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