frequency response this is an extremely important topic in ee. up until this point we have analyzed...

Frequency Response

This is an extremely important topic in EE. Up until this point we have analyzed circuits without considering the effect on the answer over a wide range of frequencies. Many circuits have frequency limitations that are very important.

Example: Discuss the frequency limitations on the following items.

1) An audio amplifier

2) An op amp circuit

Read: Ch. 14, Sect. 1-5 in Electric Circuits, 9th Edition by Nilsson

1Chapter 14 EGR 272 – Circuit Theory II

Example: Discuss the frequency limitations on the following items (continued)

3) A voltmeter (% error vs frequency)

4) The tuner on a radio (band-pass filter)


Filters

A filter is a circuit designed to have a particular frequency response, perhaps to alter the frequency characteristics of some signal. It is often used to filter out, or block, frequencies in certain ranges, much like a mechanical filter might be used to filter out sediment in a water line.

Basic Filter Types

• Low-pass filter (LPF) - passes frequencies below some cutoff frequency, wC

• High-pass filter (HPF) - passes frequencies above some cutoff frequency, wC

• Band-pass filter (BPF) - passes frequencies between two cutoff frequency, wC1 and wC2

• Band-stop filter (BSF) or band-reject filter (BRF) - blocks frequencies between two cutoff frequency, wC1 and wC2


Ideal filters

An ideal filter will completely block signals with certain frequencies and pass (with no attenuation) other frequencies. (To attenuate a signal means to decrease the signal strength. Attenuation is the opposite of amplification.)

LM

wwC

Ideal LPF

LM

w

wC

Ideal HPF

LM

wwC1

Ideal BPF

wC2

LM

wwC1

Ideal BSF

wC2


Filter order

Unfortunately, we can’t build ideal filters. However, the higher the order of a filter, the more closely it will approximate an ideal filter.

The order of a filter is equal to the degree of the denominator of H(s).

(Of course, H(s) must also have the correct form.)

LM

wwC

Ideal LPF

3rd-order LPF

2nd-order LPF

1st-order LPF

4th-order LPF

C

KH(s) =

(s + w )

2C

KH(s) =

(s + w )

3C

KH(s) =

(s + w )

4C

KH(s) =

(s + w )

C

KH(s) =

(s + w )


Defining frequency response

Recall that a transfer function H(s) is defined as:Y(s)

H(s) X(s)

Where Y(s) = some specified output and X(s) = some specified input

In general, s = + jw. For frequency applications we use s = jw (so = 0).

So now we define:s jw

H(jw) H(s)

Since H(jw) can be thought of as a complex number that is a function of frequency, it can be placed into polar form as follows:

H(jw) H(jw) ( )w


When we use the term "frequency response", we are generally referring to information that is conveyed using the following graphs:

H(jw) vs w - referred to as the or the

20log( H(jw) ) vs w - referred to as the

(w) vs w - referred to as the

magnitude response amplitude response

log - magnitude (LM) response

phase re sponse


Example: Find H(jw) for H(s) below. Also write H(jw) in polar form.

4sH(s)

(s + 10)(s + 20)

Example:

A) Find H(s) = Vo(s)/Vi(s)

B) Find H(jw)Vo(s)

+

_ Vi(s)

+

_

R 1

sC


Example: (continued)

C) Sketch the magnitude response, |H(jw)| versus w

D) Sketch the phase response, (w) versus w

E) The circuit represents what type of filter?


Example:

A) Find H(s) = V(s)/I(s)

B) Find H(jw) V(s)

+

_

1 sC

I(s) R sL


Example:

A) Find H(s) = Vo(s)/Vi(s)

B) Find H(jw)Vo(s)

+

_ Vi(s)

+

_

2k

10mH 3k


General 2nd Order Transfer Function

For 2nd order circuits, the denominator of any transfer function will take on the

following form: s2 + 2s + wo2

Various types of 2nd order filters can be formed using a second order circuit, including:

122

o

KH(s) (2nd-order low-pass filter (LPF))

s 2 s w

222

o

K sH(s) (2nd-order band-pass filter (BPF))

s 2 s w

2

322

o

K sH(s) (2nd-order high-pass filter (HPF))

s 2 s w


Series RLC Circuit (2nd Order Circuit)

Draw a series RLC circuit and find transfer functions for LPF, BPF, and HPF.

Note that the denominator is the same in each case (s2 + 2s + wo2). Also show

that:o

R 1 and w for a series RLC circuit

2L LC


Parallel RLC Circuit (2nd Order Circuit)

Draw a parallel RLC circuit and find transfer functions for LPF, BPF, and HPF.

Note that the denominator is the same in each case (s2 + 2s + wo2). Also show

that:o

1 1 and w for a parallel RLC circuit

2RC LC


2nd Order Bandpass Filter

A 2nd order BPF will now be examined in more detail. The transfer function,

H(s), will have the following form:

22o

KsH(s) (2nd-order band-pass filter (BPF))

s 2 s w

Magnitude response• Show a general sketch of the magnitude response for H(s) above• Define wo , wc1 , wc2 , Hmax , BW, and Q

• Sketch the magnitude response for various values of Q (in general)


Determining Hmax

Find H(jw) and then H(jw).

Show that o

max ow = w

KH H(jw) H(jw )

2


Determining wc1 and wc2 :

maxc

H KH(jw )

2 2 2 leads to

22c1 o

22c2 o

w - w

w w

Show that


Determining wo, BW, and Q:

Show that wo is the geometric mean of the cutoff

frequencies, not the arithmetic mean. Also find

BW and Q. Specifically, show that:

o c1 c2

c2 c1

o o

w w w

BW w - w 2

w wQ

BW 2

o

Define (zeta) damping ratio

1

w 2Q

Damping ratio is simply defined here. Its significance will be seen later in this course and in other courses (such as Control Theory). Circuits with similar values of z have similar types of responses.


Example: A parallel RLC circuit has L = 100 mH, and C = 0.1 uF

1) Find wo , , Hmax , wc1 , wc2 , Hmax , BW, Q, and z

2) Show that wo is the geometric mean of the wc1 and wc2 , not the arithmetic

mean.

A) Use R = 1 kW


Example: A parallel RLC circuit has L = 100 mH, and C = 0.1 uF

1) Find wo , , Hmax , wc1 , wc2 , Hmax , BW, Q, and z

2) Show that wo is the geometric mean of the wc1 and wc2 , not the arithmetic

mean.

B) Use R = 20 kW


Example: Plot the magnitude response, |H(jw)|, for parts A and B in the last

example. (Note that a curve with a geometric mean will appear symmetrical on

a log scale and a curve with an arithmetic mean will appear symmetrical on a

linear scale.)

0

2

4

6

8

10

1000 10000 100000

5k 6k 7k 8k 9k 10k 20kw (log scale)

|H(jw)|


frequency response this is an extremely important topic in ee. up until this point we have analyzed...

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