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1. Electricity And Electronics
Fundamentals
BTE 1113

2. Chapter 6
Parallel Circuits

3. 3
Parallel Circuits
• 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

4. 4
Parallel Circuits

5. 5
Parallel Circuits
• Elements in parallel
– When they have exactly two nodes in common
• Elements between nodes
– Any device like voltage sources, resistors, light
bulbs, etc.
• Elements connected in parallel
– Same voltage across them

6. 6
Parallel Circuits

7. 7
Kirchhoff’s Current Law (KCL)
• The algebraic sum of the currents entering and
leaving a node is equal to zero
∑ = 0I

8. 8
Kirchhoff’s Current Law (KCL)
• 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
∑ ∑= outin II

9. 9
Kirchhoff’s Current Law (KCL)
• An analogy:
– When water flows in a pipe, the amount of
water entering a point is the amount leaving
that point

10. 10
Direction of Current
• If unsure of the direction of current through
an element, assume a direction
• Base further calculations on this
assumption

11. 11
Direction of Current
• 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

12. 12
Resistors in Parallel
• Voltage across all parallel elements in a
circuit will be the same
• Total resistance of resistors in parallel will
always be less than resistance of smallest
resistor

13. 13
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

14. 14
Parallel Resistors
 The total resistance of any number of parallel
resistors can be determined using
 The total resistance of parallel resistors is
always less than the value of the smallest
resistor.
N
T
RRRR
R
1
...
111
1
321
++++
=

15. 15
Equal Resistors in Parallel
• For n equal resistors in parallel, each
resistor has the same conductance G
• GT = nG
• RT = 1/GT = 1/nG = R/n
• Total resistance of equal resistors in
parallel is equal to the resistor value
divided by the number of resistors

16. 16
Special Case: Two Resistors in
Parallel
• 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
21
21
RR
RR
R
+
=T

17. 17
Resistors in Parallel
• For a circuit with 3 resistors: IT = I1 + I2 + I3
GGGG
RRRR
R
E
R
E
R
E
R
E
T 321
321T
321T
1111
++=
++=
++=

18. 18
Three Resistors in Parallel
• For three resistors in parallel:
• Rather than memorize this long expression
– Use basic equation for resistors in parallel
323121
321
RRRRRR
RRR
R
++
=T

19. 19
Voltage Sources in Parallel
• Voltage sources with different potentials
should never be connected in parallel
• When two equal voltage sources are
connected in parallel
– Each source supplies half the required current

20. 20
Voltage Sources in Parallel
• Jump starting automobiles
• If two unequal sources are connected
– Large currents can occur and cause damage

21. 21
Current Divider Rule
• Allows us to determine how the current
flowing into a node is split between the
various parallel resistors

22. 22
Current Divider Rule
y
y
yy
I
R
R
I
I
G
G
I
RIRI
x
x
y
y
x
x
xx
=
=
=

23. 23
Current Divider Rule
• For only two resistors in parallel:
T
TT
T
I
RR
R
I
R
RI
I
RR
RR
R
21
2
1
1
1
21
21
+
=
=
+
=

24. 24
Current Divider Rule
T
x
T
x I
R
R
I =

25. 25
Current Divider Rule
• 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
– Largest resistor will have the least current

26. 26
Current Divider Rule
• Most of the current will follow the path
of least resistance
• For parallel elements of different
values, the current will split with a ratio
equal to the inverse of their resistor
values

27. 27
Analysis of Parallel Circuits
• 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

28. 28
Analysis of Parallel Circuits
• To calculate the power dissipated by each
resistor, use either VI, I2
R, or V2
/R
• Total power consumed is the sum of the
individual powers
• Compare with IT
2
RT

29. 29
Applications
 Car system
 The electrical system on a car is essentially a
parallel system.

30. 30

31. 31
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.

32. References
• Electricity and Electronics by Gerrish, Dugger
and Roberts, 10th
edition, 2009, GW Publisher
• Circuit Analysis: Theory and Practice by A. H.
Robbins, W. C. Miller, 4th
edition, 2006, Thomson
Delmar Learning
• Introductory Circuit Analysis by R. L. Boylestad,
11th
edition, 2007, Prentice Hall
32