RLC Circuits and Resonance Analog Circuits I

Series LC Circuit Characteristics IL = IC VL and VC are 180° out of phase VS =VL - VC

Series LC Circuit Characteristics Voltage Relationships and Phase Angles

Series LC Circuit Characteristics Voltage Relationships and Phase Angles (Continued) Example: VL = 6 V and VC = 2 V Circuit is inductive Example: VL = 1 V and VC = 4 V Circuit is capacitive

Series Reactance (XS) XS = j(XL – XC)= XL<90 – XC<-90

Putting It All Together Basic Series LC Circuit Characteristics Reactance Relationship Circuit Characteristic XL> XC XS has a positive phase angle (leads circuit current by 90) The source “sees” the circuit as being inductive. VS has a positive phase angle (leads circuit current by 90) XC> XL XS has a negative phase angle (lags circuit current by 90) The source “sees” the circuit as being capacitive. VS has a negative phase angle (lags circuit current by 90)

Parallel LC Circuit Characteristics VL = VC IL and IC are 180° out of phase

Parallel LC Circuit Characteristics Current Relationships and Phase Angles

Parallel LC Circuit Characteristics Current Relationships and Phase Angles (Continued) Example: IL = 5 mA and IC = 8 mA Circuit is capacitive Example: IL = 6 mA and IC = 2 mA Circuit is inductive

Parallel LC Circuit Characteristics Parallel Reactance (XP)

Putting It All Together Basic Parallel LC Circuit Characteristics Reactance Relationship Circuit Characteristic XL> XC IC> IL XP has a negative phase angle. The circuit is capacitive in nature Circuit current leads VS by 90. XC> XL IL> IC XP has a positive phase angle. The circuit is inductive in nature. Circuit current lags VS by 90.

Resonance Inductive and Capacitive Reactance Resonant Frequency: Occurs when XL = XC

Factors Affecting the Value of fr Stray Inductance Stray Capacitance Oscilloscope Input Capacitance

Factors Affecting the Value of fr Oscilloscope Input Capacitance

Series Resonant LC Circuits Total reactance of series resonant circuit is 0  Voltage across series LC circuit is 0 V Circuit current and voltage are in phase; that is the circuit is resistive in nature

Series Resonant LC Circuits (Continued)

Parallel Resonant LC Circuits The sum of the currents through the parallel LC circuit is 0 A The circuit has infinite reactance; that is, it acts as an open

Parallel Resonant LC Circuits (Continued)

Series Versus Parallel Resonance: A Comparison

Reactance Relationship Resulting Circuit Characteristics Series RLC Circuits Reactance Relationship Resulting Circuit Characteristics XL> XC The net series reactance (XS) is inductive, so the circuit has the characteristics of a series RL circuit: source voltage and circuit impedance lead the circuit current. XL= XC The net series reactance (XS) of the LC circuit is 0 . Therefore, the circuit is resistive in nature: source voltage and circuit impedance are both in phase with circuit current. XC> XL The net series reactance (XS) is capacitive, so the circuit has the characteristics of a series RC circuit: source voltage and circuit impedance both lag the circuit current.

Series Circuit Frequency Response When Fo < Fr XC>XL ZT is capacitive Current IT leads voltage VS When Fo= Fr (in resonance) XC=XL ZT is resistive Current and voltage in phase When Fo > Fr XC < XL ZT is inductive Voltage VS leads current IT

Series RLC Circuit Series Voltages: VLC = VL<90 + VC<-90

Series RLC Circuit Series Voltages (Continued) where VS = the source voltage VLC = the net reactive voltage VR = the voltage across the resistor

Parallel RLC Circuits

Resulting Circuit Characteristics Parallel RLC Circuits Current Relationship Resulting Circuit Characteristics IL> IC The net reactive current is inductive, so the circuit has the characteristics of a parallel RL circuit: source voltage leads the circuit current and lags the circuit impedance. IL= IC The resonant LC circuit has a net current of 0 A, so the circuit is resistive in nature: source voltage, current, and impedance are all in phase. IC> IL The net reactive current is capacitive, so the circuit has the characteristics of a parallel RC circuit: source voltage lags the circuit current and leads the circuit impedance.

Total Parallel Current where IS = the source current ILC = the net reactive current, ILC = IC - IL IR = the current through the resistor

Parallel RLC Circuits Frequency Response When Fo < Fr IL>IC ILC is inductive Current IT lags voltage VS When Fo= Fr (in resonance) IL=IC ILC =0 A Current and voltage in phase When Fo > Fr IL<IC ILC is Capacitive Voltage VS lags current IT

Series-Parallel RLC Circuit Analysis