1. Electric Circuits Fundamentals
Floyd
Chapter 13

2. Summary Series RLC circuits
When a circuit contains an inductor and capacitor in series, the reactance of each tend to cancel. The total reactance is given by
The total impedance is given by
The phase angle is given by R L C VS

3. Summary Variation of XL and XC with frequency
In a series RLC circuit, the circuit can be capacitive or inductive, depending on the frequency.
XC>XL
XL>XC
At the frequency where XC=XL, the circuit is at series resonance.
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. Summary Impedance of series RLC circuits Example-1
What is the total impedance and phase angle of the series RLC circuit if R= 1.0 kW, XL = 2.0 kW, and XC = 5.0 kW?
Example-1
The total reactance is
3.16 kW
The total impedance is
The phase angle is
71.6o
The circuit is capacitive, so I leads V by 71.6o. R L C VS
1.0 kW
XL = 2.0 kW
XC = 5.0 kW

5. Summary Impedance of series RLC circuits Example-2
What is the magnitude of the impedance for the circuit? R L C VS
470 W
330 mH
2000 pF
753 W
f = 100 kHz

6. Summary Impedance of series RLC circuits
Depending on the frequency, the circuit can appear to be capacitive or inductive. The circuit in the Example-2 was capacitive because
XC>XL. X XL XC XC XL f

7. Summary Impedance of series RLC circuits Example-3
What is the total impedance for the circuit when the frequency is increased to 400 Hz?
Example-3 R L C
470 W
330 mH
2000 pF
786 W VS
f = 400 kHz
The circuit is now inductive.

8. Summary Impedance of series RLC circuits
By changing the frequency, the circuit in Example-3 is now inductive because
XL>XC X XL XL XC XC f

9. Summary Voltages in a series RLC circuits
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. Summary Series resonance
At series resonance, XC and XL cancel. VC and VL also cancel because the voltages are equal and opposite. The circuit is purely resistive at resonance.
Algebraic sum is zero.

11. Summary Series resonance
The formula for resonance can be found by setting XC = XL. The result is
Example
What is the resonant frequency for the circuit? R L C VS
330 mH
470 W
2000 pF
196 kHz

12. Summary Series resonance
Ideally, at resonance the sum of VL and VC is zero.
By KVL, VR = VS VS
V = 0
Example R L C VS
330 mH
470 W
2000 pF
5.0 Vrms
What is VR at resonance?
5.0 Vrms
5.0 Vrms

13. Summary Impedance of series RLC circuits
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. Summary Series resonance
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 given by

15. Summary Series resonant filters
An application of series resonant circuits is in filters. A band-pass filter allows signals within a range of frequencies to pass.
Circuit response:
Vout
Resonant circuit L C
Vin
Vout R f
Series resonance

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

17. Summary Conductance, susceptance, and admittance
Recall that conductance, susceptance, and admittance were defined in Chapter 10 as the reciprocals of resistance, reactance and impedance.
Conductance is the reciprocal of resistance.
Susceptance is the reciprocal of reactance.
Admittance is the reciprocal of impedance.

18. Summary Impedance of parallel RLC circuits
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 VS R L C

19. Summary Impedance of parallel RLC circuits Example
What is the total impedance of the parallel RLC circuit if R= 1.0 kW, XL = 2.0 kW, and XC = 5.0 kW?
Example
First determine the conductance and total susceptance as follows:
The total admittance is:
881 W VS R
XL = 2.0 kW
XC = 5.0 kW
1.0 kW

20. Summary Sinusoidal response of parallel RLC circuits Example
A typical current phasor diagram for a parallel RLC circuit is IC
The total current is given by:
+90o IR
The phase angle is given by:
-90o IL
Example
What is Itot and q if IR = 10 mA, IC = 15 mA and IL = 5 mA?
14.1 mA

21. Summary Currents in a parallel RLC circuits
The currents in the RLC components must add to the source current in accordance with KCL. Because of the opposite phase shift due to L and C, IL and IC effectively subtract. IC
Notice that IC is out of phase with IL. When they are algebraically added, the result is…. IL

22. Summary Currents in a parallel RLC circuits Example Solution IC
Draw a diagram of the phasors if IR = 12 mA, IC = 22 mA and IL = 15 mA?
Solution
Set up a grid with a scale that will allow all of the data– say 2 mA/div. 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

23. Summary Parallel resonance
Ideally, at parallel resonance, IC and IL cancel because the currents are equal and opposite. 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

24. Summary Parallel resonance
The formula for the resonant frequency in both parallel and series circuits is the same, namely
(ideal case)
Example
What is the resonant frequency for the circuit? R L C VS
680 mH
1.0 kW
15 nF
49.8 kHz

25. Summary Parallel resonance
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.
fr is given by

26. 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.
Circuit response: R
Vout
Passband
Vout
1.0
Vin L C
0.707
Resonant circuit f
Parallel resonant band-pass filter f1 fr f2 BW

27. Summary Parallel resonant filters
For the band-stop filter, the resonant circuit and resistance are reversed as shown here.
Circuit response: C
Vout
Stopband
Vin
Vout L 1 R
0.707
Resonant circuit f
Parallel resonant band-stop filter f1 fr f2 BW

28. 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.

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

30. Key Terms 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.
Ten times the logarithmic ratio of two powers.
A measure of how effectively a resonant circuit passes desired frequencies and rejects all others. Generally, the narrower the bandwidth, the greater the selectivity.

31. 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

32. 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

33. 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 X XL XC f f

34. 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

35. Quiz
5. In a series resonant circuit, the resonant frequency can be found from the equation a. b. c. d.

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

37. 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.

38. 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.

39. Quiz
9. 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

40. Quiz 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

41. Quiz
Answers:
1. c
2. c
3. a
4. a
5. b
6. d
7. b
8. d
9. d
10. a