EEE 302 Electrical Networks II: Lecture 22 1 EEE 302 Electrical Networks II Dr. Keith E. Holbert Summer 2001
Resonant Circuits: Lecture 22 2 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
Quality Factor (Q): Lecture 22 3 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
Bandwidth (BW): Lecture 22 4 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 See Figs. 12.23 and 12.24 in textbook (p. 692 & 694)
Series RLC Circuit: Lecture 22 5 Series RLC Circuit For a series RLC circuit the quality factor is
PowerPoint Presentation: Lecture 22 6 Class Examples Extension Exercise E12.8 Extension Exercise E12.9 Extension Exercise E12.10 Extension Exercise E12.11 Extension Exercise E12.12
Parallel RLC Circuit: Lecture 22 7 Parallel RLC Circuit For a parallel RLC circuit, the quality factor is
PowerPoint Presentation: Lecture 22 8 Class Example Extension Exercise E12.13
Scaling: Lecture 22 9 Scaling Two methods of scaling: 1) Magnitude (or impedance) scaling multiplies the impedance by a scalar, K M resonant frequency, bandwidth, quality factor are un affected 2) Frequency scaling multiplies the frequency by a scalar, ω'=K F ω resonant frequency, bandwidth, quality factor are affected
Magnitude Scaling: Lecture 22 10 Magnitude Scaling Magnitude scaling multiplies the impedance by a scalar, that is, Z new = Z old K M Resistor: Z R’ = K M Z R = K M R R’ = K M R Inductor: Z L’ = K M Z L = K M j L L’ = K M L Capacitor: Z C’ = K M Z C = K M / ( j C ) C’ = C / K M
Frequency Scaling: Lecture 22 11 Frequency Scaling Frequency scaling multiplies the frequency by a scalar, that is, ω new = ω old K F but Z new = Z old Resistor: R ” = Z R = R R” = R Inductor: j ( K F ) L = Z L = j L L” = L / K F Capacitor: 1 / [ j ( K F ) C ] = Z C = 1 / ( j C ) C” = C / K F
PowerPoint Presentation: Lecture 22 12 Class Example Extension Exercise E12.15