MODULE 2
SINGLE PHASE PARALLEL AC
CIRCUIT
EE-425 Electrical Circuits II
Engr. Sarah Jane F. Fruelda, REE, RME
SINGLE – PHASE PARALLEL AC CIRCUITS
In parallel circuits, such as those shown in the figure below, the voltage is common to each
branch of the network and is thus taken as the reference phasor when drawing phasor
diagrams.
SINGLE – PHASE PARALLEL AC CIRCUITS
When impedances are joined in parallel, the values for:
Impedances:
1
1
1
1
1
= + + + ⋯+
ZT Z1 Z2 Z3
Zn
Current:
IT = I1 + I2 + I3 + ⋯ + In
Voltage:
ET = E1 = E = E3 = ⋯ = En
EXAMPLE 1
1.
A 100 µF capacitor is connected in parallel with a coil of 10 ohms
resistance and of 30 ohms reactance. Find the total impedance.
2.
A resistor of 50 ohms and an impedance of 100+j50 ohms are connected
in parallel across a 220 volts supply. What is the power factor of the
load?
3.
A coil of 50-ohm resistance and 150mH inductance is connected in
parallel with a 50 microfarad capacitor. The source voltage is 100 sin
(377t+30). What is the equation of the line current?
SOLUTIO
N:
1.
A 100 µF capacitor is connected in parallel with a coil of 10
ohms resistance and of 30 ohms reactance. Find the total
impedance.
SOLUTIO
N:
1.
A resistor of 50 ohms and an impedance of 100+j50 ohms
are connected in parallel across a 220 volts supply. What is
the power factor of the load?
1.
A coil of 50-ohm resistance and 150mH inductance is
connected in parallel with a 50 microfarad capacitor. The
source voltage is 100 sin (377t+30). What is the equation
of the line current?
CONDUCTANCE (G), SUSCEPTANCE
(β) & ADMITTANCE (Y)
Admittance of a circuit is defined as the reciprocal of its impedance. Its symbol is Y.
Its unit is Siemens (S). The old unit was mho (ʊ).
𝟏
𝜸=
𝒁
Conductance is the reciprocal of resistance, R and is given the symbol G.
Conductance is defined as the ease at which a resistor (or a set of resistors) allows
current to flow when a voltage, either AC or DC is applied.
𝟏
𝑮=
𝑹
CONDUCTANCE (G), SUSCEPTANCE
(β) & ADMITTANCE (Y)
Susceptance is the reciprocal of reactance, X and is given the symbol 𝜷. In AC circuits
susceptance is defined as the ease at which a reactance (or a set of reactances) allows current
current to flow when a voltage is applied.
𝜷=
𝟏
𝑿
As the impedance Z of a circuit has two components X and R, similarly, admittance Y also
has two components. The G-component is known as conductance and ß-component as
susceptance. The unit of G, β and Y is Siemens. The capacitive susceptance is considered as
as positive and inductive susceptance as negative.
ADMITTANCE TRIANGLE FOR A
PARALLEL RLC CIRCUIT
𝜸=
𝜸=
𝟏
𝑹
𝑮𝟐 + 𝜷𝑳 − 𝜷𝑪
𝟐
𝟐
𝟐
𝟏
+
− 𝝎𝑪
𝝎𝑳
𝜸 = 𝑮 + 𝒋𝜷 = (
𝑮𝟐
+
𝟏
=
𝒁
𝜷𝟐 )∠ 𝐭𝐚𝐧−𝟏
𝜷
𝑮
ADMITTANCE TRIANGLE FOR A
PARALLEL RL, RC & RLC CIRCUIT
General Rule:
 Impedance in series are added
 Admittance in parallel are added
Admittance
 the reciprocal of impedance
 expressed in series
 well suited in parallel circuit
For Pure Resistive circuit,
1
Y = G = R (𝑆)(ʊ)
For Pure Inductive/Capacitive circuit,
1
Y = β = ±jx
(𝑆)(ʊ)
L,C
RL CIRCUIT
RC CIRCUIT
RLC CIRCUIT
APPLICATION OF ADMITTANCE METHOD IN
SOLVING PARALLEL CIRCUITS
Consider the circuit shown. Total conductance is
found by merely adding the conductances of three
branches. Similarly, total susceptance is found by
algebraically adding the individual suscpetances of
different branches
Total conductance: G 𝑇 = G1 + G2 + G3 … (𝑆)(ʊ)
Total susceptance: β 𝑇 = β1 + β2 + 𝛽3 … (𝑆)(ʊ)
Total admittance: 𝑌𝑇 = Y1 + 𝑌2 + Y3 … (𝑆)(ʊ)
APPLICATION OF ADMITTANCE METHOD IN
SOLVING PARALLEL CIRCUITS
Ohm’s Law:
I=
E
(𝐴)
Z
Since 1/Z = Y, therefore we can say,
I = YE (A)
It should be noted that admittances are added for parallel branches, whereas for branches
in series, it is the impedances which are added. However, it is important to remember that
since both admittances and impedances are complex quantities, all additions must be in
complex form. Simple arithmetic additions must not be attempted!
EXAMPLE 2
1. An impedance (6 + j8) is connected across 200-V, 50-Hz mains
in parallel with another circuit having an impedance of (8 – j6)
ohms. Calculate (a) the admittance, the conductance, the
susceptance of the combined circuit (b) the total current taken
from the mains and its p.f.
2. The admittance of a circuit is (0.03 – j0.04) Siemens. Find the
values of the resistance and inductive reactance of the circuit if
they are joined (a) in series and (b) in parallel.
SOLUTIO
N:
1.
An impedance (6 + j8) is connected
across 200-V, 50-Hz mains in parallel
with another circuit having an
impedance of (8 – j6) ohms. Calculate
(a) the admittance, the conductance,
the susceptance of the combined circuit
(b) the total current taken from the
mains and its p.f.
SOLUTIO
N:
1.
The admittance of a circuit is (0.03 – j0.04) Siemens. Find the values of the
resistance and inductive reactance of the circuit if they are joined (a) in series and
(b) in parallel.
THANK YOU!