11: Frequency Responses • Frequency Response • Sine Wave Response • Logarithmic axes • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary 11: Frequency Responses

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 1 / 12 Frequency Response

11: Frequency Responses • Frequency Response If x(t) is a sine wave, then y(t) will also be a • Sine Wave Response sine wave but with a different amplitude and • Logarithmic axes • LogsofPowers + phase shift. X is an input phasor and Y is the • Straight Line Approximations output phasor. • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 2 / 12 Frequency Response

11: Frequency Responses • Frequency Response If x(t) is a sine wave, then y(t) will also be a • Sine Wave Response sine wave but with a different amplitude and • Logarithmic axes • LogsofPowers + phase shift. X is an input phasor and Y is the • Straight Line Approximations output phasor. • Plot Magnitude Response • Low and High Frequency 1 Asymptotes Y /jωC 1 The gain of the circuit is 1 • Phase Approximation + X = R+ /jωC = jωRC+1 • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 2 / 12 Frequency Response

11: Frequency Responses • Frequency Response If x(t) is a sine wave, then y(t) will also be a • Sine Wave Response sine wave but with a different amplitude and • Logarithmic axes • LogsofPowers + phase shift. X is an input phasor and Y is the • Straight Line Approximations output phasor. • Plot Magnitude Response • Low and High Frequency 1 Asymptotes Y /jωC 1 The gain of the circuit is 1 • Phase Approximation + X = R+ /jωC = jωRC+1 • Plot PhaseResponse + • RCR Circuit This is a complex function of ω so we plot separate graphs for: • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 2 / 12 Frequency Response

11: Frequency Responses • Frequency Response If x(t) is a sine wave, then y(t) will also be a • Sine Wave Response sine wave but with a different amplitude and • Logarithmic axes • LogsofPowers + phase shift. X is an input phasor and Y is the • Straight Line Approximations output phasor. • Plot Magnitude Response • Low and High Frequency 1 Asymptotes Y /jωC 1 The gain of the circuit is 1 • Phase Approximation + X = R+ /jωC = jωRC+1 • Plot PhaseResponse + • RCR Circuit This is a complex function of ω so we plot separate graphs for: • Summary Y 1 1 Magnitude: X = jωRC+1 = 2 | | √1+(ωRC)

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 2 / 12 Frequency Response

11: Frequency Responses • Frequency Response If x(t) is a sine wave, then y(t) will also be a • Sine Wave Response sine wave but with a different amplitude and • Logarithmic axes • LogsofPowers + phase shift. X is an input phasor and Y is the • Straight Line Approximations output phasor. • Plot Magnitude Response • Low and High Frequency 1 Asymptotes Y /jωC 1 The gain of the circuit is 1 • Phase Approximation + X = R+ /jωC = jωRC+1 • Plot PhaseResponse + • RCR Circuit This is a complex function of ω so we plot separate graphs for: • Summary Y 1 1 Magnitude: X = jωRC+1 = 2 | | √1+(ωRC)

Magnitude Response

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 2 / 12 Frequency Response

11: Frequency Responses • Frequency Response If x(t) is a sine wave, then y(t) will also be a • Sine Wave Response sine wave but with a different amplitude and • Logarithmic axes • LogsofPowers + phase shift. X is an input phasor and Y is the • Straight Line Approximations output phasor. • Plot Magnitude Response • Low and High Frequency 1 Asymptotes Y /jωC 1 The gain of the circuit is 1 • Phase Approximation + X = R+ /jωC = jωRC+1 • Plot PhaseResponse + • RCR Circuit This is a complex function of ω so we plot separate graphs for: • Summary Y 1 1 Magnitude: X = jωRC+1 = 2 | | √1+(ωRC)

Phase Shift: ∠ Y = ∠ (jωRC + 1) = arctan ωRC X − − 1

Magnitude Response

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 2 / 12 Frequency Response

11: Frequency Responses • Frequency Response If x(t) is a sine wave, then y(t) will also be a • Sine Wave Response sine wave but with a different amplitude and • Logarithmic axes • LogsofPowers + phase shift. X is an input phasor and Y is the • Straight Line Approximations output phasor. • Plot Magnitude Response • Low and High Frequency 1 Asymptotes Y /jωC 1 The gain of the circuit is 1 • Phase Approximation + X = R+ /jωC = jωRC+1 • Plot PhaseResponse + • RCR Circuit This is a complex function of ω so we plot separate graphs for: • Summary Y 1 1 Magnitude: X = jωRC+1 = 2 | | √1+(ωRC)

Phase Shift: ∠ Y = ∠ (jωRC + 1) = arctan ωRC X − − 1

Magnitude Response Phase Response

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 2 / 12 Sine Wave Response

11: Frequency Responses • Frequency Response RC = 10 ms • Sine Wave Response • Y 1 1 Logarithmic axes X = jωRC+1 = 0.01jω+1 • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes Y ω = 50 = 0.89∠ 27◦ • Phase Approximation + ⇒ X − • Plot PhaseResponse + • Y ∠ RCR Circuit ω = 100 X = 0.71 45◦ • Summary ⇒ − Y ω = 300 = 0.32∠ 72◦ ⇒ X −

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 3 / 12 Sine Wave Response

11: Frequency Responses • Frequency Response RC = 10 ms • Sine Wave Response • Y 1 1 Logarithmic axes X = jωRC+1 = 0.01jω+1 • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes Y ω = 50 = 0.89∠ 27◦ • Phase Approximation + ⇒ X − • Plot PhaseResponse + • Y ∠ RCR Circuit ω = 100 X = 0.71 45◦ • Summary ⇒ − Y ω = 300 = 0.32∠ 72◦ ⇒ X −

1 0

-20

-40 0.5 |Y/X|

Phase (°) Phase -60

-80 0 0 100 200 300 400 500 0 100 200 300 400 500 ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 3 / 12 Sine Wave Response

11: Frequency Responses RC = 10 ms X • Frequency Response 0 • Sine Wave Response Y 1 1 -0.2 Y • Logarithmic axes X-Y X = jωRC+1 = 0.01jω+1 Imag • LogsofPowers + -0.4 ω=50 • Straight Line Approximations 0 0.5 1 Real • Plot Magnitude Response • Low and High Frequency Asymptotes Y ω = 50 = 0.89∠ 27◦ • Phase Approximation + ⇒ X − • Plot PhaseResponse + • Y ∠ RCR Circuit ω = 100 X = 0.71 45◦ • Summary ⇒ − Y ω = 300 = 0.32∠ 72◦ ⇒ X −

1 0

-20

-40 0.5 |Y/X|

Phase (°) Phase -60

-80 0 0 100 200 300 400 500 0 100 200 300 400 500 ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 3 / 12 Sine Wave Response

11: Frequency Responses RC = 10 ms X • Frequency Response 0 • Sine Wave Response Y 1 1 -0.2 Y • Logarithmic axes X-Y X = jωRC+1 = 0.01jω+1 Imag • LogsofPowers + -0.4 ω=50 • Straight Line Approximations 0 0.5 1 Real • Plot Magnitude Response • Low and High Frequency Asymptotes Y w = 50 rad/s, Gain = 0.89, Phase = -27° ω = 50 = 0.89∠ 27◦ • Phase Approximation + ⇒ X − 1 • Plot PhaseResponse + 0.5 x • Y ∠ RCR Circuit ω = 100 X = 0.71 45◦ 0 y • Summary ⇒ − -0.5 Y y=red x=blue, ω = 300 = 0.32∠ 72◦ -1 X 0 20 40 60 80 100 120 ⇒ − time (ms)

1 0

-20

-40 0.5 |Y/X|

Phase (°) Phase -60

-80 0 0 100 200 300 400 500 0 100 200 300 400 500 ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 3 / 12 Sine Wave Response

11: Frequency Responses • Frequency Response RC = 10 ms • Sine Wave Response • Y 1 1 Logarithmic axes X = jωRC+1 = 0.01jω+1 • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes Y ω = 50 = 0.89∠ 27◦ • Phase Approximation + ⇒ X − • Plot PhaseResponse + • Y ∠ RCR Circuit ω = 100 X = 0.71 45◦ • Summary ⇒ − Y ω = 300 = 0.32∠ 72◦ ⇒ X −

1 0

-20

-40 0.5 |Y/X|

Phase (°) Phase -60

-80 0 0 100 200 300 400 500 0 100 200 300 400 500 ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 3 / 12 Sine Wave Response

11: Frequency Responses RC = 10 ms X • Frequency Response 0 • Sine Wave Response Y 1 1 -0.2 • Logarithmic axes Y X = jωRC+1 = 0.01jω+1 Imag X-Y • LogsofPowers + -0.4 ω=100 • Straight Line Approximations 0 0.5 1 Real • Plot Magnitude Response • Low and High Frequency Asymptotes Y ω = 50 = 0.89∠ 27◦ • Phase Approximation + ⇒ X − • Plot PhaseResponse + • Y ∠ RCR Circuit ω = 100 X = 0.71 45◦ • Summary ⇒ − Y ω = 300 = 0.32∠ 72◦ ⇒ X −

1 0

-20

-40 0.5 |Y/X|

Phase (°) Phase -60

-80 0 0 100 200 300 400 500 0 100 200 300 400 500 ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 3 / 12 Sine Wave Response

11: Frequency Responses RC = 10 ms X • Frequency Response 0 • Sine Wave Response Y 1 1 -0.2 • Logarithmic axes Y X = jωRC+1 = 0.01jω+1 Imag X-Y • LogsofPowers + -0.4 ω=100 • Straight Line Approximations 0 0.5 1 Real • Plot Magnitude Response • Low and High Frequency Asymptotes Y w = 100 rad/s, Gain = 0.71, Phase = -45° ω = 50 = 0.89∠ 27◦ • Phase Approximation + ⇒ X − 1 • Plot PhaseResponse + 0.5 x • Y ∠ RCR Circuit ω = 100 X = 0.71 45◦ 0 y • Summary ⇒ − -0.5 Y y=red x=blue, ω = 300 = 0.32∠ 72◦ -1 X 0 20 40 60 80 100 120 ⇒ − time (ms)

1 0

-20

-40 0.5 |Y/X|

Phase (°) Phase -60

-80 0 0 100 200 300 400 500 0 100 200 300 400 500 ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 3 / 12 Sine Wave Response

11: Frequency Responses • Frequency Response RC = 10 ms • Sine Wave Response • Y 1 1 Logarithmic axes X = jωRC+1 = 0.01jω+1 • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes Y ω = 50 = 0.89∠ 27◦ • Phase Approximation + ⇒ X − • Plot PhaseResponse + • Y ∠ RCR Circuit ω = 100 X = 0.71 45◦ • Summary ⇒ − Y ω = 300 = 0.32∠ 72◦ ⇒ X −

1 0

-20

-40 0.5 |Y/X|

Phase (°) Phase -60

-80 0 0 100 200 300 400 500 0 100 200 300 400 500 ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 3 / 12 Sine Wave Response

11: Frequency Responses RC = 10 ms X • Frequency Response 0

• Sine Wave Response Y Y 1 1 -0.2 X-Y • Logarithmic axes X = jωRC+1 = 0.01jω+1 Imag • LogsofPowers + -0.4 ω=300 • Straight Line Approximations 0 0.5 1 Real • Plot Magnitude Response • Low and High Frequency Asymptotes Y ω = 50 = 0.89∠ 27◦ • Phase Approximation + ⇒ X − • Plot PhaseResponse + • Y ∠ RCR Circuit ω = 100 X = 0.71 45◦ • Summary ⇒ − Y ω = 300 = 0.32∠ 72◦ ⇒ X −

1 0

-20

-40 0.5 |Y/X|

Phase (°) Phase -60

-80 0 0 100 200 300 400 500 0 100 200 300 400 500 ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 3 / 12 Sine Wave Response

11: Frequency Responses RC = 10 ms X • Frequency Response 0

• Sine Wave Response Y Y 1 1 -0.2 X-Y • Logarithmic axes X = jωRC+1 = 0.01jω+1 Imag • LogsofPowers + -0.4 ω=300 • Straight Line Approximations 0 0.5 1 Real • Plot Magnitude Response • Low and High Frequency Asymptotes Y ∠ w = 300 rad/s, Gain = 0.32, Phase = -72° ω = 50 = 0.89 27 1 • Phase Approximation + X ◦ ⇒ − x • Plot PhaseResponse + 0.5 • Y ∠ y RCR Circuit ω = 100 X = 0.71 45◦ 0 • Summary ⇒ − -0.5 Y y=red x=blue, ω = 300 = 0.32∠ 72 -1 X ◦ 0 20 40 60 80 100 120 ⇒ − time (ms)

1 0

-20

-40 0.5 |Y/X|

Phase (°) Phase -60

-80 0 0 100 200 300 400 500 0 100 200 300 400 500 ω (rad/s) ω (rad/s)

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 3 / 12 Sine Wave Response

11: Frequency Responses RC = 10 ms X • Frequency Response 0

• Sine Wave Response Y Y 1 1 -0.2 X-Y • Logarithmic axes X = jωRC+1 = 0.01jω+1 Imag • LogsofPowers + -0.4 ω=300 • Straight Line Approximations 0 0.5 1 Real • Plot Magnitude Response • Low and High Frequency Asymptotes Y ∠ w = 300 rad/s, Gain = 0.32, Phase = -72° ω = 50 = 0.89 27 1 • Phase Approximation + X ◦ ⇒ − x • Plot PhaseResponse + 0.5 • Y ∠ y RCR Circuit ω = 100 X = 0.71 45◦ 0 • Summary ⇒ − -0.5 Y y=red x=blue, ω = 300 = 0.32∠ 72 -1 X ◦ 0 20 40 60 80 100 120 ⇒ − time (ms)

1 0

-20

-40 0.5 |Y/X|

Phase (°) Phase -60

-80 0 0 100 200 300 400 500 0 100 200 300 400 500 ω (rad/s) ω (rad/s)

The output, y(t), lags the input, x(t), by up to 90◦.

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 3 / 12 Logarithmic axes

11: Frequency Responses We usually use logarithmic axes for frequency and gain (but not phase) • Frequency Response • Sine Wave Response because % differences are more significant than absolute differences. • Logarithmic axes • LogsofPowers + E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz • Straight Line Approximations even though both differences equal 5 Hz. • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 4 / 12 Logarithmic axes

11: Frequency Responses We usually use logarithmic axes for frequency and gain (but not phase) • Frequency Response • Sine Wave Response because % differences are more significant than absolute differences. • Logarithmic axes • LogsofPowers + E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz • Straight Line Approximations even though both differences equal 5 Hz. • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 4 / 12 Logarithmic axes

11: Frequency Responses We usually use logarithmic axes for frequency and gain (but not phase) • Frequency Response • Sine Wave Response because % differences are more significant than absolute differences. • Logarithmic axes • LogsofPowers + E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz • Straight Line Approximations even though both differences equal 5 Hz. • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 4 / 12 Logarithmic axes

11: Frequency Responses We usually use logarithmic axes for frequency and gain (but not phase) • Frequency Response • Sine Wave Response because % differences are more significant than absolute differences. • Logarithmic axes • LogsofPowers + E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz • Straight Line Approximations even though both differences equal 5 Hz. • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Note that 0 does not exist on a log axis and so the starting point of the axis is arbitrary.

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 4 / 12 Logarithmic axes

11: Frequency Responses We usually use logarithmic axes for frequency and gain (but not phase) • Frequency Response • Sine Wave Response because % differences are more significant than absolute differences. • Logarithmic axes • LogsofPowers + E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz • Straight Line Approximations even though both differences equal 5 Hz. • Plot Magnitude Response V2 • Low and High Frequency Logarithmic voltage ratios are specified in decibels (dB) = 20 log | | . Asymptotes 10 V1 • Phase Approximation + | | • Plot PhaseResponse + • RCR Circuit • Summary

Note that 0 does not exist on a log axis and so the starting point of the axis is arbitrary.

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 4 / 12 Logarithmic axes

11: Frequency Responses We usually use logarithmic axes for frequency and gain (but not phase) • Frequency Response • Sine Wave Response because % differences are more significant than absolute differences. • Logarithmic axes • LogsofPowers + E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz • Straight Line Approximations even though both differences equal 5 Hz. • Plot Magnitude Response V2 • Low and High Frequency Logarithmic voltage ratios are specified in decibels (dB) = 20 log | | . Asymptotes 10 V1 • Phase Approximation + | | • Plot PhaseResponse + • RCR Circuit • Summary Common voltage ratios:

|V2| 1 |V1| dB 0

Note that 0 does not exist on a log axis and so the starting point of the axis is arbitrary.

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 4 / 12 Logarithmic axes

11: Frequency Responses We usually use logarithmic axes for frequency and gain (but not phase) • Frequency Response • Sine Wave Response because % differences are more significant than absolute differences. • Logarithmic axes • LogsofPowers + E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz • Straight Line Approximations even though both differences equal 5 Hz. • Plot Magnitude Response V2 • Low and High Frequency Logarithmic voltage ratios are specified in decibels (dB) = 20 log | | . Asymptotes 10 V1 • Phase Approximation + | | • Plot PhaseResponse + • RCR Circuit • Summary Common voltage ratios:

|V2| 0.1 1 10 100 |V1| dB 20 0 20 40 −

Note that 0 does not exist on a log axis and so the starting point of the axis is arbitrary.

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 4 / 12 Logarithmic axes

11: Frequency Responses We usually use logarithmic axes for frequency and gain (but not phase) • Frequency Response • Sine Wave Response because % differences are more significant than absolute differences. • Logarithmic axes • LogsofPowers + E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz • Straight Line Approximations even though both differences equal 5 Hz. • Plot Magnitude Response V2 • Low and High Frequency Logarithmic voltage ratios are specified in decibels (dB) = 20 log | | . Asymptotes 10 V1 • Phase Approximation + | | • Plot PhaseResponse + • RCR Circuit • Summary Common voltage ratios:

|V2| 0.1 0.5 1 2 10 100 |V1| dB 20 -6 0 6 20 40 −

Note that 0 does not exist on a log axis and so the starting point of the axis is arbitrary.

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 4 / 12 Logarithmic axes

11: Frequency Responses We usually use logarithmic axes for frequency and gain (but not phase) • Frequency Response • Sine Wave Response because % differences are more significant than absolute differences. • Logarithmic axes • LogsofPowers + E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz • Straight Line Approximations even though both differences equal 5 Hz. • Plot Magnitude Response V2 • Low and High Frequency Logarithmic voltage ratios are specified in decibels (dB) = 20 log | | . Asymptotes 10 V1 • Phase Approximation + | | • Plot PhaseResponse + • RCR Circuit • Summary Common voltage ratios:

|V2| 0.1 0.5 √0.5 1 √2 2 10 100 |V1| dB 20 -6 -3 0 3 6 20 40 −

Note that 0 does not exist on a log axis and so the starting point of the axis is arbitrary.

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 4 / 12 Logarithmic axes

11: Frequency Responses We usually use logarithmic axes for frequency and gain (but not phase) • Frequency Response • Sine Wave Response because % differences are more significant than absolute differences. • Logarithmic axes • LogsofPowers + E.g. 5 kHz versus 5.005 kHz is less significant than 10 Hz versus 15 Hz • Straight Line Approximations even though both differences equal 5 Hz. • Plot Magnitude Response V2 • Low and High Frequency Logarithmic voltage ratios are specified in decibels (dB) = 20 log | | . Asymptotes 10 V1 • Phase Approximation + | | • Plot PhaseResponse + • RCR Circuit • Summary Common voltage ratios:

|V2| 0.1 0.5 √0.5 1 √2 2 10 100 |V1| dB 20 -6 -3 0 3 6 20 40 −

Note that 0 does not exist on a log axis and so the starting point of the axis is arbitrary.

Note: P V 2 decibel power ratios are given by 10 log P2 ∝ ⇒ 10 P1 E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 4 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Phase (log-lin graph): ∠H = ∠jr + ∠c = r π (+π if c < 0) × 2

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Phase (log-lin graph): ∠H = ∠jr + ∠c = r π (+π if c < 0) × 2

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Phase (log-lin graph): ∠H = ∠jr + ∠c = r π (+π if c < 0) × 2 The phase is constant ω. ∀

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Phase (log-lin graph): ∠H = ∠jr + ∠c = r π (+π if c < 0) × 2 The phase is constant ω. ∀

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes c only affects the line’s vertical position. • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Phase (log-lin graph): ∠H = ∠jr + ∠c = r π (+π if c < 0) × 2 The phase is constant ω. ∀

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes c only affects the line’s vertical position. • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Phase (log-lin graph): ∠H = ∠jr + ∠c = r π (+π if c < 0) × 2 The phase is constant ω. ∀

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes c only affects the line’s vertical position. • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Phase (log-lin graph): ∠H = ∠jr + ∠c = r π (+π if c < 0) × 2 The phase is constant ω. ∀

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes c only affects the line’s vertical position. • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Phase (log-lin graph): ∠H = ∠jr + ∠c = r π (+π if c < 0) × 2 The phase is constant ω. ∀ If c > 0, phase = 90◦ magnitude slope. ×

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes c only affects the line’s vertical position. • Phase Approximation + • Plot PhaseResponse + If H is measured in decibels, a slope of r • RCR Circuit | | • Summary is called 6r dB/octave or 20r dB/decade.

Phase (log-lin graph): ∠H = ∠jr + ∠c = r π (+π if c < 0) × 2 The phase is constant ω. ∀ If c > 0, phase = 90◦ magnitude slope. ×

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes c only affects the line’s vertical position. • Phase Approximation + • Plot PhaseResponse + If H is measured in decibels, a slope of r • RCR Circuit | | • Summary is called 6r dB/octave or 20r dB/decade.

Phase (log-lin graph): ∠H = ∠jr + ∠c = r π (+π if c < 0) × 2 The phase is constant ω. ∀ If c > 0, phase = 90◦ magnitude slope. × Negative c adds 180◦ to the phase. ±

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response Suppose we plot the magnitude and phase of H = c (jω) • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes c only affects the line’s vertical position. • Phase Approximation + • Plot PhaseResponse + If H is measured in decibels, a slope of r • RCR Circuit | | • Summary is called 6r dB/octave or 20r dB/decade.

Phase (log-lin graph): ∠H = ∠jr + ∠c = r π (+π if c < 0) × 2 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 (2018-10340) Frequency Responses: 11 – 5 / 12 Logs of Powers +

11: Frequency Responses r • Frequency Response H = c (jω) has a straight-line magnitude graph and a constant phase. • Sine Wave Response • Logarithmic axes Magnitude (log-log graph): • LogsofPowers + r • Straight Line H = cω log H = log c +r log ω Approximations | | ⇒ | | | | • Plot Magnitude Response This is a straight line with a slope of r. • Low and High Frequency Asymptotes c only affects the line’s vertical position. • Phase Approximation + • Plot PhaseResponse + If H is measured in decibels, a slope of r • RCR Circuit | | • Summary is called 6r dB/octave or 20r dB/decade.

Phase (log-lin graph): ∠H = ∠jr + ∠c = r π (+π if c < 0) × 2 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 (2018-10340) Frequency Responses: 11 – 5 / 12 Straight Line Approximations

11: Frequency Responses • Frequency Response ajω for aω b • Sine Wave Response Key idea: (ajω + b) | |≫| | • Logarithmic axes ≈ (b for aω b • LogsofPowers + | |≪| | • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 6 / 12 Straight Line Approximations

11: Frequency Responses • Frequency Response ajω for aω b • Sine Wave Response Key idea: (ajω + b) | |≫| | • Logarithmic axes ≈ (b for aω b • LogsofPowers + | |≪| | • Straight Line 1 Approximations Gain: H(jω) = jωRC+1 • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 6 / 12 Straight Line Approximations

11: Frequency Responses • Frequency Response ajω for aω b • Sine Wave Response Key idea: (ajω + b) | |≫| | • Logarithmic axes ≈ (b for aω b • LogsofPowers + | |≪| | • Straight Line 1 Approximations Gain: H(jω) = jωRC+1 • Plot Magnitude Response • Low and High Frequency Asymptotes Low frequencies ( 1 ): • Phase Approximation + ω RC H(jω) 1 • Plot PhaseResponse + ≪ ≈ • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 6 / 12 Straight Line Approximations

11: Frequency Responses • Frequency Response ajω for aω b • Sine Wave Response Key idea: (ajω + b) | |≫| | • Logarithmic axes ≈ (b for aω b • LogsofPowers + | |≪| | • Straight Line 1 Approximations Gain: H(jω) = jωRC+1 • Plot Magnitude Response • Low and High Frequency Asymptotes Low frequencies ( 1 ): • Phase Approximation + ω RC H(jω) 1 • Plot PhaseResponse + ≪ 1 ≈ 1 High frequencies (ω RC ): H(jω) jωRC • RCR Circuit ≫ ≈ • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 6 / 12 Straight Line Approximations

11: Frequency Responses • Frequency Response ajω for aω b • Sine Wave Response Key idea: (ajω + b) | |≫| | • Logarithmic axes ≈ (b for aω b • LogsofPowers + | |≪| | • Straight Line 1 Approximations Gain: H(jω) = jωRC+1 • Plot Magnitude Response • Low and High Frequency Asymptotes Low frequencies ( 1 ): • Phase Approximation + ω RC H(jω) 1 • Plot PhaseResponse + ≪ 1 ≈ 1 High frequencies (ω RC ): H(jω) jωRC • RCR Circuit ≫ ≈ • Summary Approximate the magnitude response as two straight lines

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 6 / 12 Straight Line Approximations

11: Frequency Responses • Frequency Response ajω for aω b • Sine Wave Response Key idea: (ajω + b) | |≫| | • Logarithmic axes ≈ (b for aω b • LogsofPowers + | |≪| | • Straight Line 1 Approximations Gain: H(jω) = jωRC+1 • Plot Magnitude Response • Low and High Frequency Asymptotes Low frequencies ( 1 ): • Phase Approximation + ω RC H(jω) 1 H(jω) 1 • Plot PhaseResponse + ≪ 1 ≈ ⇒|1 | ≈ High frequencies (ω RC ): H(jω) jωRC • RCR Circuit ≫ ≈ • Summary Approximate the magnitude response as two straight lines

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 6 / 12 Straight Line Approximations

11: Frequency Responses • Frequency Response ajω for aω b • Sine Wave Response Key idea: (ajω + b) | |≫| | • Logarithmic axes ≈ (b for aω b • LogsofPowers + | |≪| | • Straight Line 1 Approximations Gain: H(jω) = jωRC+1 • Plot Magnitude Response • Low and High Frequency Asymptotes Low frequencies ( 1 ): • Phase Approximation + ω RC H(jω) 1 H(jω) 1 • Plot PhaseResponse + ≪ 1 ≈ ⇒|1 | ≈ 1 1 High frequencies (ω RC ): H(jω) jωRC H(jω) RC ω− • RCR Circuit ≫ ≈ ⇒| | ≈ • Summary Approximate the magnitude response as two straight lines

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 6 / 12 Straight Line Approximations

11: Frequency Responses • Frequency Response ajω for aω b • Sine Wave Response Key idea: (ajω + b) | |≫| | • Logarithmic axes ≈ (b for aω b • LogsofPowers + | |≪| | • Straight Line 1 Approximations Gain: H(jω) = jωRC+1 • Plot Magnitude Response • Low and High Frequency Asymptotes Low frequencies ( 1 ): • Phase Approximation + ω RC H(jω) 1 H(jω) 1 • Plot PhaseResponse + ≪ 1 ≈ ⇒|1 | ≈ 1 1 High frequencies (ω RC ): H(jω) jωRC H(jω) RC ω− • RCR Circuit ≫ ≈ ⇒| | ≈ • Summary Approximate the magnitude response as two straight lines

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 6 / 12 Straight Line Approximations

11: Frequency Responses • Frequency Response ajω for aω b • Sine Wave Response Key idea: (ajω + b) | |≫| | • Logarithmic axes ≈ (b for aω b • LogsofPowers + | |≪| | • Straight Line 1 Approximations Gain: H(jω) = jωRC+1 • Plot Magnitude Response • Low and High Frequency Asymptotes Low frequencies ( 1 ): • Phase Approximation + ω RC H(jω) 1 H(jω) 1 • Plot PhaseResponse + ≪ 1 ≈ ⇒|1 | ≈ 1 1 High frequencies (ω RC ): H(jω) jωRC H(jω) RC ω− • RCR Circuit ≫ ≈ ⇒| | ≈ • Summary Approximate the magnitude response as two straight lines intersecting at the 1 corner frequency, ωc = RC .

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 6 / 12 Straight Line Approximations

11: Frequency Responses • Frequency Response ajω for aω b • Sine Wave Response Key idea: (ajω + b) | |≫| | • Logarithmic axes ≈ (b for aω b • LogsofPowers + | |≪| | • Straight Line 1 Approximations Gain: H(jω) = jωRC+1 • Plot Magnitude Response • Low and High Frequency Asymptotes Low frequencies ( 1 ): • Phase Approximation + ω RC H(jω) 1 H(jω) 1 • Plot PhaseResponse + ≪ 1 ≈ ⇒|1 | ≈ 1 1 High frequencies (ω RC ): H(jω) jωRC H(jω) RC ω− • RCR Circuit ≫ ≈ ⇒| | ≈ • Summary Approximate the magnitude response as two straight lines intersecting at the 1 corner frequency, ωc = RC .

At the corner frequency:

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

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 6 / 12 Straight Line Approximations

11: Frequency Responses • Frequency Response ajω for aω b • Sine Wave Response Key idea: (ajω + b) | |≫| | • Logarithmic axes ≈ (b for aω b • LogsofPowers + | |≪| | • Straight Line 1 Approximations Gain: H(jω) = jωRC+1 • Plot Magnitude Response • Low and High Frequency Asymptotes Low frequencies ( 1 ): • Phase Approximation + ω RC H(jω) 1 H(jω) 1 • Plot PhaseResponse + ≪ 1 ≈ ⇒|1 | ≈ 1 1 High frequencies (ω RC ): H(jω) jωRC H(jω) RC ω− • RCR Circuit ≫ ≈ ⇒| | ≈ • Summary Approximate the magnitude response as two straight lines intersecting at the 1 corner frequency, ωc = RC .

At the corner frequency:

(a) the gradient changes by 1 (= 6 dB/octave = 20 dB/decade). − − − (b) H(jω ) = 1 = 1 = 3 dB (worst-case error). | c | 1+j √2 −

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 6 / 12 Straight Line Approximations

11: Frequency Responses • Frequency Response ajω for aω b • Sine Wave Response Key idea: (ajω + b) | |≫| | • Logarithmic axes ≈ (b for aω b • LogsofPowers + | |≪| | • Straight Line 1 Approximations Gain: H(jω) = jωRC+1 • Plot Magnitude Response • Low and High Frequency Asymptotes Low frequencies ( 1 ): • Phase Approximation + ω RC H(jω) 1 H(jω) 1 • Plot PhaseResponse + ≪ 1 ≈ ⇒|1 | ≈ 1 1 High frequencies (ω RC ): H(jω) jωRC H(jω) RC ω− • RCR Circuit ≫ ≈ ⇒| | ≈ • Summary Approximate the magnitude response as two straight lines intersecting at the 1 corner frequency, ωc = RC .

At the corner frequency:

(a) the gradient changes by 1 (= 6 dB/octave = 20 dB/decade). − − − (b) H(jω ) = 1 = 1 = 3 dB (worst-case error). | c | 1+j √2 −

A linear factor (ajω + b ) has a corner frequency of ω = b . c a

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 6 / 12 Plot Magnitude Response

11: Frequency Responses The gain of a linear circuit is always a rational polynomial in jω and is • Frequency Response • Sine Wave Response called the transfer function of the circuit. For example: • Logarithmic axes 2 • LogsofPowers + 60(jω) +720(jω) • Straight Line H(jω) = 3 2 Approximations 3(jω) +165(jω) +762(jω)+600 • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 7 / 12 Plot Magnitude Response

11: Frequency Responses The gain of a linear circuit is always a rational polynomial in jω and is • Frequency Response • Sine Wave Response called the transfer function of the circuit. For example: • Logarithmic axes 2 • LogsofPowers + 60(jω) +720(jω) 20jω(jω+12) • Straight Line H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) Approximations 3(jω) +165(jω) +762(jω)+600 • Plot Magnitude Response • Low and High Frequency Step 1: Factorize the polynomials Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 7 / 12 Plot Magnitude Response

11: Frequency Responses The gain of a linear circuit is always a rational polynomial in jω and is • Frequency Response • Sine Wave Response called the transfer function of the circuit. For example: • Logarithmic axes 2 • LogsofPowers + 60(jω) +720(jω) 20jω(jω+12) • Straight Line H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) Approximations 3(jω) +165(jω) +762(jω)+600 • Plot Magnitude Response • Low and High Frequency Step 1: Factorize the polynomials Asymptotes • Phase Approximation + Step 2: Sort corner freqs: 1, 4, 12, 50 • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 7 / 12 Plot Magnitude Response

11: Frequency Responses The gain of a linear circuit is always a rational polynomial in jω and is • Frequency Response • Sine Wave Response called the transfer function of the circuit. For example: • Logarithmic axes 2 • LogsofPowers + 60(jω) +720(jω) 20jω(jω+12) • Straight Line H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) Approximations 3(jω) +165(jω) +762(jω)+600 • Plot Magnitude Response • Low and High Frequency Step 1: Factorize the polynomials Asymptotes • Phase Approximation + Step 2: Sort corner freqs: 1, 4, 12, 50 • Plot PhaseResponse + • RCR Circuit Step 3: For ω < 1 all linear factors equal • Summary their constant terms: 20ω 12 1 H 1 4×50 = 1.2ω . | | ≈ × ×

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 7 / 12 Plot Magnitude Response

11: Frequency Responses The gain of a linear circuit is always a rational polynomial in jω and is • Frequency Response • Sine Wave Response called the transfer function of the circuit. For example: • Logarithmic axes 2 • LogsofPowers + 60(jω) +720(jω) 20jω(jω+12) • Straight Line H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) Approximations 3(jω) +165(jω) +762(jω)+600 • Plot Magnitude Response • Low and High Frequency Step 1: Factorize the polynomials Asymptotes • Phase Approximation + Step 2: Sort corner freqs: 1, 4, 12, 50 • Plot PhaseResponse + • RCR Circuit Step 3: For ω < 1 all linear factors equal • Summary their constant terms: 20ω 12 1 H 1 4×50 = 1.2ω . | | ≈ × × Step 4: For 1 < ω < 4, the factor (jω + 1) jω so 20ω 12 0 ≈ H ω 4× 50 = 1.2ω | | ≈ × ×

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 7 / 12 Plot Magnitude Response

11: Frequency Responses The gain of a linear circuit is always a rational polynomial in jω and is • Frequency Response • Sine Wave Response called the transfer function of the circuit. For example: • Logarithmic axes 2 • LogsofPowers + 60(jω) +720(jω) 20jω(jω+12) • Straight Line H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) Approximations 3(jω) +165(jω) +762(jω)+600 • Plot Magnitude Response • Low and High Frequency Step 1: Factorize the polynomials Asymptotes • Phase Approximation + Step 2: Sort corner freqs: 1, 4, 12, 50 • Plot PhaseResponse + • RCR Circuit Step 3: For ω < 1 all linear factors equal • Summary their constant terms: 20ω 12 1 H 1 4×50 = 1.2ω . | | ≈ × × Step 4: For 1 < ω < 4, the factor (jω + 1) jω so 20ω 12 0 ≈ H ω 4× 50 = 1.2ω = +1.58 dB. | | ≈ × ×

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 7 / 12 Plot Magnitude Response

11: Frequency Responses The gain of a linear circuit is always a rational polynomial in jω and is • Frequency Response • Sine Wave Response called the transfer function of the circuit. For example: • Logarithmic axes 2 • LogsofPowers + 60(jω) +720(jω) 20jω(jω+12) • Straight Line H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) Approximations 3(jω) +165(jω) +762(jω)+600 • Plot Magnitude Response • Low and High Frequency Step 1: Factorize the polynomials Asymptotes • Phase Approximation + Step 2: Sort corner freqs: 1, 4, 12, 50 • Plot PhaseResponse + • RCR Circuit Step 3: For ω < 1 all linear factors equal • Summary their constant terms: 20ω 12 1 H 1 4×50 = 1.2ω . | | ≈ × × Step 4: For 1 < ω < 4, the factor (jω + 1) jω so 20ω 12 0 ≈ H ω 4× 50 = 1.2ω = +1.58 dB. | | ≈ × × 20ω 12 1 Step 5: For 4 < ω < 12, H ω ω× 50 = 4.8ω− . | | ≈ × ×

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 7 / 12 Plot Magnitude Response

11: Frequency Responses The gain of a linear circuit is always a rational polynomial in jω and is • Frequency Response • Sine Wave Response called the transfer function of the circuit. For example: • Logarithmic axes 2 • LogsofPowers + 60(jω) +720(jω) 20jω(jω+12) • Straight Line H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) Approximations 3(jω) +165(jω) +762(jω)+600 • Plot Magnitude Response • Low and High Frequency Step 1: Factorize the polynomials Asymptotes • Phase Approximation + Step 2: Sort corner freqs: 1, 4, 12, 50 • Plot PhaseResponse + • RCR Circuit Step 3: For ω < 1 all linear factors equal • Summary their constant terms: 20ω 12 1 H 1 4×50 = 1.2ω . | | ≈ × × Step 4: For 1 < ω < 4, the factor (jω + 1) jω so 20ω 12 0 ≈ H ω 4× 50 = 1.2ω = +1.58 dB. | | ≈ × × 20ω 12 1 Step 5: For 4 < ω < 12, H ω ω× 50 = 4.8ω− . | | ≈ ×20ω× ω 0 Step 6: For 12 < ω < 50, H ω ω×50 = 0.4ω = 7.96 dB. | | ≈ × × −

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 7 / 12 Plot Magnitude Response

11: Frequency Responses The gain of a linear circuit is always a rational polynomial in jω and is • Frequency Response • Sine Wave Response called the transfer function of the circuit. For example: • Logarithmic axes 2 • LogsofPowers + 60(jω) +720(jω) 20jω(jω+12) • Straight Line H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) Approximations 3(jω) +165(jω) +762(jω)+600 • Plot Magnitude Response • Low and High Frequency Step 1: Factorize the polynomials Asymptotes • Phase Approximation + Step 2: Sort corner freqs: 1, 4, 12, 50 • Plot PhaseResponse + • RCR Circuit Step 3: For ω < 1 all linear factors equal • Summary their constant terms: 20ω 12 1 H 1 4×50 = 1.2ω . | | ≈ × × Step 4: For 1 < ω < 4, the factor (jω + 1) jω so 20ω 12 0 ≈ H ω 4× 50 = 1.2ω = +1.58 dB. | | ≈ × × 20ω 12 1 Step 5: For 4 < ω < 12, H ω ω× 50 = 4.8ω− . | | ≈ ×20ω× ω 0 Step 6: For 12 < ω < 50, H ω ω×50 = 0.4ω = 7.96 dB. | 20|ω ≈ ω × × 1 − Step 7: For ω > 50, H ω ω× ω = 20ω− . | | ≈ × ×

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 7 / 12 Plot Magnitude Response

11: Frequency Responses The gain of a linear circuit is always a rational polynomial in jω and is • Frequency Response • Sine Wave Response called the transfer function of the circuit. For example: • Logarithmic axes 2 • LogsofPowers + 60(jω) +720(jω) 20jω(jω+12) • Straight Line H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) Approximations 3(jω) +165(jω) +762(jω)+600 • Plot Magnitude Response • Low and High Frequency Step 1: Factorize the polynomials Asymptotes • Phase Approximation + Step 2: Sort corner freqs: 1, 4, 12, 50 • Plot PhaseResponse + • RCR Circuit Step 3: For ω < 1 all linear factors equal • Summary their constant terms: 20ω 12 1 H 1 4×50 = 1.2ω . | | ≈ × × Step 4: For 1 < ω < 4, the factor (jω + 1) jω so 20ω 12 0 ≈ H ω 4× 50 = 1.2ω = +1.58 dB. | | ≈ × × 20ω 12 1 Step 5: For 4 < ω < 12, H ω ω× 50 = 4.8ω− . | | ≈ ×20ω× ω 0 Step 6: For 12 < ω < 50, H ω ω×50 = 0.4ω = 7.96 dB. | 20|ω ≈ ω × × 1 − Step 7: For ω > 50, H ω ω× ω = 20ω− . | | ≈ × × 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 (2018-10340) Frequency Responses: 11 – 7 / 12 Low and High Frequency Asymptotes

11: Frequency Responses You can find the low and high frequency asymptotes without factorizing: • Frequency Response 2 • Sine Wave Response 60(jω) +720(jω) 20jω(jω+12) H(jω) = 3 2 = • Logarithmic axes 3(jω) +165(jω) +762(jω)+600 (jω+1)(jω+4)(jω+50) • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 8 / 12 Low and High Frequency Asymptotes

11: Frequency Responses You can find the low and high frequency asymptotes without factorizing: • Frequency Response 2 • Sine Wave Response 60(jω) +720(jω) 20jω(jω+12) H(jω) = 3 2 = • Logarithmic axes 3(jω) +165(jω) +762(jω)+600 (jω+1)(jω+4)(jω+50) • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 8 / 12 Low and High Frequency Asymptotes

11: Frequency Responses You can find the low and high frequency asymptotes without factorizing: • Frequency Response 2 • Sine Wave Response 60(jω) +720(jω) 20jω(jω+12) H(jω) = 3 2 = • Logarithmic axes 3(jω) +165(jω) +762(jω)+600 (jω+1)(jω+4)(jω+50) • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Low Frequency Asymptote:

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 8 / 12 Low and High Frequency Asymptotes

11: Frequency Responses You can find the low and high frequency asymptotes without factorizing: • Frequency Response 2 • Sine Wave Response 60(jω) +720(jω) 20jω(jω+12) H(jω) = 3 2 = • Logarithmic axes 3(jω) +165(jω) +762(jω)+600 (jω+1)(jω+4)(jω+50) • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Low Frequency Asymptote: 20jω(12) From factors: HLF(jω) = (1)(4)(50) = 1.2jω

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 8 / 12 Low and High Frequency Asymptotes

11: Frequency Responses You can find the low and high frequency asymptotes without factorizing: • Frequency Response 2 • Sine Wave Response 60(jω) +720(jω) 20jω(jω+12) H(jω) = 3 2 = • Logarithmic axes 3(jω) +165(jω) +762(jω)+600 (jω+1)(jω+4)(jω+50) • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Low Frequency Asymptote: 20jω(12) From factors: HLF(jω) = (1)(4)(50) = 1.2jω Lowest power of jω on top and bottom: H (jω) 720(jω) = 1.2jω ≃ 600

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 8 / 12 Low and High Frequency Asymptotes

11: Frequency Responses You can find the low and high frequency asymptotes without factorizing: • Frequency Response 2 • Sine Wave Response 60(jω) +720(jω) 20jω(jω+12) H(jω) = 3 2 = • Logarithmic axes 3(jω) +165(jω) +762(jω)+600 (jω+1)(jω+4)(jω+50) • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Low Frequency Asymptote: 20jω(12) From factors: HLF(jω) = (1)(4)(50) = 1.2jω Lowest power of jω on top and bottom: H (jω) 720(jω) = 1.2jω ≃ 600 High Frequency Asymptote:

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 8 / 12 Low and High Frequency Asymptotes

11: Frequency Responses You can find the low and high frequency asymptotes without factorizing: • Frequency Response 2 • Sine Wave Response 60(jω) +720(jω) 20jω(jω+12) H(jω) = 3 2 = • Logarithmic axes 3(jω) +165(jω) +762(jω)+600 (jω+1)(jω+4)(jω+50) • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Low Frequency Asymptote: 20jω(12) From factors: HLF(jω) = (1)(4)(50) = 1.2jω Lowest power of jω on top and bottom: H (jω) 720(jω) = 1.2jω ≃ 600 High Frequency Asymptote: 20jω(jω) 1 − From factors: HHF(jω) = (jω)(jω)(jω) = 20(jω)

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 8 / 12 Low and High Frequency Asymptotes

11: Frequency Responses You can find the low and high frequency asymptotes without factorizing: • Frequency Response 2 • Sine Wave Response 60(jω) +720(jω) 20jω(jω+12) H(jω) = 3 2 = • Logarithmic axes 3(jω) +165(jω) +762(jω)+600 (jω+1)(jω+4)(jω+50) • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Low Frequency Asymptote: 20jω(12) From factors: HLF(jω) = (1)(4)(50) = 1.2jω Lowest power of jω on top and bottom: H (jω) 720(jω) = 1.2jω ≃ 600 High Frequency Asymptote: 20jω(jω) 1 − From factors: HHF(jω) = (jω)(jω)(jω) = 20(jω) 2 60(jω) 1 Highest power of jω on top and bottom: H (jω) 3 = 20(jω)− ≃ 3(jω)

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 8 / 12 Phase Approximation +

11: Frequency Responses Gain: 1 • Frequency Response H(jω) = jωRC+1 • Sine Wave Response • Logarithmic axes • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 9 / 12 Phase Approximation +

11: Frequency Responses Gain: 1 • Frequency Response H(jω) = jωRC+1 • Sine Wave Response • Logarithmic axes Low frequencies (ω 1 ): • LogsofPowers + RC • Straight Line ≪ Approximations H(jω) 1 • Plot Magnitude Response ≈ • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 9 / 12 Phase Approximation +

11: Frequency Responses Gain: 1 • Frequency Response H(jω) = jωRC+1 • Sine Wave Response • Logarithmic axes Low frequencies (ω 1 ): • LogsofPowers + RC • Straight Line ≪ Approximations H(jω) 1 ∠1 = 0 • Plot Magnitude Response ≈ ⇒ • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 9 / 12 Phase Approximation +

11: Frequency Responses Gain: 1 • Frequency Response H(jω) = jωRC+1 • Sine Wave Response • Logarithmic axes Low frequencies (ω 1 ): • LogsofPowers + RC • Straight Line ≪ Approximations H(jω) 1 ∠1 = 0 • Plot Magnitude Response ≈ ⇒ • Low and High Frequency Asymptotes High frequencies ( 1 ): 1 • Phase Approximation + ω RC H(jω) jωRC • Plot PhaseResponse + ≫ ≈ • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 9 / 12 Phase Approximation +

11: Frequency Responses Gain: 1 • Frequency Response H(jω) = jωRC+1 • Sine Wave Response • Logarithmic axes Low frequencies (ω 1 ): • LogsofPowers + RC • Straight Line ≪ Approximations H(jω) 1 ∠1 = 0 • Plot Magnitude Response ≈ ⇒ • Low and High Frequency Asymptotes High frequencies ( 1 ): 1 ∠ 1 π • Phase Approximation + ω RC H(jω) jωRC j− = 2 • Plot PhaseResponse + ≫ ≈ ⇒ − • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 9 / 12 Phase Approximation +

11: Frequency Responses Gain: 1 • Frequency Response H(jω) = jωRC+1 • Sine Wave Response • Logarithmic axes Low frequencies (ω 1 ): • LogsofPowers + RC • Straight Line ≪ Approximations H(jω) 1 ∠1 = 0 • Plot Magnitude Response ≈ ⇒ • Low and High Frequency Asymptotes High frequencies ( 1 ): 1 ∠ 1 π • Phase Approximation + ω RC H(jω) jωRC j− = 2 • Plot PhaseResponse + ≫ ≈ ⇒ − • RCR Circuit Approximate the phase response as • Summary three straight lines.

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 9 / 12 Phase Approximation +

11: Frequency Responses Gain: 1 • Frequency Response H(jω) = jωRC+1 • Sine Wave Response • Logarithmic axes Low frequencies (ω 1 ): • LogsofPowers + RC • Straight Line ≪ Approximations H(jω) 1 ∠1 = 0 • Plot Magnitude Response ≈ ⇒ • Low and High Frequency Asymptotes High frequencies ( 1 ): 1 ∠ 1 π • Phase Approximation + ω RC H(jω) jωRC j− = 2 • Plot PhaseResponse + ≫ ≈ ⇒ − • RCR Circuit Approximate the phase response as • Summary three straight lines.

By chance, they intersect close to 1 0.1ωc and 10ωc where ωc = RC .

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 9 / 12 Phase Approximation +

11: Frequency Responses Gain: 1 • Frequency Response H(jω) = jωRC+1 • Sine Wave Response • Logarithmic axes Low frequencies (ω 1 ): • LogsofPowers + RC • Straight Line ≪ Approximations H(jω) 1 ∠1 = 0 • Plot Magnitude Response ≈ ⇒ • Low and High Frequency Asymptotes High frequencies ( 1 ): 1 ∠ 1 π • Phase Approximation + ω RC H(jω) jωRC j− = 2 • Plot PhaseResponse + ≫ ≈ ⇒ − • RCR Circuit Approximate the phase response as • Summary three straight lines.

By chance, they intersect close to 1 0.1ωc and 10ωc where ωc = RC .

π Between 0.1ωc and 10ωc the phase changes by 2 over two decades. This gives a gradient = π radians/decade. − − 4

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 9 / 12 Phase Approximation +

11: Frequency Responses Gain: 1 • Frequency Response H(jω) = jωRC+1 • Sine Wave Response • Logarithmic axes Low frequencies (ω 1 ): • LogsofPowers + RC • Straight Line ≪ Approximations H(jω) 1 ∠1 = 0 • Plot Magnitude Response ≈ ⇒ • Low and High Frequency Asymptotes High frequencies ( 1 ): 1 ∠ 1 π • Phase Approximation + ω RC H(jω) jωRC j− = 2 • Plot PhaseResponse + ≫ ≈ ⇒ − • RCR Circuit Approximate the phase response as • Summary three straight lines.

By chance, they intersect close to 1 0.1ωc and 10ωc where ωc = RC .

π Between 0.1ωc and 10ωc the phase changes by 2 over two decades. This gives a gradient = π radians/decade. − − 4 (ajω + b) in denominator π 1 b ∆gradient = /decade at ω = 10∓ . ⇒ ∓ 4 a

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 9 / 12 Phase Approximation +

11: Frequency Responses Gain: 1 • Frequency Response H(jω) = jωRC+1 • Sine Wave Response • Logarithmic axes Low frequencies (ω 1 ): • LogsofPowers + RC • Straight Line ≪ Approximations H(jω) 1 ∠1 = 0 • Plot Magnitude Response ≈ ⇒ • Low and High Frequency Asymptotes High frequencies ( 1 ): 1 ∠ 1 π • Phase Approximation + ω RC H(jω) jωRC j− = 2 • Plot PhaseResponse + ≫ ≈ ⇒ − • RCR Circuit Approximate the phase response as • Summary three straight lines.

By chance, they intersect close to 1 0.1ωc and 10ωc where ωc = RC .

π Between 0.1ωc and 10ωc the phase changes by 2 over two decades. This gives a gradient = π radians/decade. − − 4 (ajω + b) in denominator π 1 b ∆gradient = /decade at ω = 10∓ . ⇒ ∓ 4 a The sign of ∆gradient is reversed for (a) numerator factors and (b) b < 0. a E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 9 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) • Frequency Response H(jω) = 3 2 • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −
Step 4: Put in ascending order and calculate gaps as ω2 decades: log10 ω1 + + + + .1− (.6) .4− (.48) 1.2 (.62) 5− (.3) 10 (.6) 40 (.48) 120− (.62) 500 .

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −
Step 4: Put in ascending order and calculate gaps as ω2 decades: log10 ω1 + + + + .1− (.6) .4− (.48) 1.2 (.62) 5− (.3) 10 (.6) 40 (.48) 120− (.62) 500 . ∠ π Step 5: Find phase of LF asymptote: 1.2jω = + 2 .

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −
Step 4: Put in ascending order and calculate gaps as ω2 decades: log10 ω1 + + + + .1− (.6) .4− (.48) 1.2 (.62) 5− (.3) 10 (.6) 40 (.48) 120− (.62) 500 . ∠ π Step 5: Find phase of LF asymptote: 1.2jω = + 2 .

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −
Step 4: Put in ascending order and calculate gaps as ω2 decades: log10 ω1 + + + + .1− (.6) .4− (.48) 1.2 (.62) 5− (.3) 10 (.6) 40 (.48) 120− (.62) 500 . ∠ π Step 5: Find phase of LF asymptote: 1.2jω = + 2 . Step 6: At ω = 0.1 gradient becomes π rad/decade. φ is still π . − 4 2

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −
Step 4: Put in ascending order and calculate gaps as ω2 decades: log10 ω1 + + + + .1− (.6) .4− (.48) 1.2 (.62) 5− (.3) 10 (.6) 40 (.48) 120− (.62) 500 . ∠ π Step 5: Find phase of LF asymptote: 1.2jω = + 2 . Step 6: At ω = 0.1 gradient becomes π rad/decade. φ is still π . − 4 2 Step 7: At ω = 0.4, φ = π 0.6 π = 0.35π. New gradient is π . 2 − 4 − 2

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −
Step 4: Put in ascending order and calculate gaps as ω2 decades: log10 ω1 + + + + .1− (.6) .4− (.48) 1.2 (.62) 5− (.3) 10 (.6) 40 (.48) 120− (.62) 500 . ∠ π Step 5: Find phase of LF asymptote: 1.2jω = + 2 . Step 6: At ω = 0.1 gradient becomes π rad/decade. φ is still π . − 4 2 Step 7: At ω = 0.4, φ = π 0.6 π = 0.35π. New gradient is π . 2 − 4 − 2 Step 8: At ω = 1.2, φ = 0.35π 0.48 π = 0.11π. New gradient is π . − 2 − 4

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −
Step 4: Put in ascending order and calculate gaps as ω2 decades: log10 ω1 + + + + .1− (.6) .4− (.48) 1.2 (.62) 5− (.3) 10 (.6) 40 (.48) 120− (.62) 500 . ∠ π Step 5: Find phase of LF asymptote: 1.2jω = + 2 . π π Step 6: At ω = 0.1 gradient becomes 4 rad/decade. φ is still 2 . π π − π Step 7: At ω = 0.4, φ = 2 0.6 4 = 0.35π. New gradient is 2 . − π − π Step 8: At ω = 1.2, φ = 0.35π 0.48 2 = 0.11π. New gradient is 4 . Steps 9-13: Repeat for each gradient− change. −

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −
Step 4: Put in ascending order and calculate gaps as ω2 decades: log10 ω1 + + + + .1− (.6) .4− (.48) 1.2 (.62) 5− (.3) 10 (.6) 40 (.48) 120− (.62) 500 . ∠ π Step 5: Find phase of LF asymptote: 1.2jω = + 2 . π π Step 6: At ω = 0.1 gradient becomes 4 rad/decade. φ is still 2 . π π − π Step 7: At ω = 0.4, φ = 2 0.6 4 = 0.35π. New gradient is 2 . − π − π Step 8: At ω = 1.2, φ = 0.35π 0.48 2 = 0.11π. New gradient is 4 . Steps 9-13: Repeat for each gradient− change. −

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −
Step 4: Put in ascending order and calculate gaps as ω2 decades: log10 ω1 + + + + .1− (.6) .4− (.48) 1.2 (.62) 5− (.3) 10 (.6) 40 (.48) 120− (.62) 500 . ∠ π Step 5: Find phase of LF asymptote: 1.2jω = + 2 . π π Step 6: At ω = 0.1 gradient becomes 4 rad/decade. φ is still 2 . π π − π Step 7: At ω = 0.4, φ = 2 0.6 4 = 0.35π. New gradient is 2 . − π − π Step 8: At ω = 1.2, φ = 0.35π 0.48 2 = 0.11π. New gradient is 4 . Steps 9-13: Repeat for each gradient− change. −

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −
Step 4: Put in ascending order and calculate gaps as ω2 decades: log10 ω1 + + + + .1− (.6) .4− (.48) 1.2 (.62) 5− (.3) 10 (.6) 40 (.48) 120− (.62) 500 . ∠ π Step 5: Find phase of LF asymptote: 1.2jω = + 2 . π π Step 6: At ω = 0.1 gradient becomes 4 rad/decade. φ is still 2 . π π − π Step 7: At ω = 0.4, φ = 2 0.6 4 = 0.35π. New gradient is 2 . − π − π Step 8: At ω = 1.2, φ = 0.35π 0.48 2 = 0.11π. New gradient is 4 . Steps 9-13: Repeat for each gradient− change. −

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −
Step 4: Put in ascending order and calculate gaps as ω2 decades: log10 ω1 + + + + .1− (.6) .4− (.48) 1.2 (.62) 5− (.3) 10 (.6) 40 (.48) 120− (.62) 500 . ∠ π Step 5: Find phase of LF asymptote: 1.2jω = + 2 . π π Step 6: At ω = 0.1 gradient becomes 4 rad/decade. φ is still 2 . π π − π Step 7: At ω = 0.4, φ = 2 0.6 4 = 0.35π. New gradient is 2 . − π − π Step 8: At ω = 1.2, φ = 0.35π 0.48 2 = 0.11π. New gradient is 4 . Steps 9-13: Repeat for each gradient− change. Final gradient is always− 0.

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 Plot Phase Response +

2 11: Frequency Responses 60(jω) +720(jω) 20jω(jω+12) • Frequency Response H(jω) = 3 2 = (jω+1)(jω+4)(jω+50) • Sine Wave Response 3(jω) +165(jω) +762(jω)+600 • Logarithmic axes • LogsofPowers + Step 1: Factorize the polynomials • Straight Line Approximations Step 2: List corner freqs: = num/den • Plot Magnitude Response + ± • Low and High Frequency ωc = 1−, 4−, 12 , 50− Asymptotes { } 1 • Phase Approximation + Step 3: Gradient changes at 10∓ ωc. • Plot PhaseResponse + Sign depends on num/den and sgn b : • RCR Circuit a + + + + • Summary .1 , 10 ; .4 , 40 ; 1.2 , 120 ; 5 , 500 − − − −
Step 4: Put in ascending order and calculate gaps as ω2 decades: log10 ω1 + + + + .1− (.6) .4− (.48) 1.2 (.62) 5− (.3) 10 (.6) 40 (.48) 120− (.62) 500 . ∠ π Step 5: Find phase of LF asymptote: 1.2jω = + 2 . π π Step 6: At ω = 0.1 gradient becomes 4 rad/decade. φ is still 2 . π π − π Step 7: At ω = 0.4, φ = 2 0.6 4 = 0.35π. New gradient is 2 . − π − π Step 8: At ω = 1.2, φ = 0.35π 0.48 2 = 0.11π. New gradient is 4 . 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 π rad/decade. ± 4

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 10 / 12 RCR Circuit

11: Frequency Responses • Frequency Response • Sine Wave Response • Logarithmic axes • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 RCR Circuit

11: Frequency Responses R+ 1 • Y jωC Frequency Response = 1 • Sine Wave Response X 3R+R+ jωC • Logarithmic axes • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 RCR Circuit

11: Frequency Responses R+ 1 • Y jωC jωRC+1 Frequency Response = 1 = • Sine Wave Response X 3R+R+ jωC 4jωRC+1 • Logarithmic axes • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 RCR Circuit

11: Frequency Responses R+ 1 • Y jωC jωRC+1 Frequency Response = 1 = • Sine Wave Response X 3R+R+ jωC 4jωRC+1 • Logarithmic axes • + LogsofPowers + Corner freqs: 0.25 − 1 • Straight Line RC , RC Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 RCR Circuit

11: Frequency Responses R+ 1 • Y jωC jωRC+1 Frequency Response = 1 = • Sine Wave Response X 3R+R+ jωC 4jωRC+1 • Logarithmic axes • + LogsofPowers + Corner freqs: 0.25 − 1 LF Asymptote: • Straight Line RC , RC H(jω) = 1 Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 RCR Circuit

11: Frequency Responses R+ 1 • Y jωC jωRC+1 Frequency Response = 1 = • Sine Wave Response X 3R+R+ jωC 4jωRC+1 • Logarithmic axes • + LogsofPowers + Corner freqs: 0.25 − 1 LF Asymptote: • Straight Line RC , RC H(jω) = 1 Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Magnitude Response:

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 RCR Circuit

11: Frequency Responses R+ 1 • Y jωC jωRC+1 Frequency Response = 1 = • Sine Wave Response X 3R+R+ jωC 4jωRC+1 • Logarithmic axes • + LogsofPowers + Corner freqs: 0.25 − 1 LF Asymptote: • Straight Line RC , RC H(jω) = 1 Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Magnitude Response: Gradient Changes: 20 dB/dec at ω = 0.25 and +20 at ω = 1 − RC RC

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 RCR Circuit

11: Frequency Responses R+ 1 • Y jωC jωRC+1 Frequency Response = 1 = • Sine Wave Response X 3R+R+ jωC 4jωRC+1 • Logarithmic axes • + LogsofPowers + Corner freqs: 0.25 − 1 LF Asymptote: • Straight Line RC , RC H(jω) = 1 Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Magnitude Response: Gradient Changes: 20 dB/dec at ω = 0.25 and +20 at ω = 1 − RC RC

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 RCR Circuit

11: Frequency Responses R+ 1 • Y jωC jωRC+1 Frequency Response = 1 = • Sine Wave Response X 3R+R+ jωC 4jωRC+1 • Logarithmic axes • + LogsofPowers + Corner freqs: 0.25 − 1 LF Asymptote: • Straight Line RC , RC H(jω) = 1 Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Magnitude Response: 0.25 1 Gradient Changes: 20 dB/dec at ω = RC and +20 at ω = RC − 1 jωRC Line equations: H(jω) = (a) 1, (b) 4jωRC , (c) 4jωRC = 0.25

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 RCR Circuit

11: Frequency Responses R+ 1 • Y jωC jωRC+1 Frequency Response = 1 = • Sine Wave Response X 3R+R+ jωC 4jωRC+1 • Logarithmic axes • + LogsofPowers + Corner freqs: 0.25 − 1 LF Asymptote: • Straight Line RC , RC H(jω) = 1 Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Magnitude Response: 0.25 1 Gradient Changes: 20 dB/dec at ω = RC and +20 at ω = RC − 1 jωRC Line equations: H(jω) = (a) 1, (b) 4jωRC , (c) 4jωRC = 0.25 Phase Response:

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 RCR Circuit

11: Frequency Responses R+ 1 • Y jωC jωRC+1 Frequency Response = 1 = • Sine Wave Response X 3R+R+ jωC 4jωRC+1 • Logarithmic axes • + LogsofPowers + Corner freqs: 0.25 − 1 LF Asymptote: • Straight Line RC , RC H(jω) = 1 Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Magnitude Response: 0.25 1 Gradient Changes: 20 dB/dec at ω = RC and +20 at ω = RC − 1 jωRC Line equations: H(jω) = (a) 1, (b) 4jωRC , (c) 4jωRC = 0.25 Phase Response: LF asymptote: φ = ∠1 = 0

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 RCR Circuit

11: Frequency Responses R+ 1 • Y jωC jωRC+1 Frequency Response = 1 = • Sine Wave Response X 3R+R+ jωC 4jωRC+1 • Logarithmic axes • + LogsofPowers + Corner freqs: 0.25 − 1 LF Asymptote: • Straight Line RC , RC H(jω) = 1 Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Magnitude Response: 0.25 1 Gradient Changes: 20 dB/dec at ω = RC and +20 at ω = RC − 1 jωRC Line equations: H(jω) = (a) 1, (b) 4jωRC , (c) 4jωRC = 0.25 Phase Response: LF asymptote: φ = ∠1 = 0 + + Gradient changes of π /decade at: ω = 0.025 −, 0.1 , 2.5 , 10 −. ± 4 RC RC RC RC

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 RCR Circuit

11: Frequency Responses R+ 1 • Y jωC jωRC+1 Frequency Response = 1 = • Sine Wave Response X 3R+R+ jωC 4jωRC+1 • Logarithmic axes • + LogsofPowers + Corner freqs: 0.25 − 1 LF Asymptote: • Straight Line RC , RC H(jω) = 1 Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

Magnitude Response: 0.25 1 Gradient Changes: 20 dB/dec at ω = RC and +20 at ω = RC − 1 jωRC Line equations: H(jω) = (a) 1, (b) 4jωRC , (c) 4jωRC = 0.25 Phase Response: LF asymptote: φ = ∠1 = 0 + + Gradient changes of π /decade at: ω = 0.025 −, 0.1 , 2.5 , 10 −. ± 4 RC RC RC RC At ω = 0.1 , φ = 0 π log 0.1 = π 0.602 = 0.15π RC − 4 10 0.025 − 4 × −

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 11 / 12 Summary

11: Frequency Responses Y • Frequency Response Frequency response: magnitude and phase of X as a function of ω • • Sine Wave Response Only applies to sine waves • Logarithmic axes ◦ • LogsofPowers + • Straight Line Approximations • Plot Magnitude Response • Low and High Frequency Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 12 / 12 Summary

11: Frequency Responses Y • Frequency Response Frequency response: magnitude and phase of X as a function of ω • • Sine Wave Response Only applies to sine waves • Logarithmic axes ◦ • LogsofPowers + • Straight Line Use log axes for frequency and gain but linear for phase Approximations ◦ V2 P2 • Plot Magnitude Response ⊲ Decibels = 20 log = 10 log • Low and High Frequency 10 V1 10 P1 Asymptotes • Phase Approximation + • Plot PhaseResponse + • RCR Circuit • Summary

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 12 / 12 Summary

11: Frequency Responses Y • Frequency Response Frequency response: magnitude and phase of X as a function of ω • • Sine Wave Response Only applies to sine waves • Logarithmic axes ◦ • LogsofPowers + • Straight Line Use log axes for frequency and gain but linear for phase Approximations ◦ V2 P2 • Plot Magnitude Response ⊲ Decibels = 20 log = 10 log • Low and High Frequency 10 V1 10 P1 Asymptotes • Phase Approximation + b Linear factor (ajω + b) gives corner frequency at ω = a . • Plot PhaseResponse + • • RCR Circuit Magnitude plot gradient changes by 20 dB/decade @ ω = b . • Summary ◦ ± a

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 12 / 12 Summary

11: Frequency Responses Y • Frequency Response Frequency response: magnitude and phase of X as a function of ω • • Sine Wave Response Only applies to sine waves • Logarithmic axes ◦ • LogsofPowers + • Straight Line Use log axes for frequency and gain but linear for phase Approximations ◦ V2 P2 • Plot Magnitude Response ⊲ Decibels = 20 log = 10 log • Low and High Frequency 10 V1 10 P1 Asymptotes • Phase Approximation + b Linear factor (ajω + b) gives corner frequency at ω = a . • Plot PhaseResponse + • • RCR Circuit Magnitude plot gradient changes by 20 dB/decade @ ω = b . • Summary ◦ ± a Phase gradient changes in two places by: ◦ ⊲ π rad/decade @ ω = 0.1 b ± 4 × a π b ⊲ rad/decade @ ω = 10 ∓ 4 × a

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 12 / 12 Summary

11: Frequency Responses Y • Frequency Response Frequency response: magnitude and phase of X as a function of ω • • Sine Wave Response Only applies to sine waves • Logarithmic axes ◦ • LogsofPowers + • Straight Line Use log axes for frequency and gain but linear for phase Approximations ◦ V2 P2 • Plot Magnitude Response ⊲ Decibels = 20 log = 10 log • Low and High Frequency 10 V1 10 P1 Asymptotes • Phase Approximation + b Linear factor (ajω + b) gives corner frequency at ω = a . • Plot PhaseResponse + • • RCR Circuit Magnitude plot gradient changes by 20 dB/decade @ ω = b . • Summary ◦ ± a Phase gradient changes in two places by: ◦ ⊲ π rad/decade @ ω = 0.1 b ± 4 × a π b ⊲ 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 (2018-10340) Frequency Responses: 11 – 12 / 12 Summary

11: Frequency Responses Y • Frequency Response Frequency response: magnitude and phase of X as a function of ω • • Sine Wave Response Only applies to sine waves • Logarithmic axes ◦ • LogsofPowers + • Straight Line Use log axes for frequency and gain but linear for phase Approximations ◦ V2 P2 • Plot Magnitude Response ⊲ Decibels = 20 log = 10 log • Low and High Frequency 10 V1 10 P1 Asymptotes • Phase Approximation + b Linear factor (ajω + b) gives corner frequency at ω = a . • Plot PhaseResponse + • • RCR Circuit Magnitude plot gradient changes by 20 dB/decade @ ω = b . • Summary ◦ ± a Phase gradient changes in two places by: ◦ ⊲ π rad/decade @ ω = 0.1 b ± 4 × a π b ⊲ 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 Ch 16 or Irwin Ch 12.

E1.1 Analysis of Circuits (2018-10340) Frequency Responses: 11 – 12 / 12