The document discusses parallel circuits, highlighting their characteristics such as the same voltage across all components and the total current being the sum of individual branch currents. It introduces the current divider rule for calculating currents in parallel resistors and emphasizes the advantages of parallel circuits, including independent operation of appliances and reduced total resistance. Additionally, it touches on series-parallel circuits and their applications in various electronic devices.

1. O B JEC
TIVES
 Par a l l e l Ci rc ui t s
 Cur re nt Di v i de r Rul
e
 Se r i e s & Par a l l e l Ci
rc ui t s
Previously we have discussed:
• Ohm’s Law & its corresponding equations
• Series circuit
• Voltage divider rule

2. Resistances in Parallel
The term parallel is used so often to describe a physical arrangement between two
elements that most individuals are aware of its general characteristics.
In general, two elements, branches, or circuits are in parallel if they have two points
in common.
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(a) Parallel resistors; (b) R1 and R2 are in parallel; (c) R3 is in parallel with the series
combination of R1 and R2.

3. Resistances in Parallel
When one end of each resistance is joined to a common point and the other end of
each resistance is joined to another common point so that there are as many paths for
current flow as the number of resistances, it is called a parallel circuit.
By Ohm’s Law
V=IR I=V/R
Here, V=V1=V2=V3 and I=I1+I2+I3
I1=V/R1, I2=V/R2, I3=V/R3
Since I=I1+I2+I3,
V/R=V/R1+V/R2+V/R3
V/R= V(1/R1+1/R2+1/R3)
1/R=1/R1+1/R2+1/R3
Hence, when a number of resistances are connected in parallel, the reciprocal of total
resistance is equal to the sum of the reciprocals of the individual resistances.
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4. The main characteristics of parallel circuit are:
• Same voltage acts across all parts of the circuit
• The total current in the circuit is equal to the sum of currents in its parallel
branches
• Conductance are additive (1/R=1/R1+1/R2+1/R3 and 1/R=G, therefore G=G1+G2+G3)
• The reciprocal of the total resistance is equal to the sum of the reciprocals of the
individual resistances
• As the number of parallel branches is increased, the total resistance of the circuit
is decreased
• The total resistance of the circuit is always less than the smallest of
the resistances
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5. Advantages of Parallel Circuits
The most useful property of a parallel circuit is the fact that potential difference has
the same value between the terminals of each branch of parallel circuit. This feature
of the parallel circuit offers the following advantages:
(i)The appliances rated for the same voltage but different powers can be connected in
parallel without disturbing each other’s performance. Thus a 230 V, 230 W TV
receiver can be operated independently in parallel with a 230 V, 40 W lamp.
(ii)If a break occurs in any one of the branch circuits, it will have no effect on other
Branch circuit
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6. Two resistances in Parallel
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7. Example: Determine the total resistance for the
network?
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8. Home Work: Find the total resistance of the network shown below:
1.
2.
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9. Par a l l e l Ci rc
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For the parallel network in Fig. :
a. Find the total resistance.
b. Calculate the source current.
c. Determine the current through each parallel branch.
d. Show that KCL is verified

10. 12
a.
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11. 13
For the parallel network in Fig.
a. Find the total resistance.
b. Calculate the source current.
c. Determine the current through each
branch.
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12. 14
a.
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13. 15
It reveals that the larger the parallel resistor, the smaller the branch
current. In general, therefore, for parallel resistors, the greatest current
will exist in the branch with the least resistance.
A more powerful statement is that current always seeks the path of
least resistance.
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14. Current Divider Rule
For series circuits we have the powerful voltage divider rule for finding
the voltage across a resistor in a series circuit. We now introduce the
equally powerful current divider rule (CDR) for finding the current
through a resistor in a parallel circuit.
• 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.
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15. 17
Deriving the current divider rule: (a) parallel network of N parallel
resistors; (b) reduced equivalent of part (a).
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16. 18
For the parallel network in Fig. 6.42, determine current I1 using Eq.
(6.14).
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17. 19
Note also that for a parallel network, the current through the smallest resistor
will be very close to the total entering current if the other parallel elements of
the configuration are much larger in magnitude.
In this example, the current through R1 is very close to the total current
because R1 is 10 times less than the next smallest resistor.
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18. 20
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19. 21
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20. 22
Special Case: Two Parallel Resistors
For the case of two parallel resistors as shown in Fig 6.43, the total
resistance is determined by
Substituting RT for current I1 results
in
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21. 23
It states that for two parallel resistors, the current through one is equal to
the other resistor times the total entering current divided by the sum of the
two resistors.
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22. 24
Determine current I2 for the network in Fig. using the current divider rule.
Using the current divider rule
to determine current I2
Solutio
n

23. 25
Determine resistor R1 in Fig. to implement the division of current shown.

24. Current Divider Rule
In parallel circuit, the voltage remains same throughout, but, the current is divided
into
number of branches.
The formula to find the current in specified branch is:
Ix=(Rt/Rx)×I
Where I is the total current
Rt is the total equivalent resistance
Ix is the current to be determined
in specified branch
Rx is the resistance of that branch
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25. Current Divider Rule
Example: (Consider the resistor values in Ohms)
1/RT=1/R1+1/R2+1/R3
RT=0.54Ω
It=V/RT
It=11.11A
I1=(Rt/R1)×It
I1=(0.54/1)×11.11
I1=6A
I2=(Rt/R2)×It
I2=2A
Similarly
I3=2.97A
Now,
I=I1+I2+I3
I=6+2+2.97 I=10.97 Approx
I=11A
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26. Se r i e s& Par a l l e l Ci
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27. Applications of Series-parallel circuits
Series-parallel circuits combine the advantages of both series and parallel. A few
common applications of series-parallel circuits are given below:
(i) In an automobile, the starting, lighting and ignition circuits are all individual
circuits
joined to make a series-parallel circuit drawing its power from one battery.
(ii)Radio and television receivers contain a number of separate circuits such as tuning
circuits, RF amplifiers, oscillator, detector and picture tube circuits. Individually, they
may be simple series or parallel circuits. However, when the receiver is considered as a
whole, the result is a series-parallel circuit.
(iii)Power supplies are connected in series to get a higher voltage and in parallel to get
a higher current.
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28. Find the equivalent resistance, REQ for the following resistor combination circuit ?
Again, at first glance this resistor ladder network may seem a complicated task, but
as before it is just a combination of series and parallel resistors connected together.
Starting from the right hand side and using the simplified equation for two parallel
resistors, we can find the equivalent resistance of the R8 to R10 combination and
call it RA.
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29. Se r i e s& Par a l l e l Ci
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30. RA is in series with R7 therefore the total resistance will be RA + R7 = 4 + 8 = 12Ω
The resistive value of 12Ω is now parallel with R6 and can be calculated as RB
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31. RB is in series with R5 therefore the total resistance will be RB + R5 = 4 + 4 = 8Ω
This resistive value of 8Ω is now in parallel with R4 and can be calculated as RC
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32. RC is in series with R3 therefore the total resistance will be RC + R3 = 4 + 4 = 8Ω
This resistive value of 8Ω is now in parallel with R2 and can be calculated as RD
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33. RD is in series with R1 therefore the total resistance will be RD + R1 = 4 + 6 = 10Ω
• When solving any combinational resistor circuit that is made up of resistors in
series and parallel branches, the first step we need to take is to identify the simple
series and parallel resistor branches and replace them with equivalent resistors
• This step will allow us to reduce the complexity of the circuit and help us
transform a complex combinational resistive circuit into a single equivalent
resistance
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34. Example 2.26-2.27 (Page-54)
Example 2.28 – 2.29 (Page-56)
Find the equivalent resistance ?
Basic Electrical Engineering by VK Mehta
Note: The book is in the files section of the Team.
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35. 37

36. 38
References &
Acknowledgments
• Introductory Circuit Analysis by
R.Bolyested