Parallel Circuits
2 February 2005
Parallel Circuits
Objectives
•
•
•
•
•
•
•
Identify a parallel circuit
Determine the voltage across each parallel branch
Apply Kirchhoff’s Current Law
Determine total parallel resistance
Apply Ohm’s law in a parallel circuit
Use a parallel circuit as a current divider
Determine power in a parallel circuit
2 February 2005
Professor Andrew H. Andersen
Parallel Circuits
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Parallel Circuits
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Characteristics of the Parallel Circuit
•
The voltage across each component (branch) is the same everywhere in
the circuit.
–
•
This means that wherever I try to measure the voltage, I will obtain the same reading,
and this is the supply voltage.
Each branch has an individual current path.
–
We may calculate the branch current using Ohm's Law if we know the voltage across
the component and the resistance.
•
Kirchoff's Current Law Applies. This means that the sum of all the
currents entering a node is equal to the sum of all the currents leaving the
node
•
The inverse of the total resistance in the circuit is equal to inverse the sum
of the inverse of the individual resistances.
IT = I1 + I2 + I3 + . . . + IN
•
1
1
1
1
1
=
+
+
+ ... +
RT
R1
R2
R3
RN
The sum of the power supplied by the source is equal to the sum of the
power dissipated in the components.
PT = P1 + P2 + P3 + . . . + PN
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Identifying Parallel Circuits
• There is more than one current path (branch) as we move from
one source terminal to the other (between two separate points)
• The voltage between these two points also appears across each
of the branches, then there is a parallel circuit between those
two points
• Each current path is called a branch
• A parallel circuit is one that has two or more branches
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Professor Andrew H. Andersen
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Parallel Circuits
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Application of a Parallel Circuit
• All lights and appliances in a home are wired in parallel
• The switches are located in series with the lights
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Application of a Parallel Circuit
• An advantage of a parallel circuit over a series circuit is that
when one component (branch) of the circuit opens the other
branches are not affected
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Professor Andrew H. Andersen
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Parallel Circuits
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Voltage in a Parallel Circuit
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Voltage in Parallel Circuits
• The voltage across any
branch of a parallel circuit is
equal to the voltage across
all of the other branches in
parallel
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Professor Andrew H. Andersen
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Parallel Circuits
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Parallel Circuit
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Voltage in a Parallel Ciruit
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Parallel Circuits
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Determining the Resistance on a Printed Circuit
Board
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Kirchhoff’s Current Law
• Kirchhoff’s current Law
(KCL) can be stated as:
ΣI = 0
The algebraic sum of all
the currents entering and
leaving a node is equal to
zero
ΣIin = ΣIout
The algebraic sum of all
the currents entering a
node is equal to the
algebraic sum of all the
currents leaving a node
2 February 2005
Professor Andrew H. Andersen
Parallel Circuits
I = 400mA
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Parallel Circuits
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Kirchhoff’s Current Law (KCL)
• The sum of the currents entering a node (total current in)
is equal to the sum of the currents leaving that node (total
current out)
ΣIIN = ΣIOUT
IIN1 + IIN2 + . . . + IINn = IOUT1 + IOUT2 + . . . + IOUTn
IT = I1 + I2 + I3 + … + In
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KCL
• KCL at Node A
ΣIIN = ΣIOUT
IT = I1 + I2 + I3
• KCL at Node B
ΣIIN = ΣIOUT
I1 + I2 + I3 = IT
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Parallel Circuits
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KCL
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Find IT
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Find I2
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What is the Reading of Meters A3 and A5
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Parallel Circuits
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What is the Reading of Meters A3 and A5
IA3 = 3.5mA
At X:
At Y:
2 February 2005
IA5 = 2.5mA
IT – IR1 – IA3 = 0
IA3 = IT – IR1 = 5mA – 1.5mA
IA3 = 3.5mA
IA3 – IR2 – IA5 = 0
IA5 = IA3 – IR2 = 3.5mA – 1mA
IA3 = 2.5mA
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Total Parallel Resistance
• When two or more resistors are connected in parallel, the total
resistance of the circuit (REQ) decreases
• The total resistance of a parallel circuit is always smaller than
the value of the smallest resistor
• The equation to find the equivalent (total) resistance of a
parallel circuit is:
1
1
1
1
1
=
+
+
+ ... +
REQ
R1
R2
R3
RN
⎛
⎜
1
REQ = ⎜
1
1
1
1
⎜
+
+
+ ... +
R2
R3
RN
⎝ R1
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Professor Andrew H. Andersen
Parallel Circuits
⎞
⎟
⎟
⎟
⎠
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Parallel Circuits
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Shorthand Notation for Parallel Resistors
• A quick way to indicate 5 resistors connected in
parallel, is
R1//R2//R3//R4//R5
• Conductance
• G <Siemens>
• Inverse of resistance
1
G=
R
1
1
1
GT =
+
+...+
R1
R2
RN
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Resistors in Parallel
• The total (equivalent) resistance for two resistors in parallel is equal to the
product of the two resistors divided by the sum of the two resistors
1
1
1
=
+
REQ
R1
R2
R1R2
REQ =
R 1 + R2
• The total (equivalent) resistance for three resistors in parallel is:
1
1
1
1
=
+
+
REQ
R1
R2
R3
R1R2
R1R3
R2R3
REQ =
+
+
R1 + R2 + R3
R1 + R2 + R3
R1 + R 2 + R3
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Professor Andrew H. Andersen
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Resistors in Parallel
• The total (equivalent) resistance for two resistors in parallel is equal to the
product of the two resistors divided by the sum of the two resistors
1
1
1
=
+
REQ
R1
R2
R1R2
REQ =
R 1 + R2
• If R1 = R2 then:
R1R1
R12
REQ =
=
R1 + R1
2R1
R1
REQ =
2
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Find REQ
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Find REQ
Since all resistors are the same value:
R
RT =
; where N is the number of resistors
N
100Ω
RT =
5
RT = 20Ω
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Current Dividers
• A parallel circuit acts as a
current divider because the
current entering the junction
of parallel branches
“divides” into several
individual branch currents
• The total current divides
among parallel resistors into
currents with values
inversely proportional to the
resistance values
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Parallel Circuits
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General Current-Divider Formula
• The current (IX) through any branch equals the total parallel
resistance (RT) divided by the resistance (RX) of that branch,
and then multiplied by the total current (IT) into the junction of
the parallel branches
IX RX = IT RT
⎛ RT ⎞
IX = IT ⎜
⎟
⎝ RX ⎠
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Current-Divider Formulas for Two Branches
• For two resistors in parallel, the current-divider formulas for
the two branches are:
⎛ R1 R2 ⎞
I1 R1 = IT ⎜
⎟
⎝ R1 + R2 ⎠
⎛ R2 ⎞
I1 = IT ⎜
⎟
⎝ R1 + R2 ⎠
⎛ R1 R2 ⎞
I2 R2 = IT ⎜
⎟
⎝ R1 + R2 ⎠
⎛ R1 ⎞
I2 = IT ⎜
⎟
⎝ R1 + R2 ⎠
I1 R1 = I2 R2
⎛ R2 ⎞
I1 = I2 ⎜
⎟
⎝ R1 ⎠
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Find All Currents
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Find All Branch Currents
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Parallel Circuits
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Find All Currents Using Current Divider
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Find All Currents Using Current Divider
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Parallel Circuits
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Find All Currents Using Current Divider
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Power in a Parallel Circuit
• The total amount of power in a series resistive circuit is equal
to the sum of the powers in each resistor in series
PS = P1 + P2 + P3 + . . . + PN
VS I = V1I + V2I + V3I + . . . + VNI
2
I RT = I2R1 + I2R2 + I2R3 + . . . + I2RN
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