1. 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

3. Series LC Circuit Characteristics
Voltage Relationships and Phase Angles

4. 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

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

6. 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)

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

8. Parallel LC Circuit Characteristics
Current Relationships and Phase Angles

9. 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

10. Parallel LC Circuit Characteristics
Parallel Reactance (XP)

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 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.

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

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

14. Factors Affecting the Value of fr
Oscilloscope Input Capacitance

15. 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

16. Series Resonant LC Circuits (Continued)

17. 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

18. Parallel Resonant LC Circuits (Continued)

19. Series Versus Parallel Resonance: A Comparison

20. 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.

21. 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

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

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

24. Parallel RLC Circuits

25. 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.

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

27. 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

28. Series-Parallel RLC Circuit Analysis