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Resonant Circuit Series Behavior  The behavior of the series i R RLC circuit is governed by the impedance. • Magnitude and phase v L 1   Z  R 2   L   C   C 1   L  C   arctan  R          2 Perfect Match  There is special behavior when XC = XL. • Vectors cancel • Impedance only from resistor VL=IXL VR=IR  This is called resonance. VC=IXC Resonant Frequency  The requirements for XC  X L 1  0 L 0C 1   LC 2 0 0  1 LC 0 1 f0   2 2 LC resonance come from the reactances.  There is a resonant frequency 0 associated with the circuit. • Angular frequency  • Can be converted into frequency f in Hz Vector Sum  The total impedance is the magnitude of Z. XC XL Z   The phase between the current and voltage is the angle  between Z and the x-axis. R Z  R2  X L  X C  2 1   2 Z  R   L   C   2 X L  XC tan   R 1   L   C   arctan  R          Peak Performance  At resonance the current is at maximum for the voltage. Circuit Example  Find the resonant frequency in the following circuit in Hz.  The problem requires the formula for the frequency f. f0  1 2 LC 100 W  Only the inductance and capacitance matter. 10 V 250 mH 0.1 mF • 1/2 (0.25 H 10-7 F)1/2 = 1 kHz Circuit Example  The behavior of the series 100 W RLC circuit is governed by the impedance. • Magnitude and phase 10 V 250 mH 0.1 mF 1   Z  R 2   L   C   1   L  C   arctan  R          2 Resonant Reactance  In the preceding circuit the voltage across each component can be found. • Current due to resistor only  The voltage across the inductor has an amplitude of 158 V. • So does the capacitor I  V / R  0.1A VL  2f 0 IL  158 V I VC   158 V 2f 0C  They are each 90° out of phase and cancel out. next

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Download Resonant Circuit