1. Principles & Applications
Electricity
Principles & Applications
Sixth Edition
Richard J. Fowler
Chapter 13
R, C, and L Circuits
© Glencoe/McGraw-Hill

2. INTRODUCTION RC, RL, and RCL Circuits Resonance (fr) Quality (Q)
Bandwidth (BW)
Filters

3. Facts About R, C, and L Circuits
Impedance (Z) is the combined opposition of R and X.
The ohm is the base unit of impedance.
Except at resonance, Z is always larger than R or X in a series circuit .
Except at resonance, Z is always smaller than R plus X in a series circuit.
Except at resonance, Z always causes phase shift.

4. More Facts About R, C, and L Circuits
The current phasor is the reference for series circuits.
For all impedance circuits,
Z = VT / IT and fr = 1 / (6.28 Ö LC ).
The voltage phasor is the reference for parallel circuits.
A given L and C are resonant at only one frequency.
At a given f, BW is inversely proportional to Q.

5. Graphical and Mathematical Addition of Phasors3 4 5
C = A2 + B2 =
= 5
Two right angle phasors
can be added graphically by
completing the rectangle
and bisecting it diagonally.
The bisecting line is the phasor resulting from the addition.
Mathematical addition can be done with this equation.
Substituting for A and B,
and solving the equation yields
5 which is the value of the resultant phasor (C).

6. Series RC Circuit VR VC VT IT V 50 V V 40 V V 30 V VR + VC
does not equal VT
because these voltages are out-of-phase.
These voltages must be added using phasor addition.
Solution:
VT = Ö VR2 + VC2 =
Ö = Ö 2500 = 50 V

7. Series RC Impedance R XC Z XC = 60 W R = 80 W V 150 V A 1.5 A
Z does not equal R + XC
because R and XC are 90° out-of-phase.
Determine Z using I
and V.
Z = V / I =
150 V / 1.5 A = 100 W
Or, using phasor addition:
Z = Ö R2 + XC2 =
Ö = Ö = 100 W

8. Parallel RC Circuit A 0.5 A IR IC IT VT A 0.4 A A 0.3 A IR + IC
does not equal IT
because these currents are out-of-phase.
These currents must be added using phasor addition.
Solution:
IT = Ö IR2 + IC2 =
Ö = Ö 0.25 = 0.5 A

9. RC-Circuits Quiz VT = ____ when VR = 20 V and VC = 15 V
in a series circuit.
25 V
IT ____ VT in a series RC circuit.
leads
Z = ____ when R = 400 W and XC = 300 W
in a series circuit.
500 W
In a parallel RC circuit, IT ____ VT.
leads
When IR = 0.5A, IC = 0.8A, and VT = 30V,
Z of a parallel RC circuit will be ____ W.
33.7

10. Series RL Circuit VR VL VT IT V 100 V V 80 V V 60 V VR + VL
does not equal VT
because these voltages are out-of-phase.
These voltages must be added using phasor addition.
Solution:
VT = Ö VR2 + VL2 =
Ö = Ö10000 = 100 V

11. Series RL Impedance R XL Z XL = 60 W R = 80 W V 150 V A 1.5 A
Z does not equal R + XL
because R and XL are 90° out-of-phase.
Determine Z using I
and V.
Z = V / I =
150 V / 1.5 A = 100 W
Or, using phasor addition:
Z = Ö R2 + XL2 =
Ö = Ö = 100 W

12. Parallel RL Circuit A 1.5 A IR IL IT VT A 0.9 A A 1.2 A IR + IL
does not equal IT
because these currents are out-of-phase.
These currents must be added using phasor addition.
Solution:
IT = Ö IR2 + IL2 =
Ö = Ö 2.25 = 1.5 A

13. RL-Circuits Quiz VT = ____ when VR = 15 V and VL = 12 V 19.2 V
in a series circuit.
19.2 V
In a series RL circuit, IT ____ VT .
lags
In a series circuit, Z = ____ when R = 600 W
and XL = 800 W.
1000 W
In a parallel RL circuit, IT ____ VT.
lags
Z of a parallel RL circuit will be ____ W
when IR = 0.7A, IL = 0.4A, and VT = 25 V.
38.5

14. Series RCL Circuits L R C V 15 V VR VL VT VC VX V 38 V V 26 V V 9 V
Notice that either (or both) VC and VL can be greater than VT . Also, VT must be greater than VX or VR.
VL +
VC + VR
does not equal VT
because these voltages are out-of-phase.
VT can be calculated using phasor addition:
VT = Ö VR2 + (VL - VC)2 =
Ö = Ö 225 = 15 V

15. Series RCL Impedance A 0.6 A R XL Z XC X XL = 50 W R = 15 W XC = 30 WV
15 V
Notice that either (or both) XC and XL can be greater than Z. Also, Z must be greater than X or R.
Z does not equal R +XC + XL
because R and X are 90° out-of-phase.
Determine Z using I
and V.
Z = V / I =
15 V / 0.6A = 25 W
Or, using phasor addition:
Z = Ö R2 + (XL -XC)2 =
Ö = Ö 625 = 25 W

16. Parallel RCL Circuits VT IR IC IT IL IX A 0.89 A A 0.6 A A 0.8 A A
Notice that either (or both) IC and IL can be greater than IT . Also, IT must be greater than IX or IR .
IL +
IC + IR
does not equal IT
because these currents are out-of-phase.
IT can be calculated using phasor addition:
IT = Ö IR2 + (IC - IL)2 =
Ö = Ö 0.8 = A

17. RCL-Circuit Quiz The current will ____ the voltage in a series RCL
circuit in which R = 20 W, XC = 45 W, and XL = 65 W.
lag
The current will ____ the voltage in a parallel RCL
circuit in which R = 30 W, XC = 50 W, and XL = 75 W.
lead
In a series RCL circuit, VT is always ____
than VX .
greater
In a series RCL circuit, if XL is ____ than XC ,
then I leads V.
less
In a parallel RCL circuit, the ____ current can be less than the ____ current or the
_____ current.
total
capacitor
inductor
Z = ____ for a parallel RCL circuit in which
R = 20 W, XC = 30 W, and XL = 60 W.
19 W

18. Series Resonance Demonstration
100 W
light
bulb
Wooden
wedge for
adjusting
the air gap
Variable inductor
( 1 to 4 H ) with
multiple taps
6 μF ac
capacitor
The next slide shows the circuit as it goes from capacitive,
to resonant, to inductive. Watch the brilliancy of the bulb.

19. Click on the image to run the display.
Notice how rapidly L changes as the air gap approaches zero.
To rerun the display, click again on the image.

20. Resonant Circuits XC3 fr Reactance XL3 XL1 fr2 XL2 XC1 XC2 Frequency
Resonance occurs when XL
equals XC.
There is only one resonant frequency for each LC combination.
However, an infinite number of LC combinations have the same fr.
L1 and C3 are resonant at fr2.

21. Effects of Q on the BW of Resonant Circuits
Frequency
Impedance or Current
0.707 BW
0.707 BW
Response curve
and BW
for a high-Q circuit.
The curve plots I for series circuits and Z for parallel circuits.
A low-Q circuit has a flatter curve
and a wider bandwidth.
A high-Q produces a narrow BW and good selectivity.

22. Low-Pass Filters XC decreases as f increases.
Frequency
Voutput
Multiple-
frequency
input
Output
voltage
XC decreases as f increases.
Thus, less V develops across C as f increases.
Multiple-
frequency
input
Output
voltage
Frequency
Voutput
XL increases as f increases.
Thus, more V develops across L as f increases.

23. High-Pass Filters XC decreases as f increases.
Multiple-
frequency
input
Output
voltage
Frequency
Voutput
XC decreases as f increases.
Thus, less V develops across C as f increases.
Multiple-
frequency
input
Output
voltage
Frequency
Voutput
XL increases as f increases.
Thus, more V develops across L as f increases.

24. Resonance and Filters Quiz
A given value of inductance and capacitance
produces ____ value(s) of resonant frequency(ies).
one
A circuit is ____ when the inductive reactance
is equal to the capacitive reactance.
resonant
The ____ the Q , the wider the BW of the circuit.
lower
The ____ the bandwidth, the more selective
the circuit.
narrower
An RC filter with the output taken across the
capacitor is a ____ pass filter.
low-
An RL filter with the output taken across the
inductor is a ____ pass filter.
high-

25. REVIEW
RC, RL, and RCL Circuits
Resonance
Quality
Bandwidth
Filters