11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
11: Frequency Responses
Frequency Responses: 11 – 1 / 12
Frequency Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
If x(t) is a sine wave, then y(t) will also be a
sine wave but with a different amplitude and
phase shift. X is an input phasor and Y is the
output phasor.
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 2 / 12
Frequency Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
If x(t) is a sine wave, then y(t) will also be a
sine wave but with a different amplitude and
phase shift. X is an input phasor and Y is the
output phasor.
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
The gain of the circuit is
E1.1 Analysis of Circuits (2016-8284)
Y
X
=
1/jωC
R+1/jωC
=
1
jωRC+1
Frequency Responses: 11 – 2 / 12
Frequency Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
If x(t) is a sine wave, then y(t) will also be a
sine wave but with a different amplitude and
phase shift. X is an input phasor and Y is the
output phasor.
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
The gain of the circuit is
Y
X
=
1/jωC
R+1/jωC
=
1
jωRC+1
This is a complex function of ω so we plot separate graphs for:
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 2 / 12
Frequency Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
If x(t) is a sine wave, then y(t) will also be a
sine wave but with a different amplitude and
phase shift. X is an input phasor and Y is the
output phasor.
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
The gain of the circuit is
Y
X
=
1/jωC
R+1/jωC
=
1
jωRC+1
This is a complex function of ω so we plot separate graphs for:
Y Magnitude: X =
E1.1 Analysis of Circuits (2016-8284)
1
|jωRC+1|
1
1+(ωRC)2
= √
Frequency Responses: 11 – 2 / 12
Frequency Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
If x(t) is a sine wave, then y(t) will also be a
sine wave but with a different amplitude and
phase shift. X is an input phasor and Y is the
output phasor.
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
The gain of the circuit is
Y
X
1/jωC
=
R+1/jωC
=
1
jωRC+1
This is a complex function of ω so we plot separate graphs for:
Y Magnitude: X =
1
|jωRC+1|
1
1+(ωRC)2
= √
1
0.5
0
0
1
2
ω RC
3
4
5
Magnitude Response
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 2 / 12
Frequency Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
If x(t) is a sine wave, then y(t) will also be a
sine wave but with a different amplitude and
phase shift. X is an input phasor and Y is the
output phasor.
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Y
X
The gain of the circuit is
1/jωC
=
R+1/jωC
=
1
jωRC+1
This is a complex function of ω so we plot separate graphs for:
Y Magnitude: X =
Phase Shift: ∠
Y
X
1
1
|jωRC+1|
1
1+(ωRC)2
= √
= −∠ (jωRC + 1) = − arctan
ωRC
1
0.5
0
0
1
2
ω RC
3
4
5
Magnitude Response
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 2 / 12
Frequency Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
If x(t) is a sine wave, then y(t) will also be a
sine wave but with a different amplitude and
phase shift. X is an input phasor and Y is the
output phasor.
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Y
X
The gain of the circuit is
1/jωC
=
R+1/jωC
=
1
jωRC+1
This is a complex function of ω so we plot separate graphs for:
Y Magnitude: X =
Phase Shift: ∠
Y
X
1
1+(ωRC)2
= √
= −∠ (jωRC + 1) = − arctan
ωRC
1
3
4
0
π
1
1
|jωRC+1|
-0.2
0.5
-0.4
0
0
1
2
ω RC
3
4
Magnitude Response
E1.1 Analysis of Circuits (2016-8284)
5
0
1
2
ω RC
5
Phase Response
Frequency Responses: 11 – 2 / 12
Sine Wave Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
RC = 10 ms
Y
X
=
1
jωRC+1
=
1
0.01jω+1
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
ω = 50 ⇒
Y
X
= 0.89∠ − 27◦
ω = 100 ⇒
Y
X
= 0.71∠ − 45◦
ω = 300 ⇒
Y
X
= 0.32∠ − 72◦
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 3 / 12
Sine Wave Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
RC = 10 ms
Y
X
=
1
jωRC+1
=
1
0.01jω+1
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
ω = 50 ⇒
Y
X
= 0.89∠ − 27◦
ω = 100 ⇒
Y
X
= 0.71∠ − 45◦
ω = 300 ⇒
Y
X
= 0.32∠ − 72◦
1
0
Phase (°)
|Y/X|
-20
0.5
-40
-60
-80
0
0
E1.1 Analysis of Circuits (2016-8284)
100
200
300
ω (rad/s)
400
500
0
100
200
300
ω (rad/s)
400
500
Frequency Responses: 11 – 3 / 12
Sine Wave Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
RC = 10 ms
Y
X
=
1
jωRC+1
=
1
0.01jω+1
Y
ω = 50 ⇒
Y
X
X-Y
ω=50
0
• Plot Magnitude Response
• Low and High Frequency
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
-0.2
-0.4
Approximations
Asymptotes
X
0
Imag
11: Frequency Responses
0.5
Real
1
= 0.89∠ − 27◦
ω = 100 ⇒
Y
X
= 0.71∠ − 45◦
ω = 300 ⇒
Y
X
= 0.32∠ − 72◦
1
0
Phase (°)
|Y/X|
-20
0.5
-40
-60
-80
0
0
E1.1 Analysis of Circuits (2016-8284)
100
200
300
ω (rad/s)
400
500
0
100
200
300
ω (rad/s)
400
500
Frequency Responses: 11 – 3 / 12
Sine Wave Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
RC = 10 ms
Y
X
=
1
jωRC+1
=
1
0.01jω+1
Y
Y
X
= 0.89∠ − 27◦
ω = 100 ⇒
Y
X
ω = 300 ⇒
Y
X
0.5
Real
1
w = 50 rad/s, Gain = 0.89, Phase = -27°
1
◦
= 0.71∠ − 45
◦
= 0.32∠ − 72
x=blue, y=red
ω = 50 ⇒
X-Y
ω=50
0
• Plot Magnitude Response
• Low and High Frequency
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
-0.2
-0.4
Approximations
Asymptotes
X
0
Imag
11: Frequency Responses
0.5
0
y
-0.5
-1
0
1
x
20
40
60
80
time (ms)
100
120
0
Phase (°)
|Y/X|
-20
0.5
-40
-60
-80
0
0
E1.1 Analysis of Circuits (2016-8284)
100
200
300
ω (rad/s)
400
500
0
100
200
300
ω (rad/s)
400
500
Frequency Responses: 11 – 3 / 12
Sine Wave Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
RC = 10 ms
Y
X
=
1
jωRC+1
=
1
0.01jω+1
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
ω = 50 ⇒
Y
X
= 0.89∠ − 27◦
ω = 100 ⇒
Y
X
= 0.71∠ − 45◦
ω = 300 ⇒
Y
X
= 0.32∠ − 72◦
1
0
Phase (°)
|Y/X|
-20
0.5
-40
-60
-80
0
0
E1.1 Analysis of Circuits (2016-8284)
100
200
300
ω (rad/s)
400
500
0
100
200
300
ω (rad/s)
400
500
Frequency Responses: 11 – 3 / 12
Sine Wave Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
RC = 10 ms
Y
X
=
1
jωRC+1
=
1
0.01jω+1
Y
ω = 50 ⇒
Y
X
X-Y
ω=100
0
• Plot Magnitude Response
• Low and High Frequency
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
-0.2
-0.4
Approximations
Asymptotes
X
0
Imag
11: Frequency Responses
0.5
Real
1
= 0.89∠ − 27◦
ω = 100 ⇒
Y
X
= 0.71∠ − 45◦
ω = 300 ⇒
Y
X
= 0.32∠ − 72◦
1
0
Phase (°)
|Y/X|
-20
0.5
-40
-60
-80
0
0
E1.1 Analysis of Circuits (2016-8284)
100
200
300
ω (rad/s)
400
500
0
100
200
300
ω (rad/s)
400
500
Frequency Responses: 11 – 3 / 12
Sine Wave Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
RC = 10 ms
Y
X
=
1
jωRC+1
=
1
0.01jω+1
-0.2
Y
X
= 0.89∠ − 27◦
ω = 100 ⇒
Y
X
ω = 300 ⇒
Y
X
0.5
Real
1
w = 100 rad/s, Gain = 0.71, Phase = -45°
1
◦
= 0.71∠ − 45
◦
= 0.32∠ − 72
x=blue, y=red
ω = 50 ⇒
X-Y
ω=100
0
• Plot Magnitude Response
• Low and High Frequency
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Y
-0.4
Approximations
Asymptotes
X
0
Imag
11: Frequency Responses
0.5
0
y
-0.5
-1
0
1
x
20
40
60
80
time (ms)
100
120
0
Phase (°)
|Y/X|
-20
0.5
-40
-60
-80
0
0
E1.1 Analysis of Circuits (2016-8284)
100
200
300
ω (rad/s)
400
500
0
100
200
300
ω (rad/s)
400
500
Frequency Responses: 11 – 3 / 12
Sine Wave Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
RC = 10 ms
Y
X
1
jωRC+1
=
=
1
0.01jω+1
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
ω = 50 ⇒
Y
X
= 0.89∠ − 27◦
ω = 100 ⇒
Y
X
= 0.71∠ − 45◦
ω = 300 ⇒
Y
X
= 0.32∠ − 72◦
1
0
Phase (°)
|Y/X|
-20
0.5
-40
-60
-80
0
0
E1.1 Analysis of Circuits (2016-8284)
100
200
300
ω (rad/s)
400
500
0
100
200
300
ω (rad/s)
400
500
Frequency Responses: 11 – 3 / 12
Sine Wave Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
RC = 10 ms
Y
X
1
jωRC+1
=
=
1
0.01jω+1
-0.2
-0.4
Approximations
0
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
ω = 50 ⇒
Y
X
X
0
Imag
11: Frequency Responses
Y
X-Y
ω=300
0.5
Real
1
= 0.89∠ − 27◦
ω = 100 ⇒
Y
X
= 0.71∠ − 45◦
ω = 300 ⇒
Y
X
= 0.32∠ − 72◦
1
0
Phase (°)
|Y/X|
-20
0.5
-40
-60
-80
0
0
E1.1 Analysis of Circuits (2016-8284)
100
200
300
ω (rad/s)
400
500
0
100
200
300
ω (rad/s)
400
500
Frequency Responses: 11 – 3 / 12
Sine Wave Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
RC = 10 ms
Y
X
1
jωRC+1
=
=
1
0.01jω+1
-0.2
-0.4
Approximations
ω = 50 ⇒
Y
X
= 0.89∠ − 27◦
X-Y
ω=300
0.5
Real
1
w = 300 rad/s, Gain = 0.32, Phase = -72°
1
ω = 100 ⇒
Y
X
= 0.71∠ − 45◦
ω = 300 ⇒
Y
X
◦
x=blue, y=red
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Y
0
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
X
0
Imag
11: Frequency Responses
= 0.32∠ − 72
y
0
-0.5
-1
0
1
x
0.5
20
40
60
80
time (ms)
100
120
0
Phase (°)
|Y/X|
-20
0.5
-40
-60
-80
0
0
E1.1 Analysis of Circuits (2016-8284)
100
200
300
ω (rad/s)
400
500
0
100
200
300
ω (rad/s)
400
500
Frequency Responses: 11 – 3 / 12
Sine Wave Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
RC = 10 ms
Y
X
1
jωRC+1
=
=
1
0.01jω+1
-0.2
-0.4
Approximations
ω = 50 ⇒
Y
X
= 0.89∠ − 27◦
X-Y
ω=300
0.5
Real
1
w = 300 rad/s, Gain = 0.32, Phase = -72°
1
ω = 100 ⇒
Y
X
= 0.71∠ − 45◦
ω = 300 ⇒
Y
X
◦
x=blue, y=red
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Y
0
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
X
0
Imag
11: Frequency Responses
= 0.32∠ − 72
y
0
-0.5
-1
0
1
x
0.5
20
40
60
80
time (ms)
100
120
0
Phase (°)
|Y/X|
-20
0.5
-40
-60
-80
0
0
100
200
300
ω (rad/s)
400
500
0
100
200
300
ω (rad/s)
400
500
The output, y(t), lags the input, x(t), by up to 90◦ .
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 3 / 12
Logarithmic axes
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
We usually use logarithmic axes for frequency and gain (but not phase)
because % differences are more significant than absolute differences.
E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz
even though both differences equal 5 Hz.
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 4 / 12
Logarithmic axes
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
We usually use logarithmic axes for frequency and gain (but not phase)
because % differences are more significant than absolute differences.
E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz
even though both differences equal 5 Hz.
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
π
0
-0.2
-0.4
0.1
E1.1 Analysis of Circuits (2016-8284)
1
ω RC
10
Frequency Responses: 11 – 4 / 12
Logarithmic axes
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
We usually use logarithmic axes for frequency and gain (but not phase)
because % differences are more significant than absolute differences.
E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz
even though both differences equal 5 Hz.
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
0
π
0
-10
-0.2
-20
-30
-0.4
0.1
E1.1 Analysis of Circuits (2016-8284)
1
ω RC
10
0.1
1
ω RC
10
Frequency Responses: 11 – 4 / 12
Logarithmic axes
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
We usually use logarithmic axes for frequency and gain (but not phase)
because % differences are more significant than absolute differences.
E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz
even though both differences equal 5 Hz.
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
exist on a log axis and so
π
-10
-0.2
-20
-30
Note that 0 does not
0
0
the starting point of the
-0.4
0.1
E1.1 Analysis of Circuits (2016-8284)
1
ω RC
10
0.1
1
ω RC
10
axis is arbitrary.
Frequency Responses: 11 – 4 / 12
Logarithmic axes
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
We usually use logarithmic axes for frequency and gain (but not phase)
because % differences are more significant than absolute differences.
E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz
even though both differences equal 5 Hz.
|V |
Logarithmic voltage ratios are specified in decibels (dB) = 20 log10 |V2 | .
1
exist on a log axis and so
π
-10
-0.2
-20
-30
Note that 0 does not
0
0
the starting point of the
-0.4
0.1
E1.1 Analysis of Circuits (2016-8284)
1
ω RC
10
0.1
1
ω RC
10
axis is arbitrary.
Frequency Responses: 11 – 4 / 12
Logarithmic axes
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
We usually use logarithmic axes for frequency and gain (but not phase)
because % differences are more significant than absolute differences.
E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz
even though both differences equal 5 Hz.
|V |
Logarithmic voltage ratios are specified in decibels (dB) = 20 log10 |V2 | .
1
Common voltage ratios:
|V2 |
|V1 |
1
0
dB
exist on a log axis and so
π
-10
-0.2
-20
-30
Note that 0 does not
0
0
the starting point of the
-0.4
0.1
E1.1 Analysis of Circuits (2016-8284)
1
ω RC
10
0.1
1
ω RC
10
axis is arbitrary.
Frequency Responses: 11 – 4 / 12
Logarithmic axes
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
We usually use logarithmic axes for frequency and gain (but not phase)
because % differences are more significant than absolute differences.
E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz
even though both differences equal 5 Hz.
|V |
Logarithmic voltage ratios are specified in decibels (dB) = 20 log10 |V2 | .
1
Common voltage ratios:
|V2 |
|V1 |
dB
1
0
0.1
−20
exist on a log axis and so
π
-10
-0.2
-20
the starting point of the
-0.4
0.1
E1.1 Analysis of Circuits (2016-8284)
1
ω RC
10
100
40
Note that 0 does not
0
0
-30
10
20
0.1
1
ω RC
10
axis is arbitrary.
Frequency Responses: 11 – 4 / 12
Logarithmic axes
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
We usually use logarithmic axes for frequency and gain (but not phase)
because % differences are more significant than absolute differences.
E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz
even though both differences equal 5 Hz.
|V |
Logarithmic voltage ratios are specified in decibels (dB) = 20 log10 |V2 | .
1
Common voltage ratios:
|V2 |
|V1 |
dB
0.1
−20
1
0
0.5
-6
10
20
exist on a log axis and so
π
-10
-0.2
-20
the starting point of the
-0.4
0.1
E1.1 Analysis of Circuits (2016-8284)
1
ω RC
10
100
40
Note that 0 does not
0
0
-30
2
6
0.1
1
ω RC
10
axis is arbitrary.
Frequency Responses: 11 – 4 / 12
Logarithmic axes
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
We usually use logarithmic axes for frequency and gain (but not phase)
because % differences are more significant than absolute differences.
E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz
even though both differences equal 5 Hz.
|V |
Logarithmic voltage ratios are specified in decibels (dB) = 20 log10 |V2 | .
1
Common voltage ratios:
|V2 |
|V1 |
dB
0.1
−20
√
0.5
-6
0.5
-3
2
3
2
6
10
20
exist on a log axis and so
π
-10
-0.2
-20
the starting point of the
-0.4
0.1
E1.1 Analysis of Circuits (2016-8284)
1
ω RC
10
100
40
Note that 0 does not
0
0
-30
1
0
√
0.1
1
ω RC
10
axis is arbitrary.
Frequency Responses: 11 – 4 / 12
Logarithmic axes
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
We usually use logarithmic axes for frequency and gain (but not phase)
because % differences are more significant than absolute differences.
E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz
even though both differences equal 5 Hz.
|V |
Logarithmic voltage ratios are specified in decibels (dB) = 20 log10 |V2 | .
1
Common voltage ratios:
|V2 |
|V1 |
dB
0.1
−20
√
0.5
-6
0.5
-3
2
3
2
6
10
20
exist on a log axis and so
π
-10
-0.2
-20
the starting point of the
-0.4
0.1
1
ω RC
10
100
40
Note that 0 does not
0
0
-30
1
0
√
0.1
1
ω RC
10
axis is arbitrary.
P2
Note: P ∝ V 2 ⇒ decibel power ratios are given by 10 log10 P
1
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 4 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Suppose we plot the magnitude and phase of H = c (jω)
r
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
Suppose we plot the magnitude and phase of H = c (jω)
r
Magnitude (log-log graph):
|H| = cω r ⇒ log |H| = log |c|+r log ω
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
Suppose we plot the magnitude and phase of H = c (jω)
r
Magnitude (log-log graph):
|H| = cω r ⇒ log |H| = log |c|+r log ω
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Suppose we plot the magnitude and phase of H = c (jω)
r
Magnitude (log-log graph):
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Suppose we plot the magnitude and phase of H = c (jω)
r
Magnitude (log-log graph):
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Phase (log-lin graph):
∠H = ∠j r + ∠c = r ×
E1.1 Analysis of Circuits (2016-8284)
π
2
(+π if c < 0)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Suppose we plot the magnitude and phase of H = c (jω)
r
Magnitude (log-log graph):
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Phase (log-lin graph):
∠H = ∠j r + ∠c = r ×
E1.1 Analysis of Circuits (2016-8284)
π
2
(+π if c < 0)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Suppose we plot the magnitude and phase of H = c (jω)
r
Magnitude (log-log graph):
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Phase (log-lin graph):
∠H = ∠j r + ∠c = r × π2 (+π if c < 0)
The phase is constant ∀ω .
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Suppose we plot the magnitude and phase of H = c (jω)
r
Magnitude (log-log graph):
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Phase (log-lin graph):
∠H = ∠j r + ∠c = r × π2 (+π if c < 0)
The phase is constant ∀ω .
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
Suppose we plot the magnitude and phase of H = c (jω)
r
Magnitude (log-log graph):
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
c only affects the line’s vertical position.
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Phase (log-lin graph):
∠H = ∠j r + ∠c = r × π2 (+π if c < 0)
The phase is constant ∀ω .
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
Suppose we plot the magnitude and phase of H = c (jω)
r
Magnitude (log-log graph):
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
c only affects the line’s vertical position.
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Phase (log-lin graph):
∠H = ∠j r + ∠c = r × π2 (+π if c < 0)
The phase is constant ∀ω .
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
Suppose we plot the magnitude and phase of H = c (jω)
r
Magnitude (log-log graph):
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
c only affects the line’s vertical position.
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Phase (log-lin graph):
∠H = ∠j r + ∠c = r × π2 (+π if c < 0)
The phase is constant ∀ω .
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
Suppose we plot the magnitude and phase of H = c (jω)
r
Magnitude (log-log graph):
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
c only affects the line’s vertical position.
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Phase (log-lin graph):
∠H = ∠j r + ∠c = r × π2 (+π if c < 0)
The phase is constant ∀ω .
If c > 0, phase = 90◦ × magnitude slope.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Suppose we plot the magnitude and phase of H = c (jω)
Magnitude (log-log graph):
Asymptotes
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
c only affects the line’s vertical position.
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
If |H| is measured in decibels, a slope of r
is called 6r dB/octave or 20r dB/decade.
Approximations
• Plot Magnitude Response
• Low and High Frequency
r
Phase (log-lin graph):
∠H = ∠j r + ∠c = r × π2 (+π if c < 0)
The phase is constant ∀ω .
If c > 0, phase = 90◦ × magnitude slope.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Suppose we plot the magnitude and phase of H = c (jω)
Magnitude (log-log graph):
Asymptotes
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
c only affects the line’s vertical position.
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
If |H| is measured in decibels, a slope of r
is called 6r dB/octave or 20r dB/decade.
Approximations
• Plot Magnitude Response
• Low and High Frequency
r
Phase (log-lin graph):
∠H = ∠j r + ∠c = r × π2 (+π if c < 0)
The phase is constant ∀ω .
If c > 0, phase = 90◦ × magnitude slope.
Negative c adds ±180◦ to the phase.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Suppose we plot the magnitude and phase of H = c (jω)
Magnitude (log-log graph):
Asymptotes
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
c only affects the line’s vertical position.
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
If |H| is measured in decibels, a slope of r
is called 6r dB/octave or 20r dB/decade.
Approximations
• Plot Magnitude Response
• Low and High Frequency
r
Phase (log-lin graph):
∠H = ∠j r + ∠c = r × π2 (+π if c < 0)
The phase is constant ∀ω .
If c > 0, phase = 90◦ × magnitude slope.
Negative c adds ±180◦ to the phase.
Note: Phase angles are modulo 360◦ , i.e.
+180◦ ≡ −180◦ and 450◦ ≡ 90◦ .
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Logs of Powers
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
r
H = c (jω) has a straight-line magnitude graph and a constant phase.
Magnitude (log-log graph):
Asymptotes
|H| = cω r ⇒ log |H| = log |c|+r log ω
This is a straight line with a slope of r .
c only affects the line’s vertical position.
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
If |H| is measured in decibels, a slope of r
is called 6r dB/octave or 20r dB/decade.
Approximations
• Plot Magnitude Response
• Low and High Frequency
Phase (log-lin graph):
∠H = ∠j r + ∠c = r × π2 (+π if c < 0)
The phase is constant ∀ω .
If c > 0, phase = 90◦ × magnitude slope.
Negative c adds ±180◦ to the phase.
Note: Phase angles are modulo 360◦ , i.e.
+180◦ ≡ −180◦ and 450◦ ≡ 90◦ .
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 5 / 12
Straight Line Approximations
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
(
ajω for |aω| ≫ |b|
Key idea: (ajω + b) ≈
b
for |aω| ≪ |b|
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 6 / 12
Straight Line Approximations
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
(
ajω for |aω| ≫ |b|
Key idea: (ajω + b) ≈
b
for |aω| ≪ |b|
1
Gain: H(jω) = jωRC+1
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
|Gain| (dB)
0
-10
-20
-30
E1.1 Analysis of Circuits (2016-8284)
0.1/RC
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 6 / 12
Straight Line Approximations
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
(
ajω for |aω| ≫ |b|
Key idea: (ajω + b) ≈
b
for |aω| ≪ |b|
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
): H(jω) ≈ 1
|Gain| (dB)
0
-10
-20
-30
E1.1 Analysis of Circuits (2016-8284)
0.1/RC
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 6 / 12
Straight Line Approximations
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
(
ajω for |aω| ≫ |b|
Key idea: (ajω + b) ≈
b
for |aω| ≪ |b|
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
): H(jω) ≈ 1
1
1
): H(jω) ≈ jωRC
High frequencies (ω ≫ RC
|Gain| (dB)
0
-10
-20
-30
E1.1 Analysis of Circuits (2016-8284)
0.1/RC
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 6 / 12
Straight Line Approximations
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
(
ajω for |aω| ≫ |b|
Key idea: (ajω + b) ≈
b
for |aω| ≪ |b|
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
): H(jω) ≈ 1
1
1
): H(jω) ≈ jωRC
High frequencies (ω ≫ RC
Approximate the magnitude response
as two straight lines
0
|Gain| (dB)
11: Frequency Responses
-10
-20
-30
E1.1 Analysis of Circuits (2016-8284)
0.1/RC
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 6 / 12
Straight Line Approximations
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
(
ajω for |aω| ≫ |b|
Key idea: (ajω + b) ≈
b
for |aω| ≪ |b|
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
): H(jω) ≈ 1⇒ |H(jω)| ≈ 1
1
1
): H(jω) ≈ jωRC
High frequencies (ω ≫ RC
Approximate the magnitude response
as two straight lines
0
|Gain| (dB)
11: Frequency Responses
-10
-20
-30
E1.1 Analysis of Circuits (2016-8284)
0.1/RC
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 6 / 12
Straight Line Approximations
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
(
ajω for |aω| ≫ |b|
Key idea: (ajω + b) ≈
b
for |aω| ≪ |b|
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
): H(jω) ≈ 1⇒ |H(jω)| ≈ 1
1
1
1
): H(jω) ≈ jωRC
⇒ |H(jω)| ≈ RC
ω −1
High frequencies (ω ≫ RC
Approximate the magnitude response
as two straight lines
0
|Gain| (dB)
11: Frequency Responses
-10
-20
-30
E1.1 Analysis of Circuits (2016-8284)
0.1/RC
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 6 / 12
Straight Line Approximations
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
(
ajω for |aω| ≫ |b|
Key idea: (ajω + b) ≈
b
for |aω| ≪ |b|
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
): H(jω) ≈ 1⇒ |H(jω)| ≈ 1
1
1
1
): H(jω) ≈ jωRC
⇒ |H(jω)| ≈ RC
ω −1
High frequencies (ω ≫ RC
Approximate the magnitude response
as two straight lines
0
|Gain| (dB)
11: Frequency Responses
-10
-20
-30
E1.1 Analysis of Circuits (2016-8284)
0.1/RC
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 6 / 12
Straight Line Approximations
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
(
ajω for |aω| ≫ |b|
Key idea: (ajω + b) ≈
b
for |aω| ≪ |b|
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
): H(jω) ≈ 1⇒ |H(jω)| ≈ 1
1
1
1
): H(jω) ≈ jωRC
⇒ |H(jω)| ≈ RC
ω −1
High frequencies (ω ≫ RC
Approximate the magnitude response
as two straight lines intersecting at the
1
corner frequency, ωc = RC
.
0
|Gain| (dB)
11: Frequency Responses
-10
-20
-30
E1.1 Analysis of Circuits (2016-8284)
0.1/RC
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 6 / 12
Straight Line Approximations
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
(
ajω for |aω| ≫ |b|
Key idea: (ajω + b) ≈
b
for |aω| ≪ |b|
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
): H(jω) ≈ 1⇒ |H(jω)| ≈ 1
1
1
1
): H(jω) ≈ jωRC
⇒ |H(jω)| ≈ RC
ω −1
High frequencies (ω ≫ RC
Approximate the magnitude response
as two straight lines intersecting at the
1
corner frequency, ωc = RC
.
At the corner frequency:
0
|Gain| (dB)
11: Frequency Responses
-10
-20
-30
0.1/RC
1/RC
ω (rad/s)
10/RC
(a) the gradient changes by −1 (= −6 dB/octave = −20 dB/decade).
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 6 / 12
Straight Line Approximations
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
(
ajω for |aω| ≫ |b|
Key idea: (ajω + b) ≈
b
for |aω| ≪ |b|
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
): H(jω) ≈ 1⇒ |H(jω)| ≈ 1
1
1
1
): H(jω) ≈ jωRC
⇒ |H(jω)| ≈ RC
ω −1
High frequencies (ω ≫ RC
Approximate the magnitude response
as two straight lines intersecting at the
1
corner frequency, ωc = RC
.
At the corner frequency:
0
|Gain| (dB)
11: Frequency Responses
-10
-20
-30
0.1/RC
1/RC
ω (rad/s)
10/RC
(a) the gradient changes
by −1 (= −6 dB/octave = −20 dB/decade).
1 √1
(b) |H(jωc )| = 1+j
= 2 = −3 dB (worst-case error).
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 6 / 12
Straight Line Approximations
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
(
ajω for |aω| ≫ |b|
Key idea: (ajω + b) ≈
b
for |aω| ≪ |b|
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
): H(jω) ≈ 1⇒ |H(jω)| ≈ 1
1
1
1
): H(jω) ≈ jωRC
⇒ |H(jω)| ≈ RC
ω −1
High frequencies (ω ≫ RC
Approximate the magnitude response
as two straight lines intersecting at the
1
corner frequency, ωc = RC
.
At the corner frequency:
0
|Gain| (dB)
11: Frequency Responses
-10
-20
-30
0.1/RC
1/RC
ω (rad/s)
10/RC
(a) the gradient changes
by −1 (= −6 dB/octave = −20 dB/decade).
1 √1
(b) |H(jωc )| = 1+j
= 2 = −3 dB (worst-case error).
b
A linear factor (ajω + b) has a corner frequency of ωc = a .
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 6 / 12
Plot Magnitude Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
The gain of a linear circuit is always a rational polynomial in jω and is
called the transfer function of the circuit. For example:
H(jω) =
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
0
-20
-40
0.1
E1.1 Analysis of Circuits (2016-8284)
1
10
ω (rad/s)
100
1000
Frequency Responses: 11 – 7 / 12
Plot Magnitude Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
The gain of a linear circuit is always a rational polynomial in jω and is
called the transfer function of the circuit. For example:
H(jω) =
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
Step 1: Factorize the polynomials
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0
-20
-40
0.1
E1.1 Analysis of Circuits (2016-8284)
1
10
ω (rad/s)
100
1000
Frequency Responses: 11 – 7 / 12
Plot Magnitude Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
The gain of a linear circuit is always a rational polynomial in jω and is
called the transfer function of the circuit. For example:
H(jω) =
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
Step 1: Factorize the polynomials
Step 2: Sort corner freqs: 1, 4, 12, 50
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0
-20
-40
0.1
E1.1 Analysis of Circuits (2016-8284)
1
10
ω (rad/s)
100
1000
Frequency Responses: 11 – 7 / 12
Plot Magnitude Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
The gain of a linear circuit is always a rational polynomial in jω and is
called the transfer function of the circuit. For example:
H(jω) =
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
Step 1: Factorize the polynomials
Step 2: Sort corner freqs: 1, 4, 12, 50
Step 3: For ω < 1 all linear factors equal
their constant terms:
20ω×12
|H| ≈ 1×4×50
= 1.2ω 1 .
E1.1 Analysis of Circuits (2016-8284)
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0
-20
-40
0.1
1
10
ω (rad/s)
100
1000
Frequency Responses: 11 – 7 / 12
Plot Magnitude Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
The gain of a linear circuit is always a rational polynomial in jω and is
called the transfer function of the circuit. For example:
H(jω) =
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
Step 1: Factorize the polynomials
Step 2: Sort corner freqs: 1, 4, 12, 50
Step 3: For ω < 1 all linear factors equal
their constant terms:
20ω×12
|H| ≈ 1×4×50
= 1.2ω 1 .
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0
-20
-40
0.1
1
10
ω (rad/s)
100
1000
Step 4: For 1 < ω < 4, the factor (jω + 1) ≈ jω so
|H| ≈
E1.1 Analysis of Circuits (2016-8284)
20ω×12
ω×4×50
= 1.2ω 0
Frequency Responses: 11 – 7 / 12
Plot Magnitude Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
The gain of a linear circuit is always a rational polynomial in jω and is
called the transfer function of the circuit. For example:
H(jω) =
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
Step 1: Factorize the polynomials
Step 2: Sort corner freqs: 1, 4, 12, 50
Step 3: For ω < 1 all linear factors equal
their constant terms:
20ω×12
|H| ≈ 1×4×50
= 1.2ω 1 .
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0
-20
-40
0.1
1
10
ω (rad/s)
100
1000
Step 4: For 1 < ω < 4, the factor (jω + 1) ≈ jω so
20ω×12
|H| ≈ ω×4×50
= 1.2ω 0 = +1.58 dB.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 7 / 12
Plot Magnitude Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
The gain of a linear circuit is always a rational polynomial in jω and is
called the transfer function of the circuit. For example:
H(jω) =
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
Step 1: Factorize the polynomials
Step 2: Sort corner freqs: 1, 4, 12, 50
Step 3: For ω < 1 all linear factors equal
their constant terms:
20ω×12
|H| ≈ 1×4×50
= 1.2ω 1 .
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0
-20
-40
0.1
1
10
ω (rad/s)
100
1000
Step 4: For 1 < ω < 4, the factor (jω + 1) ≈ jω so
20ω×12
|H| ≈ ω×4×50
= 1.2ω 0 = +1.58 dB.
20ω×12
Step 5: For 4 < ω < 12, |H| ≈ ω×ω×50
= 4.8ω −1 .
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 7 / 12
Plot Magnitude Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
The gain of a linear circuit is always a rational polynomial in jω and is
called the transfer function of the circuit. For example:
H(jω) =
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
Step 1: Factorize the polynomials
Step 2: Sort corner freqs: 1, 4, 12, 50
Step 3: For ω < 1 all linear factors equal
their constant terms:
20ω×12
|H| ≈ 1×4×50
= 1.2ω 1 .
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0
-20
-40
0.1
1
10
ω (rad/s)
100
1000
Step 4: For 1 < ω < 4, the factor (jω + 1) ≈ jω so
20ω×12
|H| ≈ ω×4×50
= 1.2ω 0 = +1.58 dB.
20ω×12
Step 5: For 4 < ω < 12, |H| ≈ ω×ω×50
= 4.8ω −1 .
20ω×ω
Step 6: For 12 < ω < 50, |H| ≈ ω×ω×50
= 0.4ω 0 = −7.96 dB.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 7 / 12
Plot Magnitude Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
The gain of a linear circuit is always a rational polynomial in jω and is
called the transfer function of the circuit. For example:
H(jω) =
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
Step 1: Factorize the polynomials
Step 2: Sort corner freqs: 1, 4, 12, 50
Step 3: For ω < 1 all linear factors equal
their constant terms:
20ω×12
|H| ≈ 1×4×50
= 1.2ω 1 .
0
-20
-40
0.1
1
10
ω (rad/s)
100
1000
Step 4: For 1 < ω < 4, the factor (jω + 1) ≈ jω so
20ω×12
|H| ≈ ω×4×50
= 1.2ω 0 = +1.58 dB.
20ω×12
Step 5: For 4 < ω < 12, |H| ≈ ω×ω×50
= 4.8ω −1 .
20ω×ω
Step 6: For 12 < ω < 50, |H| ≈ ω×ω×50
= 0.4ω 0 = −7.96 dB.
20ω×ω
= 20ω −1 .
Step 7: For ω > 50, |H| ≈ ω×ω×ω
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 7 / 12
Plot Magnitude Response
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
The gain of a linear circuit is always a rational polynomial in jω and is
called the transfer function of the circuit. For example:
H(jω) =
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
Step 1: Factorize the polynomials
Step 2: Sort corner freqs: 1, 4, 12, 50
Step 3: For ω < 1 all linear factors equal
their constant terms:
20ω×12
|H| ≈ 1×4×50
= 1.2ω 1 .
0
-20
-40
0.1
1
10
ω (rad/s)
100
1000
Step 4: For 1 < ω < 4, the factor (jω + 1) ≈ jω so
20ω×12
|H| ≈ ω×4×50
= 1.2ω 0 = +1.58 dB.
20ω×12
Step 5: For 4 < ω < 12, |H| ≈ ω×ω×50
= 4.8ω −1 .
20ω×ω
Step 6: For 12 < ω < 50, |H| ≈ ω×ω×50
= 0.4ω 0 = −7.96 dB.
20ω×ω
= 20ω −1 .
Step 7: For ω > 50, |H| ≈ ω×ω×ω
At each corner frequency, the graph is continuous but its gradient changes
abruptly by +1 (numerator factor) or −1 (denominator factor).
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 7 / 12
Low and High Frequency Asymptotes
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
You can find the low and high frequency asymptotes without factorizing:
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 8 / 12
Low and High Frequency Asymptotes
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
You can find the low and high frequency asymptotes without factorizing:
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0.5
0
π
11: Frequency Responses
-20
0
-40
-0.5
0.1
E1.1 Analysis of Circuits (2016-8284)
1
10
ω (rad/s)
100
1000
0.1
1
10
ω (rad/s)
100
1000
Frequency Responses: 11 – 8 / 12
Low and High Frequency Asymptotes
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
You can find the low and high frequency asymptotes without factorizing:
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0.5
0
π
11: Frequency Responses
-20
0
-40
-0.5
0.1
1
10
ω (rad/s)
100
1000
0.1
1
10
ω (rad/s)
100
1000
Low Frequency Asymptote:
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 8 / 12
Low and High Frequency Asymptotes
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
You can find the low and high frequency asymptotes without factorizing:
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0.5
0
π
11: Frequency Responses
-20
0
-40
-0.5
0.1
1
10
ω (rad/s)
100
1000
0.1
1
10
ω (rad/s)
100
1000
Low Frequency Asymptote:
20jω(12)
From factors: HLF (jω) = (1)(4)(50) = 1.2jω
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 8 / 12
Low and High Frequency Asymptotes
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
You can find the low and high frequency asymptotes without factorizing:
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0.5
0
π
11: Frequency Responses
-20
0
-40
-0.5
0.1
1
10
ω (rad/s)
100
1000
0.1
1
10
ω (rad/s)
100
1000
Low Frequency Asymptote:
20jω(12)
From factors: HLF (jω) = (1)(4)(50) = 1.2jω
720(jω)
Lowest power of jω on top and bottom: H (jω) ≃ 600 = 1.2jω
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 8 / 12
Low and High Frequency Asymptotes
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
You can find the low and high frequency asymptotes without factorizing:
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0.5
0
π
11: Frequency Responses
-20
0
-40
-0.5
0.1
1
10
ω (rad/s)
100
1000
0.1
1
10
ω (rad/s)
100
1000
Low Frequency Asymptote:
20jω(12)
From factors: HLF (jω) = (1)(4)(50) = 1.2jω
720(jω)
Lowest power of jω on top and bottom: H (jω) ≃ 600 = 1.2jω
High Frequency Asymptote:
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 8 / 12
Low and High Frequency Asymptotes
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
You can find the low and high frequency asymptotes without factorizing:
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0.5
0
π
11: Frequency Responses
-20
0
-40
-0.5
0.1
1
10
ω (rad/s)
100
1000
0.1
1
10
ω (rad/s)
100
1000
Low Frequency Asymptote:
20jω(12)
From factors: HLF (jω) = (1)(4)(50) = 1.2jω
720(jω)
Lowest power of jω on top and bottom: H (jω) ≃ 600 = 1.2jω
High Frequency Asymptote:
20jω(jω)
−1
From factors: HHF (jω) = (jω)(jω)(jω) = 20 (jω)
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 8 / 12
Low and High Frequency Asymptotes
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
You can find the low and high frequency asymptotes without factorizing:
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0.5
0
π
11: Frequency Responses
-20
0
-40
-0.5
0.1
1
10
ω (rad/s)
100
1000
0.1
1
10
ω (rad/s)
100
1000
Low Frequency Asymptote:
20jω(12)
From factors: HLF (jω) = (1)(4)(50) = 1.2jω
720(jω)
Lowest power of jω on top and bottom: H (jω) ≃ 600 = 1.2jω
High Frequency Asymptote:
20jω(jω)
−1
From factors: HHF (jω) = (jω)(jω)(jω) = 20 (jω)
Highest power of jω on top and bottom: H (jω) ≃
E1.1 Analysis of Circuits (2016-8284)
60(jω)2
3(jω)3
= 20 (jω)−1
Frequency Responses: 11 – 8 / 12
Phase Approximation
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
1
Gain: H(jω) = jωRC+1
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 9 / 12
Phase Approximation
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
):
H(jω) ≈ 1
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Phase/π
0
-0.2
-0.4
0.1/RC
E1.1 Analysis of Circuits (2016-8284)
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 9 / 12
Phase Approximation
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
):
H(jω) ≈ 1⇒ ∠1 = 0
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Phase/π
0
-0.2
-0.4
0.1/RC
E1.1 Analysis of Circuits (2016-8284)
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 9 / 12
Phase Approximation
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
1
Low frequencies (ω ≪ RC
):
H(jω) ≈ 1⇒ ∠1 = 0
1
1
High frequencies (ω ≫ RC
): H(jω) ≈ jωRC
0
Phase/π
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
1
Gain: H(jω) = jωRC+1
-0.2
-0.4
0.1/RC
E1.1 Analysis of Circuits (2016-8284)
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 9 / 12
Phase Approximation
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
1
Low frequencies (ω ≪ RC
):
H(jω) ≈ 1⇒ ∠1 = 0
1
1
High frequencies (ω ≫ RC
): H(jω) ≈ jωRC
⇒ ∠j −1 = − π2
0
Phase/π
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
1
Gain: H(jω) = jωRC+1
-0.2
-0.4
0.1/RC
E1.1 Analysis of Circuits (2016-8284)
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 9 / 12
Phase Approximation
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
):
H(jω) ≈ 1⇒ ∠1 = 0
1
1
High frequencies (ω ≫ RC
): H(jω) ≈ jωRC
⇒ ∠j −1 = − π2
Approximate the phase response as
three straight lines.
0
Phase/π
11: Frequency Responses
-0.2
-0.4
0.1/RC
E1.1 Analysis of Circuits (2016-8284)
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 9 / 12
Phase Approximation
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
):
H(jω) ≈ 1⇒ ∠1 = 0
1
1
High frequencies (ω ≫ RC
): H(jω) ≈ jωRC
⇒ ∠j −1 = − π2
Approximate the phase response as
three straight lines.
By chance, they intersect close to
1
.
0.1ωc and 10ωc where ωc = RC
E1.1 Analysis of Circuits (2016-8284)
0
Phase/π
11: Frequency Responses
-0.2
-0.4
0.1/RC
1/RC
ω (rad/s)
10/RC
Frequency Responses: 11 – 9 / 12
Phase Approximation
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
):
H(jω) ≈ 1⇒ ∠1 = 0
1
1
High frequencies (ω ≫ RC
): H(jω) ≈ jωRC
⇒ ∠j −1 = − π2
Approximate the phase response as
three straight lines.
By chance, they intersect close to
1
.
0.1ωc and 10ωc where ωc = RC
0
Phase/π
11: Frequency Responses
-0.2
-0.4
0.1/RC
1/RC
ω (rad/s)
10/RC
Between 0.1ωc and 10ωc the phase changes by − π2 over two decades.
This gives a gradient = − π4 radians/decade.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 9 / 12
Phase Approximation
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
):
H(jω) ≈ 1⇒ ∠1 = 0
1
1
High frequencies (ω ≫ RC
): H(jω) ≈ jωRC
⇒ ∠j −1 = − π2
Approximate the phase response as
three straight lines.
By chance, they intersect close to
1
.
0.1ωc and 10ωc where ωc = RC
0
Phase/π
11: Frequency Responses
-0.2
-0.4
0.1/RC
1/RC
ω (rad/s)
10/RC
Between 0.1ωc and 10ωc the phase changes by − π2 over two decades.
This gives a gradient = − π4 radians/decade.
(ajω + b) in denominator
π
∓1 b ⇒ ∆gradient = ∓ 4 /decade at ω = 10
a .
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 9 / 12
Phase Approximation
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
1
Gain: H(jω) = jωRC+1
1
Low frequencies (ω ≪ RC
):
H(jω) ≈ 1⇒ ∠1 = 0
1
1
High frequencies (ω ≫ RC
): H(jω) ≈ jωRC
⇒ ∠j −1 = − π2
Approximate the phase response as
three straight lines.
By chance, they intersect close to
1
.
0.1ωc and 10ωc where ωc = RC
0
Phase/π
11: Frequency Responses
-0.2
-0.4
0.1/RC
1/RC
ω (rad/s)
10/RC
Between 0.1ωc and 10ωc the phase changes by − π2 over two decades.
This gives a gradient = − π4 radians/decade.
(ajω + b) in denominator
π
∓1 b ⇒ ∆gradient = ∓ 4 /decade at ω = 10
a .
The sign of ∆gradient is reversed for (a) numerator factors and (b) ab < 0.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 9 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
H(jω) =
Approximations
• Plot Magnitude Response
• Low and High Frequency
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
0.5
π
11: Frequency Responses
0
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
-0.5
0.1
1
10
ω (rad/s)
100
1000
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
ωc = {1− , 4− , 12+ , 50− }
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0.5
π
11: Frequency Responses
0
-0.5
0.1
1
10
ω (rad/s)
100
1000
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
:
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
0.5
π
11: Frequency Responses
0
-0.5
0.1
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
0.5
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
π
11: Frequency Responses
0
-0.5
0.1
:
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
1
Step 4: Put in ascending order and calculate gaps as log10 ω
ω decades:
2
−
.1 (.6) .4 (.48) 1.2 (.62) 5 (.3) 10 (.6) 40 (.48) 120− (.62) 500+ .
E1.1 Analysis of Circuits (2016-8284)
−
+
−
+
+
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
0.5
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
π
11: Frequency Responses
0
-0.5
0.1
:
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
1
Step 4: Put in ascending order and calculate gaps as log10 ω
ω decades:
2
−
.1 (.6) .4 (.48) 1.2 (.62) 5 (.3) 10 (.6) 40 (.48) 120− (.62) 500+ .
Step 5: Find phase of LF asymptote: ∠1.2jω = + π2 .
E1.1 Analysis of Circuits (2016-8284)
−
+
−
+
+
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
0.5
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
π
11: Frequency Responses
0
-0.5
0.1
:
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
1
Step 4: Put in ascending order and calculate gaps as log10 ω
ω decades:
2
−
.1 (.6) .4 (.48) 1.2 (.62) 5 (.3) 10 (.6) 40 (.48) 120− (.62) 500+ .
Step 5: Find phase of LF asymptote: ∠1.2jω = + π2 .
E1.1 Analysis of Circuits (2016-8284)
−
+
−
+
+
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
0.5
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
π
11: Frequency Responses
0
-0.5
0.1
:
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
1
Step 4: Put in ascending order and calculate gaps as log10 ω
ω decades:
2
−
.1 (.6) .4 (.48) 1.2 (.62) 5 (.3) 10 (.6) 40 (.48) 120− (.62) 500+ .
Step 5: Find phase of LF asymptote: ∠1.2jω = + π2 .
Step 6: At ω = 0.1 gradient becomes − π4 rad/decade. φ is still π2 .
E1.1 Analysis of Circuits (2016-8284)
−
+
−
+
+
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
0.5
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
π
11: Frequency Responses
0
-0.5
0.1
:
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
1
Step 4: Put in ascending order and calculate gaps as log10 ω
ω decades:
2
−
.1 (.6) .4 (.48) 1.2 (.62) 5 (.3) 10 (.6) 40 (.48) 120− (.62) 500+ .
Step 5: Find phase of LF asymptote: ∠1.2jω = + π2 .
Step 6: At ω = 0.1 gradient becomes − π4 rad/decade. φ is still π2 .
Step 7: At ω = 0.4, φ = π2 − 0.6 π4 = 0.35π . New gradient is − π2 .
E1.1 Analysis of Circuits (2016-8284)
−
+
−
+
+
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
0.5
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
π
11: Frequency Responses
0
-0.5
0.1
:
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
1
Step 4: Put in ascending order and calculate gaps as log10 ω
ω decades:
2
−
.1 (.6) .4 (.48) 1.2 (.62) 5 (.3) 10 (.6) 40 (.48) 120− (.62) 500+ .
Step 5: Find phase of LF asymptote: ∠1.2jω = + π2 .
Step 6: At ω = 0.1 gradient becomes − π4 rad/decade. φ is still π2 .
Step 7: At ω = 0.4, φ = π2 − 0.6 π4 = 0.35π . New gradient is − π2 .
Step 8: At ω = 1.2, φ = 0.35π − 0.48 π2 = 0.11π . New gradient is − π4 .
E1.1 Analysis of Circuits (2016-8284)
−
+
−
+
+
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
0.5
π
11: Frequency Responses
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
0
-0.5
0.1
:
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
1
Step 4: Put in ascending order and calculate gaps as log10 ω
ω decades:
2
−
−
+
−
+
+
.1 (.6) .4 (.48) 1.2 (.62) 5 (.3) 10 (.6) 40 (.48) 120− (.62) 500+ .
Step 5: Find phase of LF asymptote: ∠1.2jω = + π2 .
Step 6: At ω = 0.1 gradient becomes − π4 rad/decade. φ is still π2 .
Step 7: At ω = 0.4, φ = π2 − 0.6 π4 = 0.35π . New gradient is − π2 .
Step 8: At ω = 1.2, φ = 0.35π − 0.48 π2 = 0.11π . New gradient is − π4 .
Steps 9-13: Repeat for each gradient change.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
0.5
π
11: Frequency Responses
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
0
-0.5
0.1
:
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
1
Step 4: Put in ascending order and calculate gaps as log10 ω
ω decades:
2
−
−
+
−
+
+
.1 (.6) .4 (.48) 1.2 (.62) 5 (.3) 10 (.6) 40 (.48) 120− (.62) 500+ .
Step 5: Find phase of LF asymptote: ∠1.2jω = + π2 .
Step 6: At ω = 0.1 gradient becomes − π4 rad/decade. φ is still π2 .
Step 7: At ω = 0.4, φ = π2 − 0.6 π4 = 0.35π . New gradient is − π2 .
Step 8: At ω = 1.2, φ = 0.35π − 0.48 π2 = 0.11π . New gradient is − π4 .
Steps 9-13: Repeat for each gradient change.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
0.5
π
11: Frequency Responses
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
0
-0.5
0.1
:
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
1
Step 4: Put in ascending order and calculate gaps as log10 ω
ω decades:
2
−
−
+
−
+
+
.1 (.6) .4 (.48) 1.2 (.62) 5 (.3) 10 (.6) 40 (.48) 120− (.62) 500+ .
Step 5: Find phase of LF asymptote: ∠1.2jω = + π2 .
Step 6: At ω = 0.1 gradient becomes − π4 rad/decade. φ is still π2 .
Step 7: At ω = 0.4, φ = π2 − 0.6 π4 = 0.35π . New gradient is − π2 .
Step 8: At ω = 1.2, φ = 0.35π − 0.48 π2 = 0.11π . New gradient is − π4 .
Steps 9-13: Repeat for each gradient change.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
0.5
π
11: Frequency Responses
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
0
-0.5
0.1
:
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
1
Step 4: Put in ascending order and calculate gaps as log10 ω
ω decades:
2
−
−
+
−
+
+
.1 (.6) .4 (.48) 1.2 (.62) 5 (.3) 10 (.6) 40 (.48) 120− (.62) 500+ .
Step 5: Find phase of LF asymptote: ∠1.2jω = + π2 .
Step 6: At ω = 0.1 gradient becomes − π4 rad/decade. φ is still π2 .
Step 7: At ω = 0.4, φ = π2 − 0.6 π4 = 0.35π . New gradient is − π2 .
Step 8: At ω = 1.2, φ = 0.35π − 0.48 π2 = 0.11π . New gradient is − π4 .
Steps 9-13: Repeat for each gradient change.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
0.5
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
π
11: Frequency Responses
0
-0.5
0.1
:
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
1
Step 4: Put in ascending order and calculate gaps as log10 ω
ω decades:
2
−
−
+
−
+
+
.1 (.6) .4 (.48) 1.2 (.62) 5 (.3) 10 (.6) 40 (.48) 120− (.62) 500+ .
Step 5: Find phase of LF asymptote: ∠1.2jω = + π2 .
Step 6: At ω = 0.1 gradient becomes − π4 rad/decade. φ is still π2 .
Step 7: At ω = 0.4, φ = π2 − 0.6 π4 = 0.35π . New gradient is − π2 .
Step 8: At ω = 1.2, φ = 0.35π − 0.48 π2 = 0.11π . New gradient is − π4 .
Steps 9-13: Repeat for each gradient change. Final gradient is always 0.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 10 / 12
Plot Phase Response
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
H(jω) =
60(jω)2 +720(jω)
3(jω)3 +165(jω)2 +762(jω)+600
=
Step 1: Factorize the polynomials
Step 2: List corner freqs: ± = num/den
0.5
ωc = {1− , 4− , 12+ , 50− }
Step 3: Gradient changes at 10∓1 ωc . Sign depends on
num/den and sgn ab
+
+
− −
20jω(jω+12)
(jω+1)(jω+4)(jω+50)
π
11: Frequency Responses
0
-0.5
0.1
:
1
10
ω (rad/s)
100
1000
.1− , 10+ ; .4− , 40 ; 1.2 , 120 ; 5 , 500+
1
Step 4: Put in ascending order and calculate gaps as log10 ω
ω decades:
2
−
−
+
−
+
+
.1 (.6) .4 (.48) 1.2 (.62) 5 (.3) 10 (.6) 40 (.48) 120− (.62) 500+ .
Step 5: Find phase of LF asymptote: ∠1.2jω = + π2 .
Step 6: At ω = 0.1 gradient becomes − π4 rad/decade. φ is still π2 .
Step 7: At ω = 0.4, φ = π2 − 0.6 π4 = 0.35π . New gradient is − π2 .
Step 8: At ω = 1.2, φ = 0.35π − 0.48 π2 = 0.11π . New gradient is − π4 .
Steps 9-13: Repeat for each gradient change. Final gradient is always 0.
At 0.1 and 10 times each corner frequency, the graph is continuous but its
gradient changes abruptly by ± π4 rad/decade.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 10 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Y
X
=
1
R+ jωC
1
3R+R+ jωC
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Y
X
=
1
R+ jωC
1
3R+R+ jωC
=
jωRC+1
4jωRC+1
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
Y
X
=
1
R+ jωC
1
3R+R+ jωC
=
jωRC+1
4jωRC+1
−
+
1
1
, RC
Corner freqs: 4RC
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
Y
X
=
1
R+ jωC
1
3R+R+ jωC
=
jωRC+1
4jωRC+1
−
+
1
1
, RC
Corner freqs: 4RC
LF Asymptote: H(jω) = 1
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
Y
X
=
1
R+ jωC
1
3R+R+ jωC
=
jωRC+1
4jωRC+1
−
+
1
1
, RC
Corner freqs: 4RC
LF Asymptote: H(jω) = 1
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Magnitude Response:
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
Y
X
=
1
R+ jωC
1
3R+R+ jωC
=
jωRC+1
4jωRC+1
−
+
1
1
, RC
Corner freqs: 4RC
LF Asymptote: H(jω) = 1
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Magnitude Response:
1
1
Gradient Changes: −20 dB/dec at ω = 4RC
and +20 at ω = RC
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
Y
X
=
1
R+ jωC
1
3R+R+ jωC
=
jωRC+1
4jωRC+1
−
+
1
1
, RC
Corner freqs: 4RC
LF Asymptote: H(jω) = 1
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Magnitude Response:
1
1
Gradient Changes: −20 dB/dec at ω = 4RC
and +20 at ω = RC
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
Y
X
=
1
R+ jωC
1
3R+R+ jωC
=
jωRC+1
4jωRC+1
−
+
1
1
, RC
Corner freqs: 4RC
LF Asymptote: H(jω) = 1
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Magnitude Response:
1
1
Gradient Changes: −20 dB/dec at ω = 4RC
and +20 at ω = RC
jωRC
1
, (c) 4jωRC = 14
Line equations: H(jω) = (a) 1, (b) 4jωRC
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
Y
X
=
1
R+ jωC
1
3R+R+ jωC
=
jωRC+1
4jωRC+1
−
+
1
1
, RC
Corner freqs: 4RC
LF Asymptote: H(jω) = 1
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Magnitude Response:
1
1
Gradient Changes: −20 dB/dec at ω = 4RC
and +20 at ω = RC
jωRC
1
, (c) 4jωRC = 14
Line equations: H(jω) = (a) 1, (b) 4jωRC
Phase Response:
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
Y
X
=
1
R+ jωC
1
3R+R+ jωC
=
jωRC+1
4jωRC+1
−
+
1
1
, RC
Corner freqs: 4RC
LF Asymptote: H(jω) = 1
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Magnitude Response:
1
1
Gradient Changes: −20 dB/dec at ω = 4RC
and +20 at ω = RC
jωRC
1
, (c) 4jωRC = 14
Line equations: H(jω) = (a) 1, (b) 4jωRC
Phase Response:
LF asymptote: φ = ∠1 = 0
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
Y
X
=
1
R+ jωC
1
3R+R+ jωC
=
jωRC+1
4jωRC+1
−
+
1
1
, RC
Corner freqs: 4RC
LF Asymptote: H(jω) = 1
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Magnitude Response:
1
1
Gradient Changes: −20 dB/dec at ω = 4RC
and +20 at ω = RC
jωRC
1
, (c) 4jωRC = 14
Line equations: H(jω) = (a) 1, (b) 4jωRC
Phase Response:
LF asymptote: φ = ∠1 = 0
−
+
+
−
0.1
2.5
10
Gradient changes of ± π4 /decade at: ω = 0.025
, RC
, RC
, RC
.
RC
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
RCR Circuit
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
Y
X
=
1
R+ jωC
1
3R+R+ jωC
=
jωRC+1
4jωRC+1
−
+
1
1
, RC
Corner freqs: 4RC
LF Asymptote: H(jω) = 1
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
Magnitude Response:
1
1
Gradient Changes: −20 dB/dec at ω = 4RC
and +20 at ω = RC
jωRC
1
, (c) 4jωRC = 14
Line equations: H(jω) = (a) 1, (b) 4jωRC
Phase Response:
LF asymptote: φ = ∠1 = 0
−
+
+
−
0.1
2.5
10
Gradient changes of ± π4 /decade at: ω = 0.025
, RC
, RC
, RC
.
RC
0.1
0.1
, φ = 0 − π4 log10 0.025
= −0.15π
At ω = RC
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 11 / 12
Summary
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
• Frequency response: magnitude and phase of
◦ Only applies to sine waves
Y
X
as a function of ω
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 12 / 12
Summary
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
• Frequency response: magnitude and phase of
◦ Only applies to sine waves
Y
X
as a function of ω
◦ Use log axes for frequency and gain but linear for phase
2
⊲ Decibels = 20 log10 VV21 = 10 log10 P
P1
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 12 / 12
Summary
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
• Frequency response: magnitude and phase of
◦ Only applies to sine waves
Y
X
as a function of ω
◦ Use log axes for frequency and gain but linear for phase
2
⊲ Decibels = 20 log10 VV21 = 10 log10 P
P1
b
• Linear factor (ajω + b) gives corner frequency at ω = a .
b
◦ Magnitude plot gradient changes by ±20 dB/decade @ ω = a .
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 12 / 12
Summary
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
• Frequency response: magnitude and phase of
◦ Only applies to sine waves
Y
X
as a function of ω
◦ Use log axes for frequency and gain but linear for phase
2
⊲ Decibels = 20 log10 VV21 = 10 log10 P
P1
b
• Linear factor (ajω + b) gives corner frequency at ω = a .
b
◦ Magnitude plot gradient changes by ±20 dB/decade @ ω = a .
◦ Phase gradient changes in two places by:
b
π
⊲ ± 4 rad/decade @ ω = 0.1 × a b
π
⊲ ∓ rad/decade @ ω = 10 × 4
E1.1 Analysis of Circuits (2016-8284)
a
Frequency Responses: 11 – 12 / 12
Summary
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
• Frequency response: magnitude and phase of
◦ Only applies to sine waves
Y
X
as a function of ω
◦ Use log axes for frequency and gain but linear for phase
2
⊲ Decibels = 20 log10 VV21 = 10 log10 P
P1
b
• Linear factor (ajω + b) gives corner frequency at ω = a .
b
◦ Magnitude plot gradient changes by ±20 dB/decade @ ω = a .
◦ Phase gradient changes in two places by:
b
π
⊲ ± 4 rad/decade @ ω = 0.1 × a b
π
⊲ ∓ rad/decade @ ω = 10 × 4
a
• LF/HF asymptotes: keep only the terms with the lowest/highest power
of jω in numerator and denominator polynomials
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 12 / 12
Summary
11: Frequency Responses
• Frequency Response
• Sine Wave Response
• Logarithmic axes
• Logs of Powers
• Straight Line
Approximations
• Plot Magnitude Response
• Low and High Frequency
Asymptotes
• Phase Approximation
• Plot Phase Response
• RCR Circuit
• Summary
• Frequency response: magnitude and phase of
◦ Only applies to sine waves
Y
X
as a function of ω
◦ Use log axes for frequency and gain but linear for phase
2
⊲ Decibels = 20 log10 VV21 = 10 log10 P
P1
b
• Linear factor (ajω + b) gives corner frequency at ω = a .
b
◦ Magnitude plot gradient changes by ±20 dB/decade @ ω = a .
◦ Phase gradient changes in two places by:
b
π
⊲ ± 4 rad/decade @ ω = 0.1 × a b
π
⊲ ∓ rad/decade @ ω = 10 × 4
a
• LF/HF asymptotes: keep only the terms with the lowest/highest power
of jω in numerator and denominator polynomials
For further details see Hayt et al. Chapter 16.
E1.1 Analysis of Circuits (2016-8284)
Frequency Responses: 11 – 12 / 12