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11: Frequency Responses


Frequency Response

Sine Wave Response

Logarathmic 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 (2014-5225) Frequency Responses: 11 1 / 12

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Frequency Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Ifx(t)is a sine wave, theny(t)will also be asine wave but with a different amplitude and

phase shift.Xis an input phasor andY is theoutput phasor.

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Frequency Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary




Thegainof the circuit is YX = 1/jCR+1/jC =

1jRC+1

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Frequency Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





1jRC+1

This is a complex function of so we plot separate graphs for:

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Frequency Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





1jRC+1


Magnitude: YX=

1|jRC+1| =

1

1+(RC)2

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Frequency Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





1jRC+1


Magnitude: YX=

1|jRC+1| =

1

1+(RC)2

0 1 2 3 4 50

0.5

1

RC

Magnitude Response

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Frequency Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





1jRC+1


Magnitude: YX=

1|jRC+1| =

1

1+(RC)2Phase Shift:

YX

= (jRC+ 1) = arctan RC1

0 1 2 3 4 50

0.5

1

RC

Magnitude Response

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Frequency Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





1jRC+1


Magnitude: YX=

1|jRC+1| =

1

1+(RC)2Phase Shift:

YX

= (jRC+ 1) = arctan RC1

0 1 2 3 4 50

0.5

1

RC

Magnitude Response

0 1 2 3 4 5

-0.4

-0.2

0

RC

Phase Response

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Sine Wave Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


RC= 10 ms

YX =

1jRC+1 =

10.01j+1

= 50 YX = 0.89 27

= 100 YX = 0.71 45

= 300 YX = 0.32 72

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Sine Wave Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


RC= 10 ms

YX =

1jRC+1 =

10.01j+1

= 50 YX = 0.89 27

= 100 YX = 0.71 45

= 300 YX = 0.32 72

0 100 200 300 400 5000

0.5

1

(rad/s)

|Y/X|

0 100 200 300 400 500

-80

-60

-40

-20

0

(rad/s)

Phase()

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Sine Wave Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


RC= 10 ms

YX =

1jRC+1 =

10.01j+1

0 0.5 1

-0.4

-0.2

0 X

YX-Y

=50

Real

Imag

= 50 YX = 0.89 27

= 100 YX = 0.71 45

= 300 YX = 0.32 72

0 100 200 300 400 5000

0.5

1

(rad/s)

|Y/X|

0 100 200 300 400 500

-80

-60

-40

-20

0

(rad/s)

Phase()

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Sine Wave Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


RC= 10 ms

YX =

1jRC+1 =

10.01j+1

0 0.5 1

-0.4

-0.2

0 X

YX-Y

=50

Real

Imag

= 50 YX = 0.89 27

= 100 YX = 0.71 45

= 300 YX = 0.32 72 0 20 40 60 80 100 120-1-0.5

0

0.5

1

x

y

time (ms)

x=blue,y=red

w = 50 rad/s, Gain = 0.89, Phase = -27

0 100 200 300 400 5000

0.5

1

(rad/s)

|Y/X|

0 100 200 300 400 500

-80

-60

-40

-20

0

(rad/s)

Phase()

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Sine Wave Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


RC= 10 ms

YX =

1jRC+1 =

10.01j+1

= 50 YX = 0.89 27

= 100 YX = 0.71 45

= 300 YX = 0.32 72

0 100 200 300 400 5000

0.5

1

(rad/s)

|Y/X|

0 100 200 300 400 500

-80

-60

-40

-20

0

(rad/s)

Phase()

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Sine Wave Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


RC= 10 ms

YX =

1jRC+1 =

10.01j+1

0 0.5 1

-0.4

-0.2

0 X

YX-Y

=100

Real

Imag

= 50 YX = 0.89 27

= 100 YX = 0.71 45

= 300 YX = 0.32 72

0 100 200 300 400 5000

0.5

1

(rad/s)

|Y/X|

0 100 200 300 400 500

-80

-60

-40

-20

0

(rad/s)

Phase()

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Sine Wave Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


RC= 10 ms

YX =

1jRC+1 =

10.01j+1

0 0.5 1

-0.4

-0.2

0 X

YX-Y

=100

Real

Imag

= 50 YX = 0.89 27

= 100 YX = 0.71 45

= 300 YX = 0.32 72 0 20 40 60 80 100 120-1

-0.5

0

0.5

1

x

y

time (ms)

x=blue,y=red

w = 100 rad/s, Gain = 0.71, Phase = -45

0 100 200 300 400 5000

0.5

1

(rad/s)

|Y/X|

0 100 200 300 400 500

-80

-60

-40

-20

0

(rad/s)

Phase()

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Sine Wave Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


RC= 10 ms

YX =

1jRC+1 =

10.01j+1

= 50 YX = 0.89 27

= 100 YX = 0.71 45

= 300 YX = 0.32 72

0 100 200 300 400 5000

0.5

1

(rad/s)

|Y/X|

0 100 200 300 400 500

-80

-60

-40

-20

0

(rad/s)

Phase(

)

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Sine Wave Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


RC= 10 ms

YX =

1jRC+1 =

10.01j+1

0 0.5 1

-0.4

-0.2

0 X

YX-Y

=300

Real

Imag

= 50 YX = 0.89 27

= 100 YX = 0.71 45

= 300 YX = 0.32 72

0 100 200 300 400 5000

0.5

1

(rad/s)

|Y/X|

0 100 200 300 400 500

-80

-60

-40

-20

0

(rad/s)

Phase(

)

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Sine Wave Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


RC= 10 ms

YX =

1jRC+1 =

10.01j+1

0 0.5 1

-0.4

-0.2

0 X

YX-Y

=300

Real

Imag

= 50 YX = 0.89 27

= 100 YX = 0.71 45

= 300 YX = 0.32 72 0 20 40 60 80 100 120-1

-0.5

0

0.5

1

x

y

time (ms)

x=blue,y=red

w = 300 rad/s, Gain = 0.32, Phase = -72

0 100 200 300 400 5000

0.5

1

(rad/s)

|Y/X|

0 100 200 300 400 500

-80

-60

-40

-20

0

(rad/s)

Phase(

)

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Sine Wave Response


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


RC= 10 ms

YX =

1jRC+1 =

10.01j+1

0 0.5 1

-0.4

-0.2

0 X

YX-Y

=300

Real

Imag

= 50 YX = 0.89 27

= 100 YX = 0.71 45

= 300 YX = 0.32 72 0 20 40 60 80 100 120-1

-0.5

0

0.5

1

x

y

time (ms)

x=blue,y=red

w = 300 rad/s, Gain = 0.32, Phase = -72

0 100 200 300 400 5000

0.5

1

(rad/s)

|Y/X|

0 100 200 300 400 500

-80

-60

-40

-20

0

(rad/s)

Phase(

)

The output,y(t),lagsthe input,x(t), by up to90.

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Logarathmic axes


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


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 versus5.005 kHz is less significant than10 Hz versus15 Hzeven though both differences equal5 Hz.

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Logarathmic axes


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





0.1 1 10

-0.4

-0.2

0

RC

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Logarathmic axes


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





0.1 1 10-30

-20

-10

0

RC0.1 1 10

-0.4

-0.2

0

RC

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Logarathmic axes


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





Logarithmic voltage ratios are specified in decibels (dB)=20 log10|V2||V1| .

0.1 1 10-30

-20

-10

0

RC0.1 1 10

-0.4

-0.2

0

RC

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Logarathmic axes


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary






Common ratios:1 0 dB,

0.1 1 10-30

-20

-10

0

RC0.1 1 10

-0.4

-0.2

0

RC

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Logarathmic axes


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary






Common ratios:1 0 dB,10 20 dB, 0.1 20 dB, 100 40 dB,

0.1 1 10-30

-20

-10

0

RC0.1 1 10

-0.4

-0.2

0

RC

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Logarathmic axes


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary






Common ratios:1 0 dB,10 20 dB, 0.1 20 dB, 100 40 dB,2 6 dB, 0.5 6 dB,

0.1 1 10-30

-20

-10

0

RC0.1 1 10

-0.4

-0.2

0

RC

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Logarathmic axes


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary







2 = 1.41 3 dB, 12

= 0.707 3 dB.

0.1 1 10-30

-20

-10

0

RC0.1 1 10

-0.4

-0.2

0

RC

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Logarathmic axes


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary







2 = 1.41 3 dB, 12

= 0.707 3 dB.

0.1 1 10-30

-20

-10

0

RC0.1 1 10

-0.4

-0.2

0

RC

Note:P V2

decibel power ratios:10log10P2P1

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Suppose we plot the magnitude and phase ofH=c (j)r

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



Magnitude (log-log graph):

|H| =cr log |H| = log |c|+r log

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary




|H| =cr log |H| = log |c|+r log

1 10 1001

10

100

2

(rad/s)

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary




|H| =cr log |H| = log |c|+r log This is a straight line with a slope ofr.

1 10 1001

10

100

2

(rad/s)

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





1 10 1001

10

100

2

(rad/s)

Phase (log-lin graph):

H= jr + c=r 2 (+ifc

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





1 10 1001

10

100

2

(rad/s)


H= jr + c=r 2 (+ifc

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





1 10 1001

10

100

2

(rad/s)


H= jr + c=r 2 (+ifc

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations

Plot Magnitude Response

Low and High Frequency

Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





1 10 1001

10

100

2

0.22

(rad/s)


H= jr + c=r 2 (+ifc

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary




|H| =cr log |H| = log |c|+r log This is a straight line with a slope ofr.conly affects the lines vertical position.

1 10 1001

10

100

2

0.22

(rad/s)


H= jr + c=r 2 (+ifc

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





1 10 1001

10

100

2

0.22

80

(rad/s)


H= jr + c=r 2 (+ifc

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





1 10 1001

10

100

2

0.22

80

65-1

(rad/s)


H= jr + c=r 2 (+ifc

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





1 10 1001

10

100

2

0.22

80

65-1

(rad/s)


H= jr + c=r 2 (+ifc 0, phase =90 magnitude slope.

1 10 100-1

-0.5

0

0.5

1

2 0.22

80

65-1

(rad/s)

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary





If |H| is measured in decibels, a slope ofris called6r dB/octaveor20r dB/decade.

1 10 1001

10

100

2

0.22

80

65-1

(rad/s)


H= jr + c=r 2 (+ifc 0, phase =90 magnitude slope.

1 10 100-1

-0.5

0

0.5

1

2 0.22

80

65-1

(rad/s)

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary






1 10 1001

10

100

2

0.22

80

65-1

-9-1

(rad/s)


H= jr + c=r 2 (+ifc 0, phase =90 magnitude slope.Negativec adds180 to the phase.

1 10 100-1

-0.5

0

0.5

1

2 0.22

80

65-1

-9-1

(rad/s)

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary






1 10 1001

10

100

2

0.22

80

65-1

-9-1

(rad/s)



Note:Phase angles are modulo360, i.e.450

90. 1 10 100

-1

-0.5

0

0.5

1

2 0.22

80

65-1

-9-1

(rad/s)

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Logs of Powers


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H=c (j)r has a straight-line magnitude graph and a constant phase.




1 10 1001

10

100

2

0.22

80

65-1

-9-1

(rad/s)



Note:Phase angles are modulo360, i.e.450

90. 1 10 100

-1

-0.5

0

0.5

1

2 0.22

80

65-1

-9-1

(rad/s)

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Straight Line Approximations


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Key idea:(aj+ b)

aj for |a| |b|b for |a| |b|

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Key idea:(aj+ b)


Gain:H(j) = 1jRC+1

0.1/RC 1/RC 10/RC-30

-20

-10

0

(rad/s)

|Gain|(dB)

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Key idea:(aj+ b)


Gain:H(j) = 1jRC+1

Low frequencies ( 1RC):H(j) 1

0.1/RC 1/RC 10/RC-30

-20

-10

0

(rad/s)

|Gain|(dB)

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Key idea:(aj+ b)


Gain:H(j) = 1jRC+1

Low frequencies ( 1RC):H(j) 1High frequencies ( 1RC):H(j) 1jRC

0.1/RC 1/RC 10/RC-30

-20

-10

0

(rad/s)

|Gain|(dB)

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Key idea:(aj+ b)


Gain:H(j) = 1jRC+1

Low frequencies ( 1RC):H(j) 1High frequencies ( 1RC):H(j) 1jRCApproximate the magnitude response

as two straight lines

0.1/RC 1/RC 10/RC-30

-20

-10

0

(rad/s)

|Gain|(dB)

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Key idea:(aj+ b)


Gain:H(j) = 1jRC+1

Low frequencies ( 1RC):H(j) 1 |H(j)| 1High frequencies ( 1RC):H(j) 1jRCApproximate the magnitude response


0.1/RC 1/RC 10/RC-30

-20

-10

0

(rad/s)

|Gain|(dB)

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Key idea:(aj+ b)


Gain:H(j) = 1jRC+1

Low frequencies ( 1RC):H(j) 1 |H(j)| 1High frequencies ( 1RC):H(j) 1jRC |H(j)| 1RC1

Approximate the magnitude response


0.1/RC 1/RC 10/RC-30

-20

-10

0

(rad/s)

|Gain|(dB)

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Key idea:(aj+ b)


Gain:H(j) = 1jRC+1




0.1/RC 1/RC 10/RC-30

-20

-10

0

(rad/s)

|Gain|(dB)

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Key idea:(aj+ b)


Gain:H(j) = 1jRC+1



as two straight lines intersecting at the

corner frequency, c = 1RC.

0.1/RC 1/RC 10/RC-30

-20

-10

0

(rad/s)

|Gain|(dB)

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11: Frequency Responses Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Key idea:(aj+ b)


Gain:H(j) = 1jRC+1





At the corner frequency: 0.1/RC 1/RC 10/RC-30

-20

-10

0

(rad/s)

|Gain|(dB)

(a)the gradient changes by 1(= 6 dB/octave= 20 dB/decade).

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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Key idea:(aj+ b)


Gain:H(j) = 1jRC+1






-20

-10

0

(rad/s)

|Gain|(dB)

(a)the gradient changes by 1(= 6 dB/octave= 20 dB/decade).(b) |H(jc)| =

11+j

= 1

2= 3 dB (worst-case error).

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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Key idea:(aj+ b)


Gain:H(j) = 1jRC+1






-20

-10

0

(rad/s)

|Gain|(dB)

(a)the gradient changes by 1(= 6 dB/octave= 20 dB/decade).(b) |H(jc)| =

11+j

= 1

2= 3 dB (worst-case error).

A linear factor(aj+ b)has a corner frequency ofc = ba .

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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


The gain of a linear circuit is always arational polynomial inj and iscalled thetransfer functionof the circuit. For example:

H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600

0.1 1 10 100 1000

-40

-20

0

(rad/s)

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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)

Step 1:Factorize the polynomials

0.1 1 10 100 1000

-40

-20

0

(rad/s)

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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)


Step 2:Sort corner freqs:1, 4, 12, 50

0.1 1 10 100 1000

-40

-20

0

(rad/s)

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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)


Step 2:Sort corner freqs:1, 4, 12, 50Step 3:For

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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)



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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)



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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)



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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)



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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)



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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)



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Low and High Frequency Asymptotes


Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


You can find the low and high frequency asymptotes without factorizing:

H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)

L d Hi h F A t t

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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)

0.1 1 10 100 1000

-40

-20

0

(rad/s)0.1 1 10 100 1000

-0.5

0

0.5

(rad/s)


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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)

0.1 1 10 100 1000

-40

-20

0

(rad/s)0.1 1 10 100 1000

-0.5

0

0.5

(rad/s)

Low Frequency Asymptote:


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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)

0.1 1 10 100 1000

-40

-20

0

(rad/s)0.1 1 10 100 1000

-0.5

0

0.5

(rad/s)

Low Frequency Asymptote:From factors:HLF(j) =

20j(12)(1)(4)(50) = 1.2j


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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)

0.1 1 10 100 1000

-40

-20

0

(rad/s)0.1 1 10 100 1000

-0.5

0

0.5

(rad/s)


20j(12)(1)(4)(50) = 1.2j

Lowestpower ofj on top and bottom:H(j) 720(j)600 = 1.2j


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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)

0.1 1 10 100 1000

-40

-20

0

(rad/s)0.1 1 10 100 1000

-0.5

0

0.5

(rad/s)


20j(12)(1)(4)(50) = 1.2j

Lowestpower ofj on top and bottom:H(j) 720(j)600 = 1.2jHigh Frequency Asymptote:


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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)

0.1 1 10 100 1000

-40

-20

0

(rad/s)0.1 1 10 100 1000

-0.5

0

0.5

(rad/s)


20j(12)(1)(4)(50) = 1.2j


From factors:HHF(j) = 20j(j)(j)(j)(j) = 20 (j)

1


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Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary



H(j) = 60(j)2+720(j)

3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)

0.1 1 10 100 1000

-40

-20

0

(rad/s)0.1 1 10 100 1000

-0.5

0

0.5

(rad/s)


20j(12)(1)(4)(50) = 1.2j


From factors:HHF(j) = 20j(j)(j)(j)(j) = 20 (j)

1

Highestpower ofj on top and bottom:H(j) 60(j)23(j)3

= 20(j)1

Phase Approximation

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Phase Approximation


Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Gain:H(j) = 1jRC+1

Phase Approximation

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Phase Approximation


Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Gain:H(j) = 1jRC+1

Low frequencies ( 1RC):H(j)

1

0.1/RC 1/RC 10/RC

-0.4

-0.2

0

(rad/s)

Phas

e/

Phase Approximation

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Phase Approximation


Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Gain:H(j) = 1jRC+1


1

1 = 0

0.1/RC 1/RC 10/RC

-0.4

-0.2

0

(rad/s)

Phas

e/

Phase Approximation

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Phase Approximation


Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Gain:H(j) = 1jRC+1


1

1 = 0

High frequencies ( 1RC):H(j) 1jRC

0.1/RC 1/RC 10/RC

-0.4

-0.2

0

(rad/s)

Phas

e/

Phase Approximation

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Phase Approximation


Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Gain:H(j) = 1jRC+1


1

1 = 0

High frequencies ( 1RC):H(j) 1jRC j1 = 2

0.1/RC 1/RC 10/RC

-0.4

-0.2

0

(rad/s)

Phas

e/

Phase Approximation

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Phase Approximation


Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Gain:H(j) = 1jRC+1


1

1 = 0

High frequencies ( 1RC):H(j) 1jRC j1 = 2Approximate the phase response as

three straight lines.

0.1/RC 1/RC 10/RC

-0.4

-0.2

0

(rad/s)

Phas

e/

Phase Approximation

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Phase Approximation


Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Asymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Gain:H(j) = 1jRC+1


1

1 = 0



By chance, they intersect close to

0.1c and10c wherec= 1RC. 0.1/RC 1/RC 10/RC

-0.4

-0.2

0

(rad/s)

Phas

e/

Phase Approximation

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Phase Approximation


Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations


Low and High FrequencyAsymptotes

Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Gain:H(j) = 1jRC+1


1

1 = 0





-0.4

-0.2

0

(rad/s)

Phas

e/

Between0.1c and10c the phase changes by

2over two decades.

This gives a gradient = 4 radians/decade.

Phase Approximation

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ase pp o at o


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Gain:H(j) = 1jRC+1


1

1 = 0





-0.4

-0.2

0

(rad/s)

Phas

e/


2over two decades.

This gives a gradient = 4 radians/decade.(aj+ b)in denominator gradient= 4 /decade at

= 101ba .

Phase Approximation

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pp


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


Gain:H(j) = 1jRC+1

Low frequencies ( 1RC):H(j) 1 1 = 0





-0.4

-0.2

0

(rad/s)

Phas

e/


2over two decades.

This gives a gradient = 4 radians/decade.(aj+ b)in denominator gradient= 4 /decade at

= 101ba .

The sign ofgradient is reversed for (a) numerator factors and (b) ba

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p


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)

Plot Phase Response

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p


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 =

20j(j+12)(j+1)(j+4)(j+50)


0.1 1 10 100 1000-0.5

0

0.5

(rad/s)

Plot Phase Response

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p


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 =

20j(j+12)(j+1)(j+4)(j+50)


Step 2:Gradient changes at101c.

Sign depends on num/den and sgnba

:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)

Plot Phase Response

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p


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 =

20j(j+12)(j+1)(j+4)(j+50)




:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)

Step 3:Put in ascending order and calculate gaps aslog1012

decades:

.1 (.6) .4 (.48)1.2+

(.62)5 (.3)10+

(.6)40+

(.48) 120 (.62) 500+

.

Plot Phase Response

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 =

20j(j+12)(j+1)(j+4)(j+50)




:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)


decades:

.1 (.6) .4 (.48)1.2+

(.62)5 (.3)10+

(.6)40+

(.48) 120 (.62) 500+

.Step 4:Find phase of LF asymptote: 1.2j= +2 .

Plot Phase Response

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 =

20j(j+12)(j+1)(j+4)(j+50)




:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)


decades:

.1 (.6) .4 (.48)1.2+

(.62)5 (.3)10+

(.6)40+

(.48) 120 (.62) 500+

.Step 4:Find phase of LF asymptote: 1.2j= +2 .

Plot Phase Response

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 =

20j(j+12)(j+1)(j+4)(j+50)




:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)


decades:

.1 (.6) .4 (.48)1.2+

(.62)5 (.3)10+

(.6)40+

(.48) 120 (.62) 500+

.Step 4:Find phase of LF asymptote: 1.2j= +2 .Step 5:At = 0.1gradient becomes 4 .is still 2 .

Plot Phase Response

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 =

20j(j+12)(j+1)(j+4)(j+50)




:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)


decades:

.1 (.6) .4 (.48)1.2+

(.62)5 (.3)10+

(.6)40+

(.48) 120 (.62) 500+

.Step 4:Find phase of LF asymptote: 1.2j= +2 .Step 5:At = 0.1gradient becomes 4 .is still 2 .Step 6:At = 0.4,= 2 0.64 = 0.35. New gradient is 2 .

Plot Phase Response

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 =

20j(j+12)(j+1)(j+4)(j+50)




:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)


decades:

.1 (.6) .4 (.48) 1.2+

(.62)5 (.3)10+

(.6)40+

(.48) 120 (.62) 500+

.Step 4:Find phase of LF asymptote: 1.2j= +2 .Step 5:At = 0.1gradient becomes 4 .is still 2 .Step 6:At = 0.4,= 2 0.64 = 0.35. New gradient is 2 .Step 7:At

= 1.2,= 0.35 0.48

2 = 0.11. New gradient is

4.

Plot Phase Response

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 =

20j(j+12)(j+1)(j+4)(j+50)




:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)


decades:

.1 (.6) .4 (.48)1.2+

(.62) 5 (.3)10+

(.6)40+

(.48) 120 (.62) 500+


= 1.2,= 0.35 0.48


4.

Steps 8-12:Repeat for each gradient change.

Plot Phase Response

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95/118


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 =

20j(j+12)(j+1)(j+4)(j+50)




:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)


decades:

.1 (.6) .4 (.48)1.2+

(.62)5 (.3) 10+

(.6)40+

(.48) 120 (.62) 500+


= 1.2,= 0.35 0.48


4.


Plot Phase Response

5/19/2018 01100_Freqresp

96/118


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 =

20j(j+12)(j+1)(j+4)(j+50)




:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)


decades:

.1 (.6) .4 (.48)1.2+

(.62)5 (.3)10+

(.6) 40+

(.48) 120 (.62) 500+

.Step 4:Find phase of LF asymptote: 1.2j= +2 .Step 5:At = 0.1gradient becomes 4 .is still 2 .Step 6:At = 0.4,= 2 0.64 = 0.35. New gradient is 2 .Step 7:At = 1.2,= 0.35

0.48


4.


Plot Phase Response

5/19/2018 01100_Freqresp

97/118


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 = 20

j(j+12)(j+1)(j+4)(j+50)




:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)


decades:

.1 (.6) .4 (.48)1.2+

(.62)5 (.3)10+

(.6)40+

(.48) 120 (.62) 500+


0.48


4.


Plot Phase Response

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98/118


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)




:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)


decades:

.1 (.6) .4 (.48)1.2+

(.62)5 (.3)10+

(.6)40+

(.48) 120 (.62) 500+


0.48


4.

Steps 8-12:Repeat for each gradient change. Final gradient is always 0.

Plot Phase Response

5/19/2018 01100_Freqresp

99/118


Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

Straight Line

Approximations



Phase Approximation

Plot Phase Response

RCR Circuit

Summary


H(j) = 60(j)2

+720(j)3(j)3+165(j)2+762(j)+600 = 20j(j+12)(j+1)(j+4)(j+50)




:.1, 10+; .4, 40+; 1.2+, 120; 5, 500+

0.1 1 10 100 1000-0.5

0

0.5

(rad/s)


decades:

.1 (.6) .4 (.48)1.2+

(.62)5 (.3)10+

(.6)40

+

(.48) 120 (.62) 500+


0.48


4.

Steps 8-12: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 4 rad/decade.

RCR Circuit

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Frequency Response

Sine Wave Response

Logarathmic axes

Logs of Powers

01100_freqresp

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