1. Today
Course overview and information
09/16/2010
© 2010 NTUST

2. Series RLC
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. Variation of XL and XC
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. Impedance of Series RLC
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
The total impedance is
3.16 kW
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. Impedance of Series RLC
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. Impedance of Series RLC
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. Impedance of Series RLC
What is the total impedance for the circuit when the frequency is increased to 400 Hz?
Example-3
786 W R L C
470 W
330 mH
2000 pF VS
f = 400 kHz
The circuit is now inductive.

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

9. Voltages in Series RLC 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. 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. 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. 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. Impedance of Series RLC
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 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. 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. Series Resonant Filters
The response has a peak because at the series resonant frequency, the current is maximum at resonance and falls off before and after resonance. This develops the maximum voltage across the resistor at resonance.
I or Vout
The bandwidth (BW) of the filter is the range of frequencies for which the output is equal to or greater than 70.7% of the maximum value. f1 and f2 are commonly referred to as the critical frequencies, cutoff frequencies or half-power frequencies.
Passband
1.0
0.707 f f1 fr f2 BW

17. Series Resonant Filter

18. Decibels
Filter responses are often given in terms of decibels, which is defined as
Because it is a ratio, the decibel is dimensionless. One of the most important decibel ratios occurs when the power ratio is 1:2. This is called the -3 dB frequency, because
Another useful definition for the decibel, when measuring voltages across the same impedance is

19. Selectivity
BW1
Greatest Selectivity
Selectivity describes the basic frequency response of a resonant circuit. (The -3 dB frequencies are marked.)
BW2
Medium Selectivity
BW3
Least Selectivity
The bandwidth is inversely proportional to Q in accordance with the formula, f
Question
Which curve represents the highest Q?
The one with the greatest selectivity.

20. Bandstop

21. 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

22. 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.

23. Impedance of Parallel RLC
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

24. Impedance of Parallel RLC
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

25. Sinusoidal Response of Parallel RLC
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

26. Currents in a Parallel RLC
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

27. Currents in a Parallel RLCIC
Example
0 mA
10 mA
20 mA
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

28. Parallel
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.
Notice that IC is out of phase with IL. When they are algebraically added, the result is…. IC IL

29. Parallel
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

30. Parallel Resonance in Nonideal
In practical circuits, when the coil resistance is considered, there is a small current 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.

31. Bandwidth of Resonant Circuit
At the parallel resonant frequency, impedance is maximum, so current is a minimum at resonance. The bandwidth (BW) can be defined in terms of the impedance curve.
Ztot
Zmax
A parallel resonant circuit is commonly referred to as a tank circuit because of its ability to store energy like a storage tank.
0.707Zmax f f1 fr f2 BW

32. Parallel 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

33. Parallel Resonant
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

34. Bandpass

35. Parallel Resonant
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

36. Parallel Resonant Filters: Band-stop

37. Key Ideas for Resonant
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.

38. Admittance of Parallel RLC

39. Selected 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.

40. Selected 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.

41. 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

42. 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

43. 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

44. 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

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

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

47. 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.

48. 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.

49. 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

50. 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

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