Chapter 6 – Parallel 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, and in
this chapter we will cover the parallel circuit and
all the methods and laws associated with it
6.2 - Parallel Elements
 Two
elements, branches, or networks are in parallel if
they have two points in common as in the figure below
Insert Fig 6.2
6.3 - Total Conductance and
Resistance

For parallel elements, the total conductance is
the sum of the individual conductances.
GT = + G1 + G2 + G3 +… + Gn
 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
Total Conductance and Resistance

Since G = 1/R the total resistance for a network can
be determined by the equation below

Note that the equation is for 1 divided by the total
resistance rather than the total resistance
Once the right side of the equation has been
determined, it is necessary to divide the result into 1 to
determine the total resistance

Total Conductance and Resistance
 The
total resistance of a parallel resistor is always less than
the value of the smallest resistor
Additionally, the wider the spread in numerical value between two
parallel resistors, the closer the total resistance will be to the smaller
resistor
 The equation becomes significantly easier to apply for equal resistors
in parallel

 Total
resistance of N parallel resistors of equal value is the resistance
of one resistor divided by the number (N) of parallel elements
Total Conductance and Resistance
 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 RT
Total Conductance and Resistance

Parallel elements can be interchanged without
changing the total resistance or input current

For parallel resistors, the total resistance will
always decrease as additional elements are
added in parallel
6.4 - Parallel Circuits
 Total resistance is determined
by RT = R1 R2 / (R1 + R2 )
and the source current by Is = E / RT .
The subscript s will be used to denote a property of the
source

 The voltage across parallel
elements is the same
V1 = V2 = E
 Voltage across resistor 1 is equal to the voltage across
resistor 2

Parallel Circuits

For single-source parallel networks, the source
current (I ) is equal to the sum of the individual
branch currents
s
Is = I1 + I2
6.5 - Kirchhoff’s Current Law
Kirchhoff’s voltage law provides an important relationship
among voltage levels around any closed loop of a network
 Now consider Kirchhoff’s current law (KCL)
 Kirchhoff’s current law 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

Ientering
= Ileaving
Kirchhoff’s Current Law
Most common application of the law will be at the junction
of two or more paths of current flow
 Determining whether a current is entering or leaving a
junction is sometimes the most difficult task
 One approach to understanding the flow is to picture
yourself as standing on the junction point and treating the
path currents as arrows
 If the arrow appears to be heading toward you, the current
is entering the junction


If you see the tail of the arrow as it travels down its path away
from you, the current is leaving the junction
6.6 - Current Divider Rule
 The current
divider rule (CDR) will determine how the
current entering a set of parallel branches will split
between the elements
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

Current seeks the path of least resistance
 The
current entering any number of parallel resistors divides
into these resistors as the inverse ratio of their ohmic value
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
would be to increase the current rating
 The formula to determine the total current is:

Is = I1 + I2 +… IN
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

Two isolated terminals not connected by any element:
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 - Voltmeters: Loading Effect

Voltmeters are always placed across an element to
measure the potential difference
The resistance of two parallel resistors will always be less
than the resistance of the smallest resistor
 A DMM has internal resistance which will alter, somewhat,
the network being measured
 The loading of a network by the insertion a meter is not to
be taken lightly, especially if accuracy is a primary
consideration

Voltmeters: Loading Effect
 A good
practice is to always check the meter resistance level
against the resistive elements of the network before making
a measurement
 Most DMMs have internal resistance levels in excess of 10
MW on all voltage scales
 Internal resistance of VOMs is sensitive to the scale chosen

Internal resistance is determined by multiplying the maximum
voltage of the scale setting by the ohm/volt (W / V) rating of the
meter, normally found at the bottom of the face of the meter
6.10 - 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
6.11 - 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
