Chapter 6
–
Parallel dc Circuits
Introductory Circuit Analysis
Robert L. Boylestad
6.1

Introduction
There are two network configurations
–
series
and parallel.
In Chapter 5 we covered a series network. In
this chapter we will cover the parallel circuit and
all the methods and laws associated with it.
6.2
–
Parallel Resistors
Two elements, branches, or circuits are in parallel if
they have two points in common as in the figure below
Insert Fig 6.2
Parallel Resistors
For resistors in parallel, the total resistance is
determined from
Note that the equation is for the reciprocal of R
T
rather than for R
T
.
Once the right side of the equation has been
determined, it is necessary to divide the result into 1 to
determine the total resistance
Parallel Resistors
For parallel elements, the total conductance is the
sum of the individual conductance values.
As the number of resistors in parallel increases, the input
current level will increase for the same applied voltage.
This is the opposite effect of increasing the number of
resistors in a series circuit.
Parallel Resistors
The total resistance of any number of parallel
resistors can be determined using
The total resistance of parallel resistors is always
less than the value of the smallest resistor.
Parallel Resistors
For equal resistors in parallel:
Where N = the number of parallel resistors.
Parallel Resistors
A special case: The total resistance of two
resistors is the product of the two divided by their
sum.
The equation was developed to reduce the effects
of the inverse relationship when determining R
T
Parallel Resistors
Parallel resistors can be interchanged without
changing the total resistance or input current.
For parallel resistors, the total resistance will
always decrease as additional parallel elements
are added.
6.3
–
Parallel Circuits
Voltage is always the same across parallel elements.
V
1
=
V
2
=
E
The voltage across resistor 1 equals the voltage across
resistor 2, and both equal the voltage supplies by the source.
Parallel Circuits
For single

source parallel networks, the source
current (I
s
) is equal to the sum of the individual
branch currents.
For a parallel circuit, source current equals the sum
of the branch currents. For a series circuit, the
applied voltage equals the sum of the voltage drops.
Parallel Circuits
For parallel circuits, the greatest current will
exist in the branch with the lowest resistance.
6.4
–
Power Distribution in a Parallel
Circuit
For any resistive circuit, the power applied by
the battery will equal that dissipated by the
resistive elements.
The power relationship for parallel resistive
circuits is identical to that for series resistive
circuits.
6.5

Kirchhoff’s Current Law
Kirchhoff’s voltage law provides an important relationship
among voltage levels around any closed loop of a network.
Kirchhoff’s current law (KCL)
states that the algebraic sum of
the currents entering and leaving an area, system, or junction is
zero.
The sum of the current entering an area, system or junction
must equal the sum of the current leaving the area, system, or
junction.
Kirchhoff’s Current Law
Most common application of the law will be at the
junction of two or more paths of current.
Determining whether a current is entering or
leaving a junction is sometimes the most difficult
task.
If the current arrow points toward the junction, the
current is entering the junction.
If the current arrow points away from the junction, the
current is leaving the junction.
6.6
–
Current Divider Rule
The
current divider rule (CDR)
is used to find the
current through a resistor in a parallel circuit.
General points:
For two parallel elements of equal value, the current will
divide equally.
For parallel elements with different values, the smaller the
resistance, the greater the share of input current.
For parallel elements of different values, the current will
split with a ratio equal to the inverse of their resistor values.
Current Divider Rule
6.7

Voltage Sources in Parallel
Voltage sources are placed in parallel only if they
have the same voltage rating.
The purpose for placing two or more batteries in parallel
is to increase the current rating.
The formula to determine the total current is:
at the same terminal voltage.
Voltage Sources in Parallel
Two batteries of different terminal voltages
placed in parallel
When two batteries of different terminal voltages
are placed in parallel, the larger battery tries to drop
rapidly to the lower supply
The result is the larger battery quickly discharges to
the lower voltage battery, causing the damage to both
batteries
6.8

Open and Short Circuits
An open circuit can have a potential difference
(voltage) across its terminal, but the current is always
zero amperes.
Open and Short Circuits
A short circuit can carry a current of a level determined
by the external circuit, but the potential difference
(voltage) across its terminals is always zero volts.
Insert Fig 6.44
6.9
–
Voltmeter Loading Effects
Voltmeters are always placed across an element to
measure the potential difference.
The resistance of parallel resistors will always be less
than the resistance of the smallest resistor.
A DMM has internal resistance which may alter the
resistance of the network under test.
The loading of a network by the insertion of a meter is
not to be taken lightly, especially if accuracy is a primary
consideration.
Voltmeter Loading Effects
A good practice is to always check the meter resistance
against the resistive elements of the network before making
a measurement.
Most DMMs have internal resistance levels in excess of
10 M
W
on all voltage scales.
The internal resistance of a VOM depends on the scale
chosen.
Internal resistance is determined by multiplying the
maximum voltage of the scale setting by the
ohm/volt
(
W
⼠/⤠
rating
of the meter, normally found at the bottom
of the face of the meter.
6.11
–
Troubleshooting Techniques
Troubleshooting is a process by which acquired
knowledge and experience are employed to localize
a problem and offer or implement a solution.
Experience and a clear understanding of the basic
laws of electrical circuits is vital.
First step should always be knowing what to expect
6.13
–
Applications
Car system
The electrical system on a car is essentially a
parallel system.
Parallel computer bus connections
The bus connectors are connected in parallel with
common connections to the power supply, address
and data buses, control signals, and ground.
Applications
House wiring
Except in some very special circumstances the
basic wiring of a house is done in a parallel
configuration.
Each parallel branch, however, can have a
combination of parallel and series elements.
Each branch receives a full 120 V or 208 V, with
the current determined by the applied load.
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