Lect20EEE 2021 Spectrum Representations; Frequency Response Dr. Holbert April 14, 2008

Lect20EEE 2022 Variable-Frequency Response Analysis As an extension of AC analysis, we now vary the frequency and observe the circuit behavior Graphical display of frequency dependent circuit behavior can be very useful; however, quantities such as the impedance are complex valued such that we will tend to graph the magnitude of the impedance versus frequency (i.e., |Z(j  )| v. f) and the phase angle versus frequency (i.e.,  Z(j  ) v. f)

Lect20EEE 2023 Frequency Response of a Resistor Consider the frequency dependent impedance of the resistor, inductor and capacitor circuit elements Resistor (R):Z R = R  0° –So the magnitude and phase angle of the resistor impedance are constant, such that plotting them versus frequency yields Magnitude of Z R (  ) Frequency R Phase of Z R (°) Frequency 0°

Lect20EEE 2024 Frequency Response of an Inductor Inductor (L):Z L =  L  90° –The phase angle of the inductor impedance is a constant 90°, but the magnitude of the inductor impedance is directly proportional to the frequency. Plotting them vs. frequency yields (note that the inductor appears as a short circuit at dc) Magnitude of Z L (  ) Frequency Phase of Z L (°) Frequency 90°

Lect20EEE 2025 Frequency Response of a Capacitor Capacitor (C):Z C = 1/(  C)  –90° –The phase angle of the capacitor impedance is –90°, but the magnitude of the inductor impedance is inversely proportional to the frequency. Plotting both vs. frequency yields (note that the capacitor acts as an open circuit at dc) Magnitude of Z C (  ) Frequency Phase of Z C (°) Frequency -90°

Lect20EEE 2026 Transfer Function Recall that the transfer function, H(s), is The transfer function can be shown in a block diagram as The transfer function can be separated into magnitude and phase angle information H(j  ) = |H(j  )|  H(j  ) H(j  ) = H(s) X(j  ) e j  t = X(s) e st Y(j  ) e j  t = Y(s) e st

Lect20EEE 2027 Poles and Zeros The transfer function is a ratio of polynomials The roots of the numerator, N(s), are called the zeros since they cause the transfer function H(s) to become zero, i.e., H(z i )=0 The roots of the denominator, D(s), are called the poles and they cause the transfer function H(s) to become infinity, i.e., H(p i )= 

Lect20EEE 2028 Resonant Circuits Resonant frequency: the frequency at which the impedance of a series RLC circuit or the admittance of a parallel RLC circuit is purely real, i.e., the imaginary term is zero (ωL=1/ωC) For both series and parallel RLC circuits, the resonance frequency is At resonance the voltage and current are in phase, (i.e., zero phase angle) and the power factor is unity

Lect20EEE 2029 Quality Factor (Q) An energy analysis of a RLC circuit provides a basic definition of the quality factor (Q) that is used across engineering disciplines, specifically: The quality factor is a measure of the sharpness of the resonance peak; the larger the Q value, the sharper the peak where BW=bandwidth

Lect20EEE Bandwidth (BW) The bandwidth (BW) is the difference between the two half-power frequencies BW = ω HI – ω LO =  0 / Q Hence, a high-Q circuit has a small bandwidth Note that:  0 2 = ω LO ω HI

Lect20EEE Quality Factor: RLC Circuits For a series RLC circuit the quality factor is For a parallel RLC circuit, the quality factor is

Lect20EEE Class Examples Drill Problems P9-3, P9-4, P9-5 Use MATLAB or Excel to create the Bode plots (both magnitude and phase) for the above; we’ll make hand plots next time –Start Excel and open the file BodePlot.xls from the class webpage, -or- –Start MATLAB and open the file EEE202BodePlt.m from the class webpage