1. electronics fundamentals
circuits, devices, and applications
THOMAS L. FLOYD
DAVID M. BUCHLA
chapter 12

2. Summary Sinusoidal response of RL circuits
When both resistance and inductance are in a series circuit, the phase angle between the applied voltage and total current is between 0 and 90, depending on the values of resistance and reactance.

3. Summary Impedance of series RL circuits
In a series RL circuit, the total impedance is the phasor sum of R and XL.
R is plotted along the positive x-axis.
XL is plotted along the positive y-axis. Z Z
Z is the diagonal XL XL R R
It is convenient to reposition the phasors into the impedance triangle.

4. Summary Impedance of series RL circuits Example
Sketch the impedance triangle and show the values for R = 1.2 kW and XL = 960 W.
Z = 1.33 kW
XL = 960 W
39o
R = 1.2 kW

5. Summary Analysis of series RL circuits
Ohm’s law is applied to series RL circuits using quantities of Z, V, and I.
Because I is the same everywhere in a series circuit, you can obtain the voltage phasors by simply multiplying the impedance phasors by the current.

6. Summary Analysis of series RL circuits Example
Assume the current in the previous example is 10 mArms. Sketch the voltage phasors. The impedance triangle from the previous example is shown for reference.
The voltage phasors can be found from Ohm’s law. Multiply each impedance phasor by 10 mA.
x 10 mA =
Z = 1.33 kW
VS = 13.3 V
VL = 9.6 V
XL = 960 W
39o
39o
R = 1.2 kW
VR = 12 V

7. Summary Variation of phase angle with frequency
Phasor diagrams that have reactance phasors can only be drawn for a single frequency because X is a function of frequency.
As frequency changes, the impedance triangle for an RL circuit changes as illustrated here because XL increases with increasing f. This determines the frequency response of RL circuits.

8. Summary
Phase shift
For a given frequency, a series RL circuit can be used to produce a phase lead by a specific amount between an input voltage and an output by taking the output across the inductor. This circuit is also a basic high-pass filter, a circuit that passes high frequencies and rejects all others. R
Vout
Vin
Vout
Vin
Vin
Vout L VR

9. Summary
Phase shift
Reversing the components in the previous circuit produces a circuit that is a basic lag network. This circuit is also a basic low-pass filter, a circuit that passes low frequencies and rejects all others. L VL
Vin
Vin
Vout
Vin R
Vout
Vout

10. Summary Sinusoidal response of parallel RL circuits
For parallel circuits, it is useful to review conductance, susceptance and admittance, introduced in Chapter 10.
Conductance is the reciprocal of resistance.
Inductive susceptance is the reciprocal of inductive reactance.
Admittance is the reciprocal of impedance.

11. Summary Sinusoidal response of parallel RL circuits
In a parallel RL circuit, the admittance phasor is the sum of the conductance and inductive susceptance phasors.
The magnitude of the susceptance is
The magnitude of the phase angle is G VS G BL BL Y

12. Summary Sinusoidal response of parallel RL circuits
Some important points to notice are:
G is plotted along the positive x-axis.
BL is plotted along the negative y-axis.
Y is the diagonal G VS G BL Y BL

13. Summary Sinusoidal response of parallel RL circuits Example
Draw the admittance phasor diagram for the circuit.
The magnitude of the conductance and susceptance are:
G = 1.0 mS VS R
1.0 kW L
25.3 mH
f = 10 kHz
BL =
0.629 mS
Y =
1.18 mS

14. Summary Analysis of parallel RL circuits
Ohm’s law is applied to parallel RL circuits using quantities of Y, V, and I.
Because V is the same across all components in a parallel circuit, you can obtain the current in a given component by simply multiplying the admittance of the component by the voltage as illustrated in the following example.

15. Summary Analysis of parallel RL circuits Example
Assume the voltage in the previous example is 10 V. Sketch the current phasors. The admittance diagram from the previous example is shown for reference.
The current phasors can be found from Ohm’s law. Multiply each admittance phasor by 10 V.
G = 1.0 mS
x 10 V =
IR = 10 mA
BL =
0.629 mS
Y =
1.18 mS
IL =
6.29 mA
IS =
11.8 mA

16. Summary Phase angle of parallel RL circuits
Notice that the formula for inductive susceptance is the reciprocal of inductive reactance. Thus BL and IL are inversely proportional to f:
As frequency increases, BL and IL decrease, so the angle between IR and IS must decrease as well. IR q IS IL

17. Summary Series-Parallel RL circuits
Series-parallel RL circuits are combinations of both series and parallel elements. The solution of these circuits is similar to resistive combinational circuits but you need to combine reactive elements using phasors.
The components in the yellow box are in series and those in the green box are also in series. R2 R1 Z1 Z2 L1 L2
and
The two boxes are in parallel. You can find the branch currents by applying Ohm’s law to the source voltage and the branch impedance.

18. Summary The power triangle
Recall that in a series RC or RL circuit, you could multiply the impedance phasors by the current to obtain the voltage phasors. The earlier example from this chapter is shown for review:
x 10 mA =
Z = 1.33 kW
VS = 13.3 V
VL = 9.6 V
XL = 960 W
39o
39o
R = 1.2 kW
VR = 12 V

19. Summary The power triangle Example
Multiplying the voltage phasors by Irms gives the power triangle (equivalent to multiplying the impedance phasors by I2). Apparent power is the product of the magnitude of the current and magnitude of the voltage and is plotted along the hypotenuse of the power triangle.
Example
The rms current in the earlier example was 10 mA. Show the power triangle.
x 10 mA =
VR = 12 V
VL = 9.6 V
VS = 13.3 V
39o
Pr =
96 mVAR
Pa = 133 mVA
39o
Ptrue = 120 mW

20. Summary
Power factor
The power factor was discussed in Chapter 15 and applies to RL circuits as well as RC circuits. Recall that it is the relationship between the apparent power in volt-amperes and true power in watts. Volt-amperes multiplied by the power factor equals true power.
Power factor is defined as
PF = cos 

21. Summary Apparent power
Apparent power consists of two components; a true power component, that does the work, and a reactive power component, that is simply power shuttled back and forth between source and load.
Power factor corrections for an inductive load (motors, generators, etc.) are done by adding a parallel capacitor, which has a canceling effect.
Ptrue (W)
Pr (VAR)
Pa (VA)

22. Summary Frequency Response of RL Circuits
Series RL circuits have a frequency response similar to series RC circuits. In the case of the low-pass response shown here, the output is taken across the resistor.
Plotting the response:

23. Summary Frequency Response of RL Circuits
Reversing the position of the R and L components, produces the high-pass response. The output is taken across the inductor.
Plotting the response:

24. Key Terms Inductive susceptance (BL)
The ability of an inductor to permit current; the reciprocal of inductive reactance. The unit is the siemens (S).

25. Quiz
1. If the frequency is increased in a series RL circuit, the phase angle will
a. increase
b. decrease
c. be unchanged

26. Quiz
2. If you multiply each of the impedance phasors in a series RL circuit by the current, the result is the
a. voltage phasors
b. power phasors
c. admittance phasors
d. none of the above

27. Quiz 3. For the circuit shown, the output voltage
a. is in phase with the input voltage
b. leads the input voltage
c. lags the input voltage
d. none of the above
Vin
Vout

28. Quiz
4. In a series RL circuit, the phase angle can be found from the equation a. b.
c. both of the above are correct
d. none of the above is correct

29. Quiz
5. In a series RL circuit, if the inductive reactance is equal to the resistance, the source current will lag the source voltage by
a. 0o
b. 30o
c. 45o
d. 90o

30. Quiz 6. Susceptance is the reciprocal of a. resistance b. reactance
c. admittance
d. impedance

31. Quiz
7. In a parallel RL circuit, the magnitude of the admittance can be expressed as a. b.
c. Y = G + BL d.

32. Quiz 8. If you increase the frequency in a parallel RL circuit,
a. the total admittance will increase
b. the total current will increase
c. both a and b
d. none of the above

33. Quiz 9. The unit used for measuring true power is the a. volt-ampere
b. watt
c. volt-ampere-reactive (VAR)
d. kilowatt-hour

34. Quiz 10. A power factor of zero implies that the
a. circuit is entirely reactive
b. reactive and true power are equal
c. circuit is entirely resistive
d. maximum power is delivered to the load

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