AC Circuit

iii) RLC in series circuit XL ω

VL ω XL  XC  Z

VL VC  V

 I R
VC VR
XC

Impedance diagram

Phasor diagram • From the phasor diagrams,

• The impedance in RLC V leads I by Φ
circuit,
tan  VL VC tan  I X L  X C 
Z  Vrms  I R2  X L  X C 2
VR IR
Irms I
tan  X L  X C  ωL  1 
Z  R2  X L  X C 2  ωC 
R tan  
R 51

Resonance in RLC series circuit

• Resonance is defined as the phenomenon that

occurs when the frequency of the applied voltage

is equal to the frequency of the LRC series circuit.

XC , X L , R, Z Z

The series resonance XL  f
circuit is used for

tuning a radio receiver. R

0 fr XC  1
f f

Graph of impedance Z, inductive reactance

XL, capacitive reactance XC and resistance52R
with frequency.

Resonance in RLC circuit XC , X L, R, Z Z

The graph shows that :

• at low frequency, impedance Z XL  f
is large because 1/ωC is large.

• at high frequency, impedance Z R

is high because ωL is large. 0 fr XC  1
f f

• at resonance, impedance Z is minimum (Z=R)

which is XL  XC Z  R2  X L  XC 2

resonant 2fr L  1 Zmin  R2  0
frequency Zmin  R
2frC

fr  2 1
LC

and I is maximum Irms  Vrms  Vrms 53
Z R

54

iii) RLC in series circuit

EXERCISE 18.3.5

A series circuit contains a 50 Ω resistor adjacent
to a 200 mH inductor attached to a 0.050μF
capacitor, all connected across an ac generator

with a terminal sinusoidal voltage of 150 V

effective.

a)What is the resonant frequency ? (1.59 kHz)
b)What voltages will be measured by voltmeters

across each element at resonance ? (150V,6kV)
c) What is the voltage across the series

combination of the inductor and capacitor ?
a)Write the equation for the supply voltage at fr.

55

Example 18.3.6 iii) RLC in series circuit

A 200  resistor, a 0.75 H inductor and a capacitor
of capacitance C are connected in series to an
alternating source 250 V, fr = 600 Hz.
Calculate

a. the inductive reactance and capacitive
reactance when resonance is occurred.

b. the capacitance C.
c. the impedance of the circuit at resonance.
d. the current flows through the circuit at

resonance.
e. Sketch the phasor diagram.

56

Solution 18.3.6 iii) RLC in series circuit

R = 200  , L = 0.75 H ,Vrms = 250 V, f = 600 Hz.

a) X L  L  2.83 k

XC  2.83 k

b) 2.83 x103  1 , C  93.9 nF

2fC

c) Z = R = 200  VL

 Vrms  Vrms e) VR I
Z R VC
d) I rms  1.25 A 57

Exercise 18.3 iii) RLC in series circuit

A series RLC circuit has a resistance of 25.0 Ω, a
capacitance of 50.0 μF, and an inductance of

0.300 H. If the circuit is driven by a 120 V, 60 Hz

source, calculate

a)The total impedance of the circuit
b)The rms current in the circuit
c) The phase angle between the voltage and the

current.

64.9 Ω , 1.85 A, 67.3o

58

SUBTOPIC :
18.4 Power and power factor (1 hour)

LEARNING OUTCOMES :

At the end of this lesson, students should
be able to :

a) Apply
i) average power, Pav  IrmsVrms cos 

ii) instantaneous power, P  IV

iii) power factor, cos   Pr  Pav
Pa IrmsVrms

in AC circuit consisting of R, RC, RL and RLC in
series

59

18.4 Power and power factor

• In an ac circuit , the power is only dissipated by
a resistance, none is dissipated by inductance or
capacitance.

• Therefore, the real power (Pr) that is used or gone
is given by the average power (Pave) i.e :

Pave  I 2 R  Pr Pave  IrmsVRrms … (1)
rms

rms voltage
across resistor

60

(for RLC circuit)
VL ω
XL ω

VL VC  V XL  XC  Z

 
I
VC VR
R
XC

Phasor diagram Impedance diagram

• From the diagrams above,

cos  VR and cos  R ….. (2)

VZ

(2) into (1)

V=rms Pave  I 2 Z cos cos  Pave
supply rms
voltage Pa
or Papparent  Pa
61
Pave  IrmsV cos

• The term cos  is called the power factor.
• The power factor (cos  ) can vary from a

maximum of +1 to a minimum of 0.

• When  = 0o (cos  =+1) ,the circuit is completely

resistive or when the circuit is in resonance (RCL).

• When  = +90o (cos  =0) ,the circuit is completely

inductive.

• When  = -90o (cos  =0) ,the circuit is completely

capacitive.

62

• The power factor can be expressed either as a
percentage or a decimal.

• A typical circuit has a power factor of less than 1
(less than 100%).

• Example :
A motor has a power factor of 80% and the motor
consumes 800 W to operate. In order to operate
properly, the motor must be supplied with more power
than it consumes i.e.1000 W .

power factor cos  Pave 800 W (consume)
(80%)
Pa 1000 W (supply)

63

Example 18.4.1

An oscillator set for 500 Hz puts out a sinusoidal
voltage of 100 V effective. A 24.0 Ω resistor, a
10.0μF capacitor, and a 50.0 mH inductor in

series are wired across the terminals of the

oscillator.

a) What will an ammeter in the circuit read ?

b) What will a voltmeter read across each

element ?

c) What is the real power dissipated in the

circuit?

d) Calculate the power supply.

e) Find the power factor.

f) What is the phase angle? 64

Solution 18.4.1

f=500 Hz , V=100 V , R=24.0 Ω , C=10.0μF,

L=50.0 mH.

a) I rms  Vrms  784 mA
Z

b) VR  IR  18.8 V
VL  IXL  123 V
VC  IXC  24.9 V

c) Real power ?

Pave  IrmsV cos 65

Pave  0.7841000.188

Pave  14.7 W  Pr

d) Power supply,Pa  I Vrms rms

 0.784(100)
 78.4 W

e) Power factor,

cos  R  0.188

Z

f ) cos  0.188
  cos1(0.188)
  79.16o

66

Exercise 18.4

1. A coil having inductance 0.14 H and
resistance of 12  is connected to an
alternating source 110 V, 25 Hz. Calculate
a. the rms current flows in the coil.
b. the phase angle between the current and
supply voltage.
c. the power factor of the circuit.
d. the average power loss in the coil.

4.4 A, 61.3o , 0.48, 0.23 kW

67

Exercise 18.4

2. A series RCL circuit contains a 5.10 μF
capacitor and a generator whose voltage is
11.0 V. At a resonant frequency of 1.30 kHz t
he power dissipated in the circuit is 25.0 W.
Calculate
a. the inductance
b. the resistance
c. the power factor when the generator
frequency is 2.31 kHz.

2.94 x 10-3 H , 4.84 Ω , 0.163

68