MAE140 Linear Circuits152

Designing Circuits – Synthesis - Lego

Port = a pair of terminals to a cctOne-port cct; measure I(s) and V(s) at same port

Driving point impedance or input impedance Z(s)

Two-portsTransfer function; measure input at one port, output at

another

I(s)

V(s)+

-sCR

sLsI

sVsZ

++==

!1

1

)(

)()(

I1(s)

V1(s)+

-V2(s)

I2(s)+

-

InputsOutputs

MAE140 Linear Circuits153

Cascade Connections

We want to apply a chain rule of processing

When can we do this by cascade connection of OpAmp ccts?Cascade means output of ccti is input of ccti+1

This makes the design and analysis much easier

This rule works if stage i+1 does not load stage iVoltage is not changed because of next stage

EitherOutput impedance of source stage is zero

OrInput impedance of load stage is infinite

Works well if Zout,source<<Zin,load

)(...)()()()( 321 sTsTsTsTsT VkVVVV !!!!=

MAE140 Linear Circuits154

Cascade Connections

Look for chain rule+_ R1V1(s)R2

2

1

sC

1

1

sC

V2(s) V3(s)

+++

Mesh analysis( ) ( ) 11

1

1)()()(

22112

2121

11

2211

1121

?

total+++

=+

!+

=!=sCRCRsCCRR

sCR

sCRsCR

sCRsTsTsT VVV

I2(s)I1(s)

0)(1

)(

)()()(1

22

2111

121111

=!!"

#$$%

&+++'

='!!"

#$$%

&+

sIsC

RRsIR

sVsIRsIRsC

!"

#$%

&

!!!!

"

#

$$$$

%

&

++'

'+

=!!"

#$$%

&

'

0

)(

1

1

)(

)( 1

1

212

1

111

2

1 sV

RRsC

R

RRsC

sI

sI

( ) ( )

( ) ( ))(

1)(

1)(

)(1

)(

1222111

22121

112

23

1221211

22121

1212

2

sV

sCRCRCRsCCRR

sCRsI

sCsV

sV

sRCRCRCsCCRR

RCCssI

++++==

++++=

MAE140 Linear Circuits155

Cascade Connections – OpAmp ccts

OpAmps can be used to achieve the chain ruleproperty for cascade connectionsThe input to the next stage needs to be driven by the

OpAmp outputConsider standard configurations

Noninverting amplifierNo current drawn from V1 – no load

Inverting amplifierCurrent provided by V1(s)

Need to make sure that stage isdriven by OpAmp output toavoid loading V1(s)

V1(s)+-

V2(s)+

Z2+

Z1

V1(s) +-

Z1 Z2

V2(s)

++I(s)

)(

)()(

1

1

sZ

sVsI =

MAE140 Linear Circuits156

OpAmp Ccts and transfer functionsNode B:

Node B:

+_V1(s) +-

Z1 Z2

V2(s)

+A B C

)(

)(

)(

)()(0)(

0)(

)()(

)(

)()(

1

2

1

2

2

2

1

1

sZ

sZ

sV

sVsTsV

sZ

sVsV

sZ

sVsV

VB

BB

!=="=

=!

+!

+_V1(s)

+-V2(s)+A

B

C

Z2

Z1

)(

)()()()()(

0)(

)(

)(

)()(

1

211

12

2

sZ

sZsZsTsVsV

sZ

sV

sZ

sVsV

VB

BB

+=!=

=+"

MAE140 Linear Circuits157

Example 11-4 T&R p511

Find the transfer function from V1(s) to V2(s)

111

11

11)(sCCsR

sCRsZ1

+=+=

2211 )1)(1()( 12

++−=

CsRCsRCsRsTV

R2

R1

V1(s)

+-

1

1

sC

2

1

sCV2(s)

++

11)(222

22

22

2 +=

+=

sCRR

sCRsC

RsZ

MAE140 Linear Circuits158

Circuits as Signal Processors

Design a circuit with transfer function

R1=R2=100Ω, C1=C2=1µF, C3=100µF, L=70mH, R3=1Ω

( )( )

( )24842

52

10

60260210102

1042.1

+

−+=

+×+

×+

s

jsjsss

s ππ

VO(s)

R2

R1

VI(s)

+-1

1

sC

2

1

sCV2(s)

+

+-+

3

1

sC

R3

sL

( )( )( )LssLCR

CsRCsRCsRsTV

111

)(2

332211

12 +−×

++−

=

MAE140 Linear Circuits159

Transfer Function Design – OpAmp Stages

First order stages

α

+-

1 Κγ Κ

αγ

++

−=ssKsTV )(

Series RL design

Series RC design

1

+-

Κ !K

1

!

1

MAE140 Linear Circuits160

First-order stages

αγ

++

−=ssKsTV )(

1

+-

!K

1

!

1

K

1

Parallel RL design

1

+-

!K

1

!

1

Κ

Parallel RC design

MAE140 Linear Circuits161

Design Example 11-20 T&R p 542

Design two ccts to realize

Unrealistic component values – scaling needed

)4000)(1000(3000)( ++

=ssssTV

1000µF

+-

1Ω1Ω

+-250µF

1Ω 3Ω

Stage 1 Stage 2

[ ]1000

10001

10

11

10

1

)( 1

3

3

1+

!=

"""

#

$

%%%

&

'

+!= !

!

!

s

s

ssTV [ ][ ]

4000

3400013)(1

2+

!=+!=

!

s

s

ssTV

MAE140 Linear Circuits162

Design Example 11-19 T&R p 539

Non-inverting amplifier design

Less OpAmps but more difficult designThree stage: last stage not driven

Unrealistic component values still – scaling needed

1Ω+-

1000µF

1Ω2Ω

250µF

1Ω

Stage 1VoltageDivider

Stage 2OpAmp

Stage 3VoltageDivider

)4000)(1000(3000)(

++=

ssssTV

MAE140 Linear Circuits163

Scaled Design Example 11-21 T&R p 544

More realistic values for components

Need to play games with elements to scaleThe ratio formulas for TV help permit this scaling

It certainly is possible to demand a design TV which isunrealizable with sensible component values

Like a pole at 10-3 Hz

100nF

+-

10ΚΩ10ΚΩ

+-25nF

10ΚΩ 30ΚΩ

MAE140 Linear Circuits164

Second-order Stage Design

Circuit stages to yield 20

20

2)(

ωζω ++=

ssKsTV

02!"=R 1=LK

C

!=

20

1

1"

KC

1

2=2

0!"K

+-

12

0

!"

K

1Ω Ω

1Η !02"#

F20

1

! +-

!02

1

"#

H2

0

1

!

F1HK

1

MAE140 Linear Circuits165

Circuit Synthesis

Given a stable transfer function TV(s), realize it via acct using first-order and second-order stages

We are limited to stable transfer functions to keep within thelinear range of the OpAmpsThere is an exception

When the unstable TV(s) is part of a stable feedbacksystem

Come to MAE143B to find out

Transistor cct design is conceptually similar

cbsas

sssTV

++

++=

2

2

)(!"#

bas

ssTV

+

+=

!")(