Chapter 06
Parallel Circuits
Source:
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Tsai
Circuit Analysis: Theory and Practice Delmar Cengage Learning
Parallel Circuits
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

House circuits contain parallel circuits
The parallel circuit will continue to operate even though
one component may be open
Only the open or defective component will no longer
continue to operate
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Parallel Circuits
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
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Elements in parallel
 When they have exactly two nodes in common
Elements between nodes
 Any device like resistors, light bulbs, etc.
Elements connected in parallel, Same voltage across
them
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Series-Parallel Circuits
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Circuits may contain a combination of series and
parallel components
Being able to recognize the various connections in a
network is an important step in analyzing these circuits
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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
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Kirchhoff’s Current Law (KCL)

The algebraic sum of the currents entering and
leaving a node is equal to zero
I  0


Currents entering the node are taken to be positive,
leaving are taken to be negative
Sum of currents entering a node is equal to the sum
of currents leaving the node
I

in
  I out
An analogy:
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When water flows in a pipe, the amount of water
entering a point is the amount leaving that point
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Example of KCL
I  I
in
out
I1 + I5 = 5A + 3A
= I2 + I3 + I4
= 2A + 4A + 2A
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Example of KCL
I  I
in
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3mA+6mA+1mA
= 4mA+4mA+2mA
out
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Direction of Current
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If unsure of the direction of current through
an element, assume a direction
Base further calculations on this assumption, 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
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Example of KCL
Determine I3 and I5
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Example of KCL
Determine I1, I3, and I4
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Example of KCL
Determine unknown currents
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Example of KCL
Determine unknown currents
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Resistors in Parallel
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Voltage across all parallel elements in a
circuit will be the same
For a circuit with 3 resistors: IT = I1 + I2 + I3
E
E
E
E



RT
R1
R2
R3
1
1
1
1



RT
R1
R2
R3
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GT  G1  G 2  G 3
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Example: Resistors in Parallel
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RT = 4 // 1 = 1 / (1/4 + 1/1) = 4/5 
or
GT = 1/4 + 1/1 = 5/4, RT = 1/GT = 4/5 
Total resistance of resistors in parallel will
always be less than resistance of smallest
resistor
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Example: Resistors in Parallel
Determine RT and GT
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Example: Resistors in Parallel
Determine RT and GT
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Equal Resistors in Parallel
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For n equal resistors in parallel, each resistor has the same
conductance G
Total resistance of equal resistors in parallel is equal
to the resistor value divided by the number of resistors
GT = nG
RT = 1/GT = 1/nG = R/n
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Two Resistors in Parallel
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For only two resistors connected in parallel, the
equivalent resistance may be found by the product of the
two values divided by the sum
Often referred to as “product over the sum” formula
RT 
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R1R 2
R1  R 2
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Example: Two Resistors in Parallel
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Three Resistors in Parallel

Rather than memorize this long expression

Use basic equation for resistors in parallel
RT 
R1R 2 R3
R1R 2  R1R3  R 2 R3
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Voltage Sources in Parallel
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When two equal sources are connected in
parallel. Each source supplies half the required
current
Voltage sources with different potentials
should never be connected in parallel,
otherwise large currents can occur and cause
damage
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Example: Voltage Sources in Parallel
Determine the current I
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Current Divider Rule

Allows us to determine how the current flowing
into a node is split between the various parallel
resistors. Larger resistor smaller current
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Current Divider Rule
For only two resistors in parallel:

RT 
R1R 2
R1  R 2
I1 
I T RT
R1
I1 
R2
IT
R1  R 2
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Current Divider Rule
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If current enters a parallel network with a
number of equal resistors, current will split
equally between resistors
In a parallel network, the smallest value
resistor will have the largest current, the
largest resistor will have the least current
Most of the current will follow the path of least
resistance
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Example: Current Divider Rule
RT = 1 / (1/1 + 1/2 + 1/4) = 4/7
I1 = RT / R1 * I = 4/7 / 1 * 14 = 8A
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Example: Current Divider Rule
Determine unknown currents
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Examples: Current Divider Rule
Determine unknown currents
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Examples: Current Divider Rule
Determine unknown currents
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Analysis of Parallel Circuits
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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
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Example: Analysis of Parallel Circuits
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To calculate the power dissipated by each
resistor, use either VI, I 2R, or V 2/R
Total power consumed is the sum of the individual
powers
Compare with IT2RT
Verify that power dissipated is equal to power delivered
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Example: Analysis of Parallel Circuits
Verify that power dissipated is equal to power delivered
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Ammeter Design
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Coil of the meter can only handle a small amount of current
A voltmeter


Meter movement in series with a current-limiting resistance (Rm)
A shunt resistor (Rshunt) in parallel allows most of current
to bypass the coil
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Voltmeter Loading Effects
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If resistance is large compared with
the resistance across which the
voltage is to be measured, the
voltmeter will have a very small
loading effect Rm > R2
If this resistance is more than 10
times the resistance across which
the voltage is being measured, the
loading effect can generally be
ignored.
Rm > 10 R2
Rm
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Example: Voltmeter Loading Effects
Case1: Rm >>R
Actual value = 10V
Reading value
=10V * 200K/(200K+0.1K)
= 9.995V
Loading effect
= (10-9.995) / 10
= 0.05%
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Example: Voltmeter Loading Effects
Case2: Rm < R
Actual value = 10V
Reading value
=10V*200K/(200K+1000K)
= 1.667V
Loading effect
= (10-1.667) / 10
= 83.33%
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Circuit Analysis using Multisim
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Kernel abilities
1. What is total resistance Rt of resistors R1~Rn in
parallel? Please give an example.
2. What is KCL? Please give an example.
3. What is current divider? Please give an example.
4. What is loading effect of measuring voltage? Please
give an example.
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Problem 7: R3 = ?
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Problem 27:
Determine unknown currents
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Problem 45:
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Determine the loading effect
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