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 RT
rather than for RT.
 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.
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.
Parallel Resistors
The total resistance of any number of parallel
resistors can be determined using
1
RT 
1
1
1
1


 ... 
R1 R2 R3
RN
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 RT
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.
V1 = V2 = 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 ) is equal to the sum of the individual
branch currents.
s
Is  I1  I2
 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.
E E
Is  I1  I 2 

R1 R2
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.
PE  PR1  PR2  PR3  ... PRN
 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.
I
in
 Iout
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
RT
Ix 
IT
Rx
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:
E1  E2
I
Rint1  Rint 2

 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 MW 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 / V) 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.