1. RLC Circuits and Resonance
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
David M. Buchla
RLC Circuits and Resonance
Chapter 13

2. Series RLC Circuits Ch.13 Summary
When a circuit contains an inductor and capacitor in series, the reactance of each opposes (i.e., cancels) the other. Total series LC reactance is found using:
The total impedance is found using:
The phase angle is found using:

3. XL and XC Vs. Frequency Ch.13 Summary
A series RLC circuit can be capacitive, inductive, or resistive, depending on the frequency.
XC > XL
XL > XC
The frequency where XC=XL is called the resonant frequency.
Reactance
Below the resonant frequency, the circuit is predominantly capacitive. XC XL
XC = XL
Above the resonant frequency, the circuit is predominantly inductive. f
Series resonance

4. Series RLC Circuit Impedance
Ch.13 Summary
Series RLC Circuit Impedance
What is the total impedance and phase angle of the series RLC circuit below?
The total reactance is
3.16 kW
The total impedance is
The phase angle is
71.6o
The circuit is capacitive, and I leads V by 71.6o.

5. Series RLC Circuit Impedance
Ch.13 Summary
Series RLC Circuit Impedance
What is the magnitude of the impedance for the circuit below?
753 W

6. Series RLC Circuit Impedance
Ch.13 Summary
Series RLC Circuit Impedance
Depending on the frequency, the circuit can appear to be capacitive or inductive. The circuit in the previous slide was capacitive because XC > XL. X XL XC XC XL f
100 kHz

7. Series RLC Circuit Impedance
Ch.13 Summary
Series RLC Circuit Impedance
What is the total impedance for the circuit when the frequency is increased to 400 kHz?
786 W
The circuit is now inductive.

8. Impedance of Series RLC Circuits
Ch.13 Summary
Impedance of Series RLC Circuits
By changing the frequency, the circuit in the previous slide inductive (because XL > XC). X XL XL XC XC f
400 kHz

9. Series RLC Circuit Voltages
Ch.13 Summary
Series RLC Circuit Voltages
The voltages across the RLC components must add to the source voltage in accordance with KVL. Because of the opposite phase shift due to L and C, VL and VC effectively subtract.
Notice that VC is out of phase with VL. When they are algebraically added, the result is…. VL VC
This example is inductive.

10. Series Resonance Ch.13 Summary
At series resonance, XC and XL cancel. VC and VL also cancel because they are equal in magnitude and opposite in polarity. The circuit is purely resistive at resonance.
Algebraic sum is zero.

11. Series Resonance Ch.13 Summary 196 kHz
A formula for resonance can be found by assuming XC = XL and solving. The result is:
What is the resonant frequency for the circuit?
196 kHz

12. Series Resonance Ch.13 Summary
Ideally, at resonance the sum of VL and VC is zero.
By KVL, VR = VS VS
0 V
What is VR at resonance?
5.0 Vrms
5.0 Vrms

13. Series RLC Circuit Impedance
Ch.13 Summary
Series RLC Circuit Impedance
The general shape of the impedance versus frequency for a series RLC circuit is superimposed on the curves for XL and XC. Notice that at the resonant frequency, the circuit is resistive, and Z = R. X XL Z XC
Z = R f
Series resonance

14. Series Resonance Ch.13 Summary
Summary of important concepts for series resonance:
Capacitive and inductive reactances are equal.
Total impedance is a minimum and is resistive.
The current is maximum.
The phase angle between VS and IS is zero.
fr is calculated using:

15. Series Resonant Filters
Ch.13 Summary
Series Resonant Filters
Series resonant circuits can be used as filters. A band-pass filter allows signals within a range of frequencies to pass.
Vout
Circuit response
Resonant circuit L C
Vin
Vout R f
Series resonance

16. Series Resonant Filters
Ch.13 Summary
Series Resonant Filters
The response curve has a peak; meaning the current is maximum at resonance and falls off at frequencies below and above resonance. The maximum current (at resonance) develops maximum voltage across the series resistor(s).
The bandwidth (BW) of the filter is the range of frequencies over which the output is equal to or greater than 70.7% of its maximum value. f1 and f2 are commonly referred to as the critical frequencies, cutoff frequencies or half-power frequencies.
I or Vout
Passband
1.0
0.707 f f1 fr f2 BW

17. Ch.13 Summary
Decibels
Filter responses are often described in terms of decibels (dB). The decibel is defined as:
Example: When output power is half the input power, the ratio of Pout/Pin = ½, and
When circuit input and output voltages are known, the filter response can be calculated using:

18. Selectivity Ch.13 Summary The one with the greatest selectivity.
Selectivity describes the basic frequency response of a resonant circuit. (The -3 dB frequencies are marked by the dots.)
Greatest Selectivity
Medium Selectivity
Least Selectivity
The greater the Q of a filter at a given resonant frequency, the higher it’s selectivity. f
BW1
Which curve represents the highest Q?
BW2
The one with the greatest selectivity.
BW3

19. Series Resonant Filters
Ch.13 Summary
Series Resonant Filters
By taking the output across the resonant circuit, a band-stop (or notch) filter is produced.
Circuit response
Vout R
Stop band 1
Vin
Vout
0.707
Resonant circuit L C f f1 fr f2 BW

20. Conductance, Susceptance, and Admittance
Ch.13 Summary
Conductance, Susceptance, and Admittance
Conductance, susceptance, and admittance were defined in Chapter 10 as the reciprocals of resistance, reactance and impedance. As a review:
Conductance is the reciprocal of resistance.
Susceptance is the reciprocal of reactance.
Admittance is the reciprocal of impedance.

21. Parallel RLC Circuit Impedance
Ch.13 Summary
Parallel RLC Circuit Impedance
The admittance can be used to find the impedance. Start by calculating the total susceptance:
The admittance is given by:
The impedance is the reciprocal of the admittance:
The phase angle is:

22. Parallel RLC Circuit Impedance
Ch.13 Summary
Parallel RLC Circuit Impedance
What is the total impedance of the parallel RLC circuit below?
First, determine the conductance and total susceptance as follows:
The total admittance is:
881 W

23. AC Response of Parallel RLC Circuits
Ch.13 Summary
AC Response of Parallel RLC Circuits
The total current is given by: IC
A typical current phasor diagram for a parallel RLC circuit is shown.
+90o IR
The phase angle is given by:
-90o IL

24. Parallel RLC Circuit Currents
Ch.13 Summary
Parallel RLC Circuit Currents
The currents in the RLC components must add to the source current in accordance with KCL. Because of the opposite phase shifts of IL and IC (relative to VS) they effectively subtract.
Notice that IC is out of phase with IL. When they are algebraically added, the result is…. IC IL

25. Parallel RLC Circuit Currents
Ch.13 Summary
Parallel RLC Circuit Currents
Draw a diagram of the phasors having values of IR = 12 mA, IC = 22 mA and IL = 15 mA. IC
0 mA
10 mA
20 mA
Set up a grid. IR
Plot the currents on the appropriate axes.
Combine the reactive currents.
Use the total reactive current and IR to find
the total current. IL
In this case, Itot =
16.6 mA

26. Parallel Resonance Ch.13 Summary
Ideally, IC and IL cancel at parallel resonance because they are equal and opposite. Thus, the circuit is purely resistive at resonance.
The algebraic sum is zero. IC
Notice that IC is out of phase with IL. When they are algebraically added, the result is…. IL

27. Parallel Resonance Ch.13 Summary
The formula for the resonant frequency in both parallel and series circuits is the same:
(ideal case)
What is the resonant frequency for the circuit?
49.8 kHz

28. Parallel Resonance in Non-ideal Circuits
Ch.13 Summary
Parallel Resonance in Non-ideal Circuits
In practical circuits, there is a small current through the coil at resonance and the resonant frequency is not exactly given by the ideal equation. The Q of the coil affects the equation for resonance:
(non-ideal)
For Q >10, the difference between the ideal and the non-ideal formula is less than 1%, and generally can be ignored.

29. Bandwidth of Resonant Circuits
Ch.13 Summary
Bandwidth of Resonant Circuits
In a parallel resonant circuit, impedance is maximum and current is minimum. The bandwidth (BW) can be defined in terms of the impedance curve.
Ztot
A parallel resonant circuit is commonly referred to as a tank circuit because of its ability to store energy like a storage tank.
Zmax
0.707Zmax f f1 fr f2 BW

30. Parallel Resonance Ch.13 Summary
Summary of important concepts for parallel resonance:
Capacitive and inductive susceptance are equal.
Total impedance is a maximum (ideally infinite).
The current is minimum.
The phase angle between VS and IS is zero.
The resonant frequency (fr) is given by

31. Parallel Resonant Filters
Ch.13 Summary
Parallel Resonant Filters
Parallel resonant circuits can also be used for band-pass or band-stop filters. A basic band-pass filter is shown below. R
Vout
Passband
Vout
1.0
Vin L C
0.707
Resonant circuit f
Parallel resonant
band-pass filter f1 fr f2 BW

32. Parallel Resonant Filters
Ch.13 Summary
Parallel Resonant Filters
For the band-stop filter, the positions of the resonant circuit and resistance are reversed as shown here. C
Vout
Stop band
Vin
Vout L 1 R
0.707
Resonant circuit f
Parallel resonant
band-stop filter f1 fr f2 BW

33. Key Ideas for Resonant Filters
Ch.13 Summary
Key Ideas for Resonant Filters
A band-pass filter allows frequencies between two critical frequencies and rejects all others.
A band-stop filter rejects frequencies between two critical frequencies and passes all others.
Band-pass and band-stop filters can be made from both series and parallel resonant circuits.
The bandwidth of a resonant filter is determined by the Q and the resonant frequency.
The output voltage at a critical frequency is 70.7% of the maximum.

34. Key Terms Ch.13 Summary Series resonance Resonant frequency (fr)
A condition in a series RLC circuit in which the reactances ideally cancel and the impedance is a minimum.
Resonant frequency (fr)
The frequency at which resonance occurs; also known as the center frequency.
Parallel resonance
A condition in a parallel RLC circuit in which the reactances ideally are equal and the impedance is a maximum.
Tank circuit
A parallel resonant circuit.

35. Key Terms Ch.13 Summary Half-power frequency Decibel Selectivity
The frequency at which the output power of a resonant circuit is 50% of the maximum value (the output voltage is 70.7% of maximum); another name for critical or cutoff frequency.
Decibel
Ten times the logarithmic ratio of two powers.
Selectivity
A measure of how effectively a resonant circuit passes desired frequencies and rejects all others. Generally, the narrower the bandwidth, the greater the selectivity.

36. Ch.13 Summary
Quiz
In practical series and parallel resonant circuits, the total impedance of the circuit at resonance will be
a. capacitive
b. inductive
c. resistive
d. none of the above

37. Ch.13 Summary
Quiz
2. In a series resonant circuit, the current at the half-power frequency is
a. maximum
b. minimum
c. 70.7% of the maximum value
d. 70.7% of the minimum value

38. Ch.13 Summary
Quiz
3. The frequency represented by the red dashed line is the
a. resonant frequency
b. half-power frequency
c. critical frequency
d. all of the above XL f XC X

39. Ch.13 Summary
Quiz
4. In a series RLC circuit, if the frequency is below the resonant frequency, the circuit will appear to be
a. capacitive
b. inductive
c. resistive
d. answer depends on the particular
components

40. Ch.13 Summary
Quiz
5. In a series resonant circuit, the resonant frequency can be found from the equation a. b. c. d.

41. Ch.13 Summary
Quiz
6. In an ideal parallel resonant circuit, the total impedance at resonance is
zero
equal to the resistance
equal to the reactance
infinite

42. Ch.13 Summary
Quiz
7. In a parallel RLC circuit, the magnitude of the total current is always the
a. same as the current in the resistor.
b. phasor sum of all of the branch currents.
c. same as the source current.
d. difference between resistive and reactive
currents.

43. Ch.13 Summary
Quiz
8. If you increase the frequency in a parallel RLC circuit, the total current
a. will not change
b. will increase
c. will decrease
d. can increase or decrease depending on if it
is above or below resonance.

44. Quiz Ch.13 Summary The phase angle between the source voltage and
current in a parallel RLC circuit will be positive if
a. IL is larger than IC
b. IL is larger than IR
c. both a and b
d. none of the above

45. Quiz Ch.13 Summary 10. A highly selectivity circuit will have a
a. small BW and high Q.
b. large BW and low Q.
c. large BW and high Q.
d. none of the above

46. Answers Ch.13 Summary 1. c 2. c 6. d 3. a 7. b 4. a 8. d 5. b 9. d