ISU EE C.Y. Lee

Chapter 5Parallel Circuits

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Objectives

Identify a parallel circuitDetermine the voltage across each parallel branchApply Kirchhoff’s current lawDetermine total parallel resistanceApply Ohm’s law in a parallel circuitUse a parallel circuit as a current dividerDetermine power in a parallel circuit

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Resistors in Parallel

A parallel circuit is one that has more than one branch

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Voltage in Parallel Circuits

The voltage across any given branch of a parallel circuit is equal to the voltage across each of the other branches in parallel

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Voltage in Parallel Circuits

Example: Determine the voltage across each resistor

V1 = V2 = V3 = V4 = V5 = 25 V

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Kirchhoff’s Current Law (KCL)

The sum of the currentsinto a node is equal to the sum of the currents out of that node

IIN(1) + IIN(2) + . . . + IIN(n)

= IOUT(1) + IOUT(2) +. . . + IOUT(m)

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Generalized Circuit Node Illustrating KCL

IIN(1) + IIN(2) + . . . + IIN(n) = IOUT(1) + IOUT(2) + . . . + IOUT(m)

or IIN(1) + IIN(2) + . . . + IIN(n) − IOUT(1) − IOUT(2) − . . . − IOUT(m) = 0

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Kirchhoff’s Current Law

Example: Determine the current through R2

I2 = 100 − 30 − 20 = 50 mA

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Total Parallel ResistanceThe total resistance of a parallel circuit is always less than the value of the smallest resistor

1/RT = 1/R1 + 1/R2 + 1/R3 + . . . + 1/Rn

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Total Parallel Resistance

Example: Calculate the total parallel resistance

321

1111RRRRT

++=

321

1111

RRR

RT

++=

321 RRRRT =

RT = 1/((1/100) + (1/47) + (1/22)) = 13.0 Ω

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Application of a Parallel Circuit

One advantage of a parallel circuit over a series circuit is that when one branch opens, the other branches are not affected

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Application of a Parallel Circuit

All lights and appliances in a home are wired in parallelThe switches are located in series with the lights

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Ohm’s Law in Parallel Circuits

V = IR, I = V/R, and R = V/I

Example: Find the each current in below circuit

RT = 1/(1/100 + 1/56)= 35.9 Ω

IT = VS/RT= 10/35.9 = 279 mA

IR1 = 10/100 = 100 mAIR2 = 10/56 = 179 mA

IR1 IR2

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Current Dividers

A parallel circuit acts as a current divider because the current entering the junction of parallel branches “divides” up into several individual branch currents

ITI1

I2

IT

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Current-Divider Formulas for Two Branches

ITI1

I2

IT

• The total current divides among parallel resistors into currents with values inversely proportional to the resistance values

21

2111 RR

RRIRIRI TTT +==

21

21

RRRRRT +

=⇒= 21 RRRT

( )TIRR

RI21

21 +=

( )TIRR

RI21

12 +=

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Current-Divider Formulas for Two Branches

Example: Find the each current in below circuit

I1 = 10(1000/(100+1000)) = 9.09 mA

I2 = 10(100/(100+1000)) = 0.909 mA

I1

I2

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General Current-Divider Formula

General current-divider formula for any number of parallel branches

( )TX

TX I

RRI =

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Power in Parallel CircuitsTotal power in a parallel circuit is found by adding up the powers of all the individual resistors, the same as for series circuitsPT = P1 + P2 + . . . + Pn

ST IVP =

TT RIP 2=

T

ST R

VP2

= RT = 1/(1/68 + 1/33 + 1/22) = 11.1 Ω

PT = (2)2(11.1) = 44.4 W

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Open Branches

When an open circuit occurs in a parallel branch, the total resistance increases, the total currentdecreases, and the same current continues through each of the remaining parallel paths

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Open Branches

When a lamp filament opens, total current decreases, the other branch currents remain unchanged

4I 3I

I I II I II

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Summary

A parallel circuit provides more than one path for currentThe total parallel resistance is less than the lowest-value parallel resistorThe voltages across all branches of a parallel circuit are the sameKirchhoff’s Current Law: The sum of the currents into a node equals the sum of the currents out of the nodeKirchhoff’s Current Law may also be stated as: The algebraic sum of all the currents entering and leaving a node is zero

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Summary

A parallel circuit is a current dividerThe total power in a parallel-resistive circuit is the sum of all the individual powers of the resistors making up the parallel circuitIf one of the branches of a parallel circuit opens, the total resistance increases, and therefore the total current decreasesIf a branch of a parallel circuit opens, there is no change in current through the remaining branches