Lesson 4:
Parallel Circuits
Learning Objectives

Restate the definition of a node and demonstrate how
to measure voltage and current in parallel circuits.

Solve for total circuit resistance of a parallel circuit.

State and apply KCL in the analysis of parallel circuits.

Solve for branch currents in a parallel circuit using the
current divider rule.

Compute the power dissipated by each element in a
parallel circuit, and calculate the total circuit power.
Learning Objectives

Describe the effect of connecting DC voltage
sources (e.g. battery) in series and in parallel.

Determine the net effect of parallel combining
voltage sources.
Parallel

Two or more elements are in parallel if they are
connected to the same two nodes and
consequently have the same voltage across
them.
2-, 3-, and 2A
are in parallel
Parallel Circuits

Remember: nodes are connection points
between components

Notice each component has two terminals
and each is connected to one of the nodes
above
Parallel Circuits


House circuits contain parallel circuits
The parallel circuit will continue to operate even
though one component may fail open
Parallel Circuits vs. Series Circuits

In a series circuit, failure of a single
components can disable all components in
the circuit.

In a parallel circuit, failure of one component
will still allow other components to operate.
Circuit Breaker panel in house


Homes and ships are usually wired in parallel
instead of series.
All components can operate at rated voltage
independent of other loads when wired in parallel.
AC
Series - Parallel Circuits

Circuits may contain a combination of series
and parallel components
Parallel Circuits

To analyze a particular circuit
 First
identify the node
 Next, label the nodes with a letter or number
 Then, identify types of connections
Example Problem 1
Determine which elements are connected in
parallel and which are connected in series
Parallel vs. Series current flow
Kirchhoff’s Current Law (KCL)

The algebraic sum of the currents
entering and leaving a node is equal
to zero
I  0
I  0

Currents entering a node are
positive and those leaving a node
are negative.
N
I
n 1
n
 I1  ( I 2 )  ( I 3 )  ( I 4 )  I 5  0
Kirchhoff’s Current Law (KCL)
KCL can also be stated as “the sum of currents
entering a node is equal to the sum of currents
leaving the node.”

I in   I out
I1  I5  I 2  I3  I 4
KIRCHHOFF’S CURRENT LAW
FIG. 6.31 (a) Demonstrating Kirchhoff ’s current law; (b)
the water analogy for the junction in (a).
KIRCHHOFF’S CURRENT
LAW

In technology, the term node is commonly
used to refer to a junction of two or more
branches.
FIG. 6.32 Two-node configuration for
Example 6.16.
KIRCHHOFF’S CURRENT LAW
FIG. 6.33 Four-node configuration for
Example 6.17.
KIRCHHOFF’S CURRENT LAW
FIG. 6.34 Network for
Example 6.18.
KIRCHHOFF’S CURRENT LAW
FIG. 6.35 Parallel network for
Example 6.19.
Direction of Current

Assume a current direction and draw current arrows.

If this assumption is incorrect, calculations will show
that the current has a negative sign

Negative sign simply indicates that the current flows
in the opposite direction to the arrow you drew
Example Problem 2
Determine the magnitude and direction of each
current:
Example 3
Determine the currents I2 and I3
Resistors in Parallel

For a circuit with 3 resistors:
I T  I1  I 2  I 3
E V1 V2 V3

  
RT R1 R2 R3
Resistors in Parallel

Since voltage across all parallel elements in a
circuit are the same (E = V1 = V2=V3):
E
E E E
  
RT R1 R2 R3

1
1 1
1
  
RT R1 R2 R3
Resistors in Parallel

For a circuit with 3 resistors:
1
1
1
1



RT
R1
R2
R3
Resistors in Parallel

Total resistance of any number of resistors in
parallel:
1
RT 
1
1
1

 ... 
R1 R2
Rn
1
1
1
 1
RT  


 
180 90 60 60 
1
 20
Current Divider Rule
E = IT RT
Ix
 E 


 RX 
 RT 
 I T RT 
Ix
 IT  I x
 
 RX 
 Rx 
Example Problem 4
Use the current divider rule to determine all
unknown currents:
Analysis of Parallel Circuits
Voltage across all branches is the same
as the source voltage
 Determine current through each branch
using Ohm’s Law
 Find the total current using Kirchhoff’s
Current Law

Example Problem 5
a.
b.
Determine all unknown currents and total
resistance.
Verify KCL for node a
Power Calculations
To calculate the power dissipated by each
resistor, use either VI, I2R, or V2/R
 Total power consumed is the sum of the
individual powers
 Compare with IT2RT

Example Problem 6
a.
b.
c.
Solve for indicated currents.
Determine power dissipated
by each resistor
Verify total power = sum of
all power dissipated
Battery cells in series vs parallel
Cells connected in series
increases available voltage.
Cells connected in parallel
increases available current.
Voltage Sources in Parallel

When two equal sources are connected in
parallel
 Each
source supplies half the required current
VOLTAGE SOURCES IN PARALLEL
FIG. 6.46 Demonstrating the effect of placing two ideal
supplies of the same voltage in parallel.
VOLTAGE SOURCES IN PARALLEL
Because the voltage is the same across
parallel elements, voltage sources can
be placed in parallel only if they have
the same voltage.
 The primary reason for placing two or
more batteries or supplies in parallel is to
increase the current rating above that of a
single supply.

Voltage Sources in Parallel
Voltage sources with different potentials
should never be connected in parallel
 Large currents can occur and cause
damage

VOLTAGE SOURCES IN PARALLEL

If for some reason two batteries of different
voltages are placed in parallel, both will become
ineffective or damaged because the battery with
the larger voltage will rapidly discharge through
the battery with the smaller terminal voltage.
FIG. 6.47 Examining the
impact of placing two lead-acid
batteries of different terminal
voltages in parallel.
Backup Slides
Two Resistors in Parallel

For only two resistors connected in parallel, the
equivalent resistance may be found by the
product of the two values divided by the sum
R1R 2
RT 
R1  R 2

Often referred to as “product over the sum”
formula
(98)(2)
RT 
 1.96
98  2
Current Divider Rule

For only two resistors in parallel:
 R2
I1  
 R1  R2

 IT

 R1
I2  
 R1  R2

 IT

Equal Resistors in Parallel
Total resistance of equal resistors in
parallel is equal to the resistor value
divided by the number of resistors
 RT = R/n
