass

2 sC1 G2 G3 sC4+G4 0 G6 G7 sC8+G8

( ) ( )87323286426412

2

6732

1

1)(

GGGGGGCGGCsGCCsGG

GGGsT

++++

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

Low-pass

3 G1 G2 sC3 G4 0 G6 sC7 sC8+G8 ( )( ) 4168233287

26273

2

1 )(GGGGGsCCGCCs

GGCCssT

++++

= High-pass

4 G1 G2 sC3 G4 0 G6

77

7

1 RsCsC

+0 ( )

416764172732

62732

1 )(GGGRGGGsCGCCs

GGCCssT

+++

= High-pass

5 G1 sC2 sC3 G4 sC5 G6 0 G8

6414158322

4

8415

2

1)(

GGGGGsCGCCsGG

GGsCsT

++

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

Band -pass

6 sC1 G2 G3 sC4+G4 0 G6 sC7 G8

( ) 8328413276412

2

6327

1

1)(

GGGGGCGGCsGCCsGG

GGsCsT

+++

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

Band-pass

7 G1 G2 sC3 G4 0 G6 G7 sC8+G8 ( )( ) 6418723283

26273

1 )(GGGGGGsCGCCs

GGGsCsT

++++

= Band-pass

8 sC1 G2 G3 sC4 G5 0 G7 sC8 ( )732328541

27328451

2

2 )(GGGGGsCGCCs

GGGCCGCssT

++++

= Notch

9 G1 G2 sC3 G4 G5 0 sC7 G8 ( )541823273

28451273

2

2 )(GGGGGsCGCCs

GGGGGCCssT

++++

= Notch

10 G1 G2 sC3 G4 G5 G6 sC7 sC8+G8

( )( )8723

6541

87

82

87

7

273

5412

3 )(

CCGCGGGG

CCGss

CCC

GCCGGGs

sT

++

++

+

+⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

Notch

11 sC1 G2 G3 sC4 G5 G6 G7 sC8 ( )( ) 7323286541

26273385541

2

1 )(GGGGGsCGGCCs

GGGGGCsGGGCssT

+++++−

= For all-pass: 06 =G

25 GG =

12 G1 G2 sC3 G4 G5 G6 sC7 G8 ( )( ) 4165823573

25418536273

2

1 )(GGGGGGsCGCCsGGGGGsCGGCCs

sT++++−+

= For all-pass: 06 =G

25 GG =

Table 2 Design values and tuning procedure

Tuning sequence Circuit Number

(from Table 1)

Design values Transfer function realized

nω 0ω Q

1 GGGGG ==== 8541 , QGG =3 , CCC == 32 , where 0ωGC =)(

2)(20

2 sDsT ω=

8G 3G

3 GGGGG ==== 6421 , QGG =8 , 08 =C , CCC == 73 , where

0ωGC = )(2)(

2

1 sDssT =

4G 8G

7 GGGGG ==== 6421 , QGG =7 , 08 =G , CCC == 83 , where

0ωGC = ( )

)(2)( 0

1 sDsQsT ω

= 2G 7G

10 GGGGGG =+=== 65421 , QGG =8 , CCCC =+= 873 , where CG=0ω and 750

2 CGn ωω = ( )

)()(

227

3 sDs

CCsT nω+

= 2G 6G 8G

12 GGGGG ==== 5421 , 06 =G , CCC == 73 , QGG =8 , where

0ωGC = )()()(1 sD

sDsT −= 4G 4G 8G

Note: ( ) 200

2)( ωω ++= sQssD

Study Guide C


Biquads II: The current Generalized Immittance (CGIC) Structure


High-Order Low-Pass Filter Design I. Objective

To study the cascade design method for realizing high-order filters. Current Generalized Immittance Structure (CGIC) Biquads are employed to build high-order low-pass filters.

II. Introduction The previous experiments have discussed the design of first-order and second-order filters.

In this experiment, one design method for the high-order filter is introduced: the cascade design method. This method is the most commonly used method because of its simplicity. It is based on cascading first-order and second-order filter sections in such a manner that each section does not interact with others.

For the cascade design method, the effect of component variations on the overall frequency response is large. Thus, the cascade design method may be tolerated for the applications where only a few filters are constructed and manual tuning is used, or where filter specifications are not particularly tight. However, when filters are produced in large quantities and are subjected to stringent specifications, the cascade design method is not recommended.

The cascade design method usually consists of three steps: decomposing transfer functions into poles and zeros, grouping poles and zeros into first-order or second-order filter sections, and synthesizing these filter sections.

Step 1: Decomposing transfer functions

In general, any form of transfer functions can be expressed in term of poles and zeros. Poles and zeros can be either real numbers or complex numbers. Complex zeros and poles, however, must occur in conjugate pairs.

Step 2: Grouping poles and zeros into first-order or second-order filter sections

In determining the order of sections and pairing of poles and zeros, it is important to consider the occurrence of “internal resonances.” This refers to the phenomenon of the large-voltage buildup at certain frequencies at internal nodes of filters (usually at the output terminal of an Op Amp). Thus, the gain to one of these nodes from the input of the filter may be higher than the gain to the output of the filter. Consequently, clipping may occur at output of an internal Op Amp before the filter output voltage shows clipping. This internal clipping will

show up as a level-dependent change in the overall frequency characteristic, and will restrict the range of input signal levels that the filter can handle.

In order to achieve maximum dynamic range, the filter should be designed such that the clipping will first occur at the output of the last Op Amp of the filter. The following rules of thumb are useful in maximizing the dynamic range of cascaded design filters.

Rule 1: Place the sections in the order of increasing values. (The Q value of the first-order

section is assumed to be zero.)

Q

Rule 2: Group the pole and the zero, which are closest in the S plate.

Rule 3: Distribute the overall gain equally among the sections.

Step 3: Synthesizing these first-order and second-order filter sections

For each filter section decided in step 2, we choose the appropriate circuit to synthesize it. When we cascade these sections together, we must consider effects of the input and output impedance. If the output impedance of the last section is not zero, and the input impedance of the next section can not be assumed as infinite, a buffer is needed to insert between these two sections. Otherwise, we can connect these sections directly.

In this experiment, the low-pass CGIC Biquad, which was studied in the last experiment, is used as the build-up circuit block for the high-order low-pass filter. The circuit of the CGIC low-pass Biquad is shown in Fig. 1. In this Biquad, the resistor and capacitor values are chosen as:

RRRRR ==== 8541 (1)

and

CCC == 32 (2)

Based on these resistance and capacitance values in Equation (1) and (2), the transfer function is given by:

223

2

22

1

2

)(

CRCRss

CRsTCGIC

++= (3)

From Equation (3), we can obtain the cutoff frequency 0ω and the quality factor as: Q

RC1 0 =ω (4)

and

RRQ 3= (5)

When we use this circuit to realize a second-order transfer function, we first choose the capacitor value . Then, we can compute the resistor value C R using Equation (4). Finally,

we can compute the resistance value of using Equation (5). 3R

+

-1R

)(sVi)(sVo

4R

3C

2C

3R

+

-

8R

5R

Fig. 1 The second-order low-pass CGIC Biquad

Design Procedure for the High-Order Low-Pass Filter

1. Design a six-order Chebyshev low-pass filter, which has a maximum pass-band attenuation of 1.0dB and the bandwidth 3979 Hz. That is

srad /000,250 =ω (6)

2. The normalized six-order Chebyshev low-pass transfer function with a maximum pass-band attenuation of 1.0dB is given as:

( )( )( )12471.046413.055772.033976.099073.012436.0061415.0)( 222 ++++++

=ssssss

sTnorm (7)

3. From (7), it can be seen that this high-order filter consists of three second-order sections.

)(sTnorm is decomposed into three second-order transfer functions as:

( )99073012436099073.0*2

21 .s.s(s)Tnorm ++

= (8)

( )55772.033976.055772.0*2)( 22 ++

=ss

sTnorm (9)

and

( )12471.046413.012471.0*2)( 23 ++

=ss

sTnorm (10)

To denormalize these three transfer functions, we replace s by 0ω

ss = in Equations (8), (9),

and (10). Consequently, we obtain the following denormalized transfer functions as:

( )832

9

1 10*192.610*11310*24.1++

=s.s

(s)T (11)

( )832

9

2 10*49.310*49.810*97.6)(++

=ss

sT (12)

and

( )742

8

3 10*79.710*16.110*56.1)(++

=ss

sT (13)

For the denormalized transfer function , the cutoff frequency )(1 sT 1,0ω and the quality

factor are given as follows: 1Q

srad /10*48.2 41,0 =ω (14)

and

00.81 =Q (15)

In the same way, the cutoff frequency 2,0ω and the quality factor corresponding to the 2Q

transfer function are given by: )(2 sT

srad /10*87.1 42,0 =ω (16)

and

20.22 =Q (17)

For the transfer function , the cutoff frequency )(3 sT 3,0ω and the quality factor are given

by:

3Q

srad /10*83.8 33,0 =ω (18)

and

76.03 =Q (19)

4. Notice that

321 QQQ >> (20)

According to Rule 1, we arrange these sections in the increasing order of the quality factor. The

first section is used to synthesize . and are synthesized in the second

section and the third section, respectively.

)(3 sT )(2 sT )(1 sT

5. Synthesize , , and using the CGIC low-pass Biquad as Fig.1. For all

three sections, choose the capacitor value as

)(1 sT )(2 sT )(3 sT

uFC 01.0= .

For the first section,

Ω=== kC

R 02.4883.24*01.0

11

1,0ω (21)

and

Ω=== kkQRR 16.328*02.4* 13 (22)

For the second section,

Ω=== kC

R 36.567.18*01.0

11

2,0ω (23)

and

Ω=== kkRQR 79.112.2*36.523 (24)

Similarly, the following resistance values are chosen for the third section.

Ω=== kC

R 33.11829.8*01.0

11

3,0ω (25)

and

Ω=== kkRQR p 61.876.0*33.11*1,3 (26)

6. Cascade these three filter sections into a six-order Chebyshev low-pass filter. In this application, the output of each section is the Op Amp output. Thus, the output resistance is assumed to be zero. Consequently, there is no need to insert buffer here. The final schematic circuit of this high-order filter is shown in Fig. 2.

III. Design Design example: the six-order Butterworth low-pass filter. The normalized transfer function is given as:

( )( )( )193186.1141421.1151764.01)( 222 ++++++

=ssssss

sT (27)

The cutoff frequency of this filter is given as .20 kHzf =

Following the above design procedure, use the cascade method to design this filter, and the low-pass CGIC Biquad as the build-up circuit block. In these Biquads, choose the capacitor value , and compute the appropriate resistor values. uFC 01.0= IV. Computer Simulations 1. Simulate the above design filter with the calculated resistance values.

2. Plot the magnitude and phase responses, for the frequency range 10Hz-10kHz.

V. Experiments 1. Build the six-order Butterworth filter designed in part III, using LF 351 Op Amps with a split

power supply voltage of ±15V.


3. Measure the magnitude response of this high-order filter. The frequency range is from 10Hz to 10kHz.

VI. Lab Report In the report, present the experiment results and compare them with the simulation results. Comment on deviations from expected results, if any, and the reasons for these deviations. Your report should include the following: 1. The complete circuit design processes. 2. The computer simulation results: the magnitude and phase responses for this high-order filter. 3. The experiment results: the magnitude responses for the high-order filter and the recorded graphs. 4. Summary and conclusions. References [1]. M. E. Van Valkenburg, Analog Filter Design, Oxford University Press, 1982.

[2]. Wasfy B. Mikhael, “Biquad II: The Current Generalized Immittance (CGIC) Stucture,” Chapter 9, in RC Active Filter Design Handbook, Edited by F. W. Stephenson, John Wiley & Sons, Inc., 1985.

[3]. Bert D. Nelin, “Design of High-Order Active Filters,” Chapter 10, in RC Active Filter Design Handbook, Edited by F. W. Stephenson, John Wiley & Sons, Inc., 1985.

[4]. Wasfy B. Mikhael, “The Current Generalized Immittance (CGIC Biquad),” Chapter 82, in Circuits and Filter Design Handbook, Edited by F. W. Stephenson, CRC Press, 2003, pp 2495-2514.

+

-Ωk33.11

)(sVi)(sVo

uF01.0+

-

+

-

+

-

+

-

+

-

uF01.0

uF01.0

uF01.0

uF01.0

uF01.0

Ωk33.11

Ωk61.8

Ωk33.11

Ωk33.11

Ωk36.5

Ωk36.5

Ωk36.5

Ωk36.5

Ωk79.11

Ωk02.4

Ωk02.4

Ωk02.4

Ωk02.4

Ωk16.32

Fig.2 The six-order Chebyshev low-pass filter

Study Guide C


Biquads II: The current Generalized Immittance (CGIC) Structure


Butterworth Filter Approximation I. Objective

To study Butterworth approximations of low-pass, band-pass, and high-pass filters.

II. Introduction The magnitude response of the low-pass Butterworth filter is expressed as:

( )( ) n

jT2

01

1

ωωω

+= (1)

Where is the filter order and n 0ω is the cutoff frequency.

From Equation (1), poles of low-pass Butterworth filters can be derived.

Poles of Low-Pass Butterworth Filters

From Equation (1), the following equation can be derived as:

( )( )

( )n

n

n

s

jssTsT

2

0

20

11

1

11)(

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

=

+=−

ω

ω (2)

The poles of equation (2) are the roots of its denominator.

( ) 0112

0

=⎟⎟⎠

⎞⎜⎜⎝

⎛−+

nn sω

(3)

The poles of Equation (2) are given by:

( ) 12,...,1,0 ,

even isn ,

odd isn ,

212

0

22

0 −=

⎪⎪⎩

⎪⎪⎨

⎧

=+

nk

e

es

nkj

nkj

π

π

ω

ω (4)

As well known, poles in the right half-plane correspond to an unstable system. Thus, poles

in the left half-plane are selected to associate with . The poles of is given by: )(sT )(sT

( )⎪⎪

⎩

⎪⎪

⎨

⎧

++=

−+++=

=+

1)-(n2n1,...,

2n,

2nk even, isn ,

2)12(n,...,

23n,

21nk odd, isn ,

212

0

22

0

nkj

nkj

e

nes

π

π

ω

ω (5)

From Equation (5), it is can be seen that the poles of the low-pass Butterworth filter are

located on the circle with the radius 0ω , and are separated by nn πφ = . If n is odd, there is a

pole on the real axis. If n is even, there are poles at 2nφπφ ±= . As examples, the pole

locations of the 4th and 5th order low-pass Butterworth filters are shown in Fig. 1 and Fig.2, respectively.

nφ2nφ

o45

4

=

=

n

n

φ

ωj

σ

0ωj

0ωj−

Fig. 1 The pole locations of the 4th order low-pass Butterworth filter

nφ

o36

5

=

=

n

n

φ

nφ

ωj

0ωj

0ωj−

σ

Fig. 2 The pole locations of the 5th order low-pass Butterworth filter

Low-Pass, High-Pass, and Band-Pass Filter Specifications

Filter specifications are usually given in term of attenuation characteristics. The

attenuation )(ωα is defined as:

dBjT )(log20)( ωωα −= (6)

The specifications for low-pass, high-pass, and band-pass filters are shown in Fig. 3, Fig. 4, and Fig. 5, respectively.

1. Low-pass filter specifications: the attenuation at the pass-band (from 0=ω to pωω = )

should be smaller the maxα ; the attenuation at the stop-band (from sωω = to ∞=ω ) should

be larger than the minα .

2. High-pass filter specifications: the attenuation at the stop-band (from 0=ω to sωω = )

should be larger than the minα ; the attenuation at the pass-band (from pωω = to ∞=ω )

should be smaller than the maxα .

3. Band-pass filter specifications: the attenuation at two stop-bands (from 0=ω to

1,sωω = , and from 2,sωω = to ∞=ω ) should be larger than minα ; the attenuation at the

pass-band (from 1,pωω = to 2,pωω = ) should be smaller than maxα .

α

pass-band stop-band

pωsω

maxα

minα

ω

Fig.3 Low-pass filter specifications

α

pass-bandstop-band

pωsω

maxα

minα

ω

Fig.4 High-pass filter specifications

α

pass-bandstop-band

1,pω1,sω

maxα

minα

ω

stop-band

minα

2,pω 2,sω

Fig.5 Band-pass filter specifications

Design Procedure for Low-Pass Butterworth Filter Approximation

1. The low-pass filter specifications are given by dB20min =α , dB5.0max =α ,

sradp /1000=ω , and srads /2000=ω .

2. Decide the filter order as: n

( )[ ]( ) 4.8321

log2110110log 1010 maxmin

=−−

=ps

nωω

αα

(7)

Round up the next integer. n5=n (8)

3. Compute the cutoff frequency 0ω . First, calculate the stop-band cutoff frequency as:

[ ]srad.

n

s /21263

110 21

101,0

max

=

−

=α

ωω (9)

Then, calculate the pass-band cutoff frequency as:

[ ]srad.

n

p /11234

110 21

102,0

min

=

−

=α

ωω (10)

The actual cutoff frequency is the geometric average of the pass-band and stop-band cutoff frequencies as:

srad /1248.62,01,00 == ωωω (11)

Using this way, the excess attenuations are achieved at the frequency sω and pω .

4. Obtain the normalized low-pass Butterworth transfer function, according to the filter order . n

( )

162.1

162.0

1

1)(22 ⎟

⎠⎞

⎜⎝⎛ ++⎟⎠⎞

⎜⎝⎛ +++

=sssss

sT (12)

5. Denormalize the transfer function by substituting in Equation (12) withs 0ωs . The

transfer function satisfying the given specifications is given by:

( )( )( )6262

15

020

2

020

2

0

10*1.56770.7 10*1.562013.81248.610*5.6566

162.1

162.0

1

1)(

+++++=

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟

⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=

sssss

ssssssT

ωωωωω (13)

Design Procedure for High-Pass Butterworth Filter Approximation

1. The high-pass filter specifications are given by dB25min =α , dB1max =α ,

sradp /75.43=ω , and srads /50.12=ω .

2. Change the high-pass specifications to the corresponding low-pass specifications. The pass and stop frequencies of the corresponding low-pass filter are given by:

rad/s.s

lowp 022901_ ==

ωω (14)

and

sradp

lows /08.01_ ==

ωω (15)

3. Design the corresponding low-pass Butterworth filter. Its specifications are as follows: the

attenuation should be at most dB1max =α at the frequency sradlowp /0229.0_ =ω , and the

attenuation should be at least dB25min =α at the frequency sradlows /08.0_ =ω .

Then, the order of the corresponding low-pass filter is 3=n , and the cutoff frequency is

. The denomalized corresponding low-pass filter transfer function is given

by

rad/sω 0.02960 =

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟

⎠⎞

⎜⎝⎛ +

=1

0.02960.02961

0.0296

1)(

2

2mod_ ssssT ellow (16)

4. Change the corresponding low-pass transfer function into the high-pass transfer function by

substituting in Equation (16) withss1 .

( )( )1141.333.7833.78

1*0.0296

1*0.0296

11*0.0296

11)(

2

3

22

+++=

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟

⎠⎞

⎜⎝⎛ +

=

ssss

sss

sT

(17)

Design Procedure for Band-Pass Butterworth Filter Approximation

1. The band-pass filter specifications: dB25min =α , dB5.0max =α , sradp /5001, =ω ,

sradp /10002, =ω , srads /2501, =ω , and srads /20002, =ω . The center frequency cω is

given by:

rad/s.ss

ppc

1707

2,1,

2,1,

=

=

=

ωω

ωωω

(18)

2. Change the band-pass specifications to the corresponding low-pass specifications. The pass and stop frequencies of the corresponding low-pass filter are given by:

sradpplowp /5001,2,_ =−= ωωω (19)

and

sradsslows /17501,2,_ =−= ωωω (20)

3. Design the corresponding low-pass Butterworth filter. Its specifications are as follows: the

attenuation must be at most dB5.0max =α at frequency sradlowp /500_ =ω , and the

attenuation must be at least dB25min =α at the frequency sradlows /1750_ =ω

The order of the corresponding low-pass filter is 4=n and the cutoff frequency

is . The denomalized corresponding low-pass filter transfer function is given

by:

rad/sω 744.60 =

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟

⎟⎠

⎞⎜⎜⎝

⎛++

=

131.1*744.6744.6

154.0*744.6744.6

1

2

2

2

2mod_ssss

T ellow (21)

4. Change the corresponding low-pass filter model into the band-pass filter by substituting

in Equation (21) with

s

ss c

22 ω+.

( )

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛ ++⎟

⎟⎠

⎞⎜⎜⎝

⎛ +

×

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛ ++⎟

⎟⎠

⎞⎜⎜⎝

⎛ +=

131.16744170767441707

1

15406744170767441707

1)(

222

222

222

222

*.s.s.

s.s

.*.s.s.

s.s

sT

(22)

III. Design 1. Find the poles of the normalized 9th and 10th order low-pass Butterworth transfer function

( srad /10 =ω ). Compare your results with the poles given in the textbook.

2. Design a low-pass Butterworth transfer function. The specifications are given

by dB20min =α , dB1max =α , sradp /1000=ω , and srads /1725=ω .

3. Design a high-pass Butterworth transfer function. The specifications are given

by dB55min =α , dB5.0max =α , sradp /1500=ω , and srads /300=ω .

4. Design a pass-pass Butterworth transfer function. The specifications are given

by dB22min =α , dB5.0max =α , sradp /5001, =ω , sradp /10002, =ω , srads /3331, =ω ,

and srads /15002, =ω .

IV. Computer Simulations 1. For three designed Butterworth filters, plot the magnitude and phase responses according to their transfer functions (using MATLAB, MATHCAD, or other languages).

2. For the low-pass filter, record the attenuation values at the frequencies sradp /1000=ω ,

and srads /1725=ω .

3. For the high-pass filter, record the attenuation values at the frequencies sradp /1500=ω ,

and srads /300=ω .

4. For the band-pass filter, record the attenuation value at the frequencies sradp /5001, =ω ,

sradp /10002, =ω , srads /3331, =ω , and srads /15002, =ω .

V. Experiments This lab is a computer simulation lab. No actual experiment.

VI. Lab Report In the report, present the simulation results. Comment on deviations from expected results, if any, and the reasons for these deviations. Your report should include the following: 1. The design steps. 2. The computer simulation results: the magnitude and phase responses for all designed Butterworth filters.

3. Compare the specification attenuations dB20min =α and dB1max =α of the low-pass

Butterworth filter with the corresponding attenuations of the approximated transfer function. To see if the designed transfer function satisfies the requirements.

4. Compare the specification attenuations dB55min =α and dB5.0max =α of the high-pass

Butterworth filter with the corresponding attenuations of the approximated transfer function. To see if the designed transfer function satisfies the requirements.

5. Compare the specification attenuations dB22min =α , and dB5.0max =α of the band-pass

Butterworth filter with the corresponding attenuations of the approximated transfer function.

To see if the designed transfer function satisfies the requirements. 6. Summary and conclusions. References [1]. M. E. Van Valkenburg, Analog Filter Design, Oxford University Press, 1982.

APPENDIX

How to read Resistor Color Codes First the code

Black Brown Red Orange Yellow Green Blue Violet Gray White

0 1 2 3 4 5 6 7 8 9

How to read the code

First find the tolerance band, it will typically be gold (5%) and sometimes silver (10%). Starting from the other end, identify the first band - write down the number associated with that color; in this case Blue is 6. Now 'read' the next color, here it is red so write down a '2' next to the six. (you should have '62' so far.) Now read the third or 'multiplier' band and write down that number of zeros. In this example it is two so we get '6200' or '6,200'. If the 'multiplier' band is Black (for zero) don't write any zeros down. If the 'multiplier' band is Gold move the decimal point one to the left. If the 'multiplier' band is Silver move the decimal point two places to the left. If the resistor has one more band past the tolerance band it is a quality band. Read the number as the '% Failure rate per 1000 hour' This is rated assuming full wattage being applied to the resistors. (To get better failure rates, resistors are typically specified to have twice the needed wattage dissipation that the circuit produces) 1% resistors have three

bands to read digits to the left of the multiplier. They have a different temperature coefficient in order to provide the 1% tolerance. Examples Example 1: You are given a resistor whose stripes are colored from left to right as brown, black, orange, gold. Find the resistance value. Step One: The gold stripe is on the right so go to Step Two. Step Two: The first stripe is brown which has a value of 1. The second stripe is black which has a value of 0. Therefore the first two digits of the resistance value are 10. Step Three: The third stripe is orange which means x 1,000. Step Four: The value of the resistance is found as 10 x 1000 = 10,000 ohms (10 kilohms = 10 kohms). The gold stripe means the actual value of the resistor mar vary by 5% meaning the actual value will be somewhere between 9,500 ohms and 10,500 ohms. (Since 5% of 10,000 = 0.05 x 10,000 = 500) Example 2: You are given a resistor whose stripes are colored from left to right as orange, orange, brown, silver. Find the resistance value. Step One: The silver stripe is on the right so go to Step Two. Step Two: The first stripe is orange which has a value of 3. The second stripe is orange which has a value of 3. Therefore the first two digits of the resistance value are 33. Step Three: The third stripe is brown which means x 10. Step Four: The value of the resistance is found as 33 x 10 = 330 ohms. The silver stripe means the actual value of the resistor mar vary by 10% meaning the actual value will be between 297 ohms and 363 ohms. (Since 10% of 330 = 0.10 x 330 = 33) Example 3: You are given a resistor whose stripes are colored from left to right as blue, gray, red, gold. Find the resistance value. Step One: The gold stripe is on the right so go to Step Two. Step Two: The first stripe is blue which has a value of 6. The second stripe is gray which has a value of 8. Therefore the first two digits of the resistance value are 68. Step Three: The third stripe is red which means x 100. Step Four: The value of the resistance is found as 68 x 100 = 6800 ohms (6.8 kilohms = 6.8 kohms). The gold stripe means the actual value of the resistor mar vary by 5% meaning the actual value will be somewhere between 6,460 ohms and 7,140 ohms. (Since 5% of 6,800 = 0.05 x 6,800 = 340)

2

Example 4: You are given a resistor whose stripes are colored from left to right as green, brown, black, gold. Find the resistance value. Step One: The gold stripe is on the right so go to Step Two. Step Two: The first stripe is green which has a value of 5. The second stripe is brown which has a value of 1. Therefore the first two digits of the resistance value are 51. Step Three: The third stripe is black which means x 1. Step Four: The value of the resistance is found as 51 x 1 = 51 ohms. The gold stripe means the actual value of the resistor mar vary by 5% meaning the actual value will be somewhere between 48.45 ohms and 53.55 ohms. (Since 5% of 51 = 0.05 x 51 = 2.5

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eel 4140 lab manual

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