### Chapter 14 – Frequency Response

Chapter 14
EGR 272 – Circuit Theory II
Read: Ch. 14, Sect. 1-5 in Electric Circuits, 9th Edition by Nilsson
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
1
Chapter 14
EGR 272 – Circuit Theory II
2
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)
Chapter 14
EGR 272 – Circuit Theory II
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
3
Chapter 14
EGR 272 – Circuit Theory II
4
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
LM
Ideal LPF
Ideal HPF
w
w
wC
LM
wC
LM
Ideal BPF
wC1
wC2
Ideal BSF
w
wC1
wC2
w
Chapter 14
EGR 272 – Circuit Theory II
5
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
Ideal LPF
K
H (s) =
(s + w C )
4th-order LPF

K
H (s) =
(s + w C )
3rd-order LPF
H (s) =
4
K
(s + w C )
2nd-order LPF
3
K
H (s) =
(s + w C )
1st-order LPF
H (s) =
2
K
(s + w C )
wC
w
Chapter 14
EGR 272 – Circuit Theory II
6
Defining frequency response
Y (s)
Recall that a transfer function H(s) is defined as: 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:
H(jw)  H(s) s  jw
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 )
Chapter 14
EGR 272 – Circuit Theory II
7
Example: Find H(jw) for H(s) below. Also write H(jw) in polar form.
H (s) 
4s
(s + 10)(s + 20)
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 m agn itu de respon se or the am plitu de respon se

20log( H (jw ) ) vs w - referred to as the log - m agn itu de (L M ) respon se

 (w ) vs w - referred to as the ph ase re spon se
Chapter 14
EGR 272 – Circuit Theory II
8
Example:
A) Find H(s) = Vo(s)/Vi(s)
B) Find H(jw)
+
Vi (s)
_
R
1
sC
+
Vo(s)
_
Chapter 14
EGR 272 – Circuit Theory II
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?
9
Chapter 14
EGR 272 – Circuit Theory II
10
Example:
A) Find H(s) = V(s)/I(s)
B) Find H(jw)
I(s)
R
sL
1
sC
+
V(s)
_
Chapter 14
EGR 272 – Circuit Theory II
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?
11
Chapter 14
EGR 272 – Circuit Theory II
12
Example:
A) Find H(s) = Vo(s)/Vi(s)
B) Find H(jw)
+
Vi (s)
_
+
2kW
3kW
10mH
Vo(s)
_
Chapter 14
EGR 272 – Circuit Theory II
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?
13
Chapter 14
EGR 272 – Circuit Theory II
14
General 2nd Order Transfer Function
For 2nd order circuits, the denominator of any transfer function will take on the
following form: s2 + 2s + wo2
Various types of 2nd order filters can be formed using a second order circuit,
including:
K1
H (s) 
s
2
 2 s  w o
K 2s
H (s) 
s
2
 2 s  w o
K 3s
H (s) 
s
2
2
2
(2nd-order low -pass filter (LPF))
(2nd-order band-pass filter (BPF))
2
 2 s  w o
2
(2nd-order high-pass filter (H P F))
Chapter 14
EGR 272 – Circuit Theory II
15
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 + 2s + wo2). Also show
R
1
that:
 
2L
and w o 
for a series R LC circuit
LC
Chapter 14
EGR 272 – Circuit Theory II
16
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 + 2s + wo2). Also show
that:
1
1
 
2R C
and w o 
for a parallel R LC circu it
LC
Chapter 14
EGR 272 – Circuit Theory II
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:
Ks
H (s) 
s
2
 2 s  w o
2
(2nd-order band-pass filter (BPF))
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)
17
Chapter 14
EGR 272 – Circuit Theory II
18
Determining Hmax
Find H(jw) and then H(jw).
Show that
H m ax  H (jw ) w = w

o
H (jw o )

K
2
Chapter 14
EGR 272 – Circuit Theory II
19
Determining wc1 and wc2 :
Show that
H (jw c )

H m ax
2

K
2 2
w c1  - 
w c2   



 wo
2

2
2
 wo
2
Chapter 14
EGR 272 – Circuit Theory II
Determining wo, BW, and Q:
20
wo

w c1  w c2
Show that wo is the geometric mean of the cutoff
B W  w c2 - w c1  2 
frequencies, not the arithmetic mean. Also find
wo
wo
BW and Q. Specifically, show that:
Q 

BW
2
Damping ratio is simply defined here. Its
significance will be seen later in this
D efine  (zeta)  dam ping ratio course and in other courses (such as

1
Control Theory). Circuits with similar
 

values of  have similar types of responses.
wo
2Q
Chapter 14
EGR 272 – Circuit Theory II
Example: A parallel RLC circuit has L = 100 mH, and C = 0.1 uF
1) Find wo ,  , Hmax , wc1 , wc2 , Hmax , BW, Q, and 
2) Show that wo is the geometric mean of the wc1 and wc2 , not the arithmetic
mean.
A) Use R = 1 kW
21
Chapter 14
EGR 272 – Circuit Theory II
Example: A parallel RLC circuit has L = 100 mH, and C = 0.1 uF
1) Find wo ,  , Hmax , wc1 , wc2 , Hmax , BW, Q, and 
2) Show that wo is the geometric mean of the wc1 and wc2 , not the arithmetic
mean.
B) Use R = 20 kW
22
EGR 272 – Circuit Theory II
Chapter 14
23
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.)
|H(jw)|
5k
6k
7k
8k
9k 10k
10000
20k
w (log scale)