RLC Circuits and Resonance

Analog Circuits I

2

Series LC Circuit Characteristics

– IL = IC

– VL and VC are 180° out of phase

– VS =VL - VC

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Series LC Circuit Characteristics• Voltage Relationships and Phase Angles

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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

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Series Reactance (XS)

XS = j(XL – XC)= XL<90 – XC<-90

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Putting It All Together

• Basic Series LC Circuit Characteristics

Reactance Relationsh

ip

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)

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Parallel LC Circuit Characteristics

– VL = VC

– IL and IC are 180° out of phase

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Parallel LC Circuit Characteristics

• Current Relationships and Phase Angles

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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

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Parallel LC Circuit Characteristics

• Parallel Reactance (XP)

CL

CLP

CL

P XX

XXX

XX

X

or 11

1

11

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 natureCircuit 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.

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Resonance

• Inductive and Capacitive Reactance

• Resonant Frequency: Occurs when XL = XC

902

1 and 902

fCXfLX CL

LCf r 2

1

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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

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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.

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• 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 Circuit Frequency Response

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Series RLC Circuit

• Series Voltages: VLC = VL<90 + VC<-90

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Series RLC Circuit

• Series Voltages (Continued)

22RLCS VVV

where VS = the source voltage VLC = the net reactive voltage VR = the voltage across the resistor

R

LC

V

V1tan

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Parallel RLC Circuits

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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.

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Total Parallel Current

22RLCT III

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

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Parallel RLC CircuitsFrequency 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

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Series-Parallel RLC Circuit Analysis