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1. Chapter 6 – Parallel dc Circuits
Introductory Circuit Analysis
Robert L. Boylestad

2. 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.

3. 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 6.2Insert Fig 6.2

4. Parallel Resistors
 For resistors in parallel, the total resistance is
determined from

Note that the equation is for the reciprocal of R
T
rather than for R
T.
 Once the right side of the equation has been
determined, it is necessary to divide the result into 1 to
determine the total resistance

5. Parallel Resistors
 For parallel elements, the total conductance is the
sum of the individual conductance values.
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.
NT
GGGGG  ...
321

6. 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



7. Parallel Resistors
For equal resistors in parallel:
Where N = the number of parallel resistors.

9. 1/R
T
= 1/1 + ¼ + 1/5 = 1 + 0.25 + 0.2 = 1.45
R
T
= 1/1.45 = 0.69

10. 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 R
T
R
T = PRODUCT/SUM

11. R
T
= (3 x 6)/(3 + 6) = 18/9 = 2

12. 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.

13. Using a protoboard to set up the circuit

14. 6.3 – Parallel Circuits
Voltage is always the same across parallel elements.
V
1 = V
2 = E

The voltage across resistor 1 equals the voltage across
resistor 2, and both equal the voltage supplies by the source.

15. Measuring the voltages of a parallel dc network.

16. Parallel Circuits
 For single-source parallel networks, the source
current (I
s) is equal to the sum of the individual branch
currents.
21
III
s

 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.

17. Parallel Circuits

For parallel circuits, the greatest current will
exist in the branch with the lowest resistance.
21
21
R
E
R
E
III
s


18. 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.
N
RRRRE PPPPP  ...
321

The power relationship for parallel resistive
circuits is identical to that for series resistive
circuits.

19. Measuring the source current of a parallel network.

20. Measuring the current through resistor R
1.

21. 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.

outin II

22. 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.

23. Kirchhoff’s current law.

24. (a) Demonstrating Kirchhoff’s current law; (b) the water
analogy for the junction in (a).

25. I
3
= 5A and I
4
= 4A

26. I
1
= 1A; I
3
= I
1
= 1A; I
4
= I
2
= 4A; I
5
= I
3
+ I
4
= 5A

27. 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.

28. Current Divider Rule
T
x
T
x I
R
R
I

29. Using the current divider rule to calculate current I
1

1/R
T
= 1/1
k
+ 1/10
k
+ 1/22
k
R
T
= 873
I
1
= (R
T
/R
1
)I
T
= (873/1000)(12 mA) = 10.5 mA

30. 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:

 at the same terminal voltage.
21 intint
21
RR
EE
I




31. 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

32. Examining the impact of placing two lead-acid batteries
of different terminal voltages in parallel.
I = (12 – 6)/(0.03 + 0.02) = 120A

33. 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.

34. 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 6.44Insert Fig 6.44

35. I = (6V)/(12) = 0.5A and V = (0.5A)(10) = 5V

36. I = (6V)/(2) = 3A and V = 0

37. 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.

38. 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
M 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 (
/ V) rating of the meter, normally found at the bottom of
the face of the meter.

39. V
ab
= 20V
Vab = (11M)/(12M)(20V)

= 18.33V

40. 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

41. 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.

42. Expanded view of an automobile’s electrical system.

43. 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.

44. Single phase of house wiring: (a) physical details; (b)
schematic representation.

45. Continuous ground connection in a duplex outlet.