Download Currents in a parallel circuit

Contents
Introduction
3
Parallel connections
4
Voltage in a parallel circuit
5
Equivalent resistance of a parallel circuit
5
Currents in a parallel circuit
7
Branch currents
7
Total Current
8
Kirchhoff’s current law
8
The current divider equation
Equivalent Resistance
10
12
Equal value resistors in parallel
13
Product over sum formula
14
Rules for calculating total resistance
15
Power in a parallel circuit
16
Summary
21
Answers
28
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
1
2
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
Introduction
This section looks at resistors connected in parallel and describes the effect
of this connection on circuit resistance, current, voltage, and power. As was
the case with series connections, Ohm’s law is used extensively in
calculating these effects.
The aim of this section is to give you an understanding of parallel circuits
and the skills to determine the conditions in a dc parallel circuit.
After completing this topic, you should be able to:

set-up and connect a single-source dc parallel circuit

take measurements of resistance, voltage and current in a single-source
dc parallel circuit

determine voltages, currents, resistances or power dissipation from
measured or given values of any two of these quantities

describe the relationships between currents entering a junction and
currents leaving the junction

show the relationship between branch currents and resistances in a
simple current divider network.
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
3
Parallel connections
A parallel circuit is where every component is connected identically across
the supply voltage. Some examples of parallel circuits are shown in Figure 1
below.
Figure 1: Parallel circuit examples
These circuits in can also be drawn as in Figure 2.
Figure 2: Parallel circuits (horizontal arrangement)
Most lighting and power circuits of domestic, commercial and industrial
buildings are parallel circuits. This is done because:
4

In a series circuit, the failure of one component interrupts the current
flow to every other component, whereas if connected in parallel, the
failure of one component will not affect the others. This makes fault
finding easier too, because the faulty component is the only one that
does not work.

It is convenient to manufacture components to the same nominal
operating voltage (230 V). If the components were series connected,
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
they would each receive a portion of this supply voltage, depending
on how many components were in series.
Batteries are connected in parallel, for example when ‘jump starting’ a car,
so that voltage of the ‘good’ battery will replace the voltage of the ‘flat’
battery.
Voltage in a parallel circuit
In a parallel circuit, every component is connected across the voltage source.
Obviously therefore, every component has an identical voltage across it,
equal to the supply voltage.
Refer to Figure 1(a). If a voltage V is applied to points A and B, then this
voltage must appear across all three resistors. Similarly when a voltage is
applied to the circuit in Figure 2(b), all three lamps will receive the same
voltage.
Figure 3 below makes the parallel layout a little clearer, where three separate
paths (called branches) exist between to terminals A and B.
Figure 3: Wiring diagram
Equivalent resistance of a parallel
circuit
The equation for combining resistors in parallel is not as simple as for the
series circuit, where we simply add the values. We will return to this later.
For the moment, you should be able to see that the equivalent resistance
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
5
must be less than any of the individual resistances, because the parallel
paths provide additional paths for the current to flow through. This is like
putting water pipes in parallel – the total resistance in reduced.
Activity 1
Work through the questions below.
1
What is a parallel circuit?
_____________________________________________________________________
_____________________________________________________________________
2
List some common examples of parallel circuits.
_____________________________________________________________________
_____________________________________________________________________
3
How are house lights connected and controlled?
_____________________________________________________________________
_____________________________________________________________________
4
What is the advantage of lamps being connected in parallel?
_____________________________________________________________________
_____________________________________________________________________
Check your answers with those given at the end of the section.
6
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
Currents in a parallel circuit
Branch currents
The branch currents are just the currents flowing in the individual parallelconnected components. By applying Ohm’s Law, we can easily calculate the
individual currents through each resistor.
Example 1
Suppose we have three resistors of R1 = 3 Ω, R2=4 Ω, and R3 = 6 Ω
connected across a 24 V dc supply. The currents are:
I1 
V
R1
24
3
8A

I2 
V
R2
24
4
6A

I3 
V
R3
24
6
4A

Note that the smallest current flows through the largest resistor, as you
would expect.
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
7
Total Current
The current in a parallel circuit has more than one path (or branch) to follow
and can be compared with water mains diverging into customer service
pipes as shown in Figure 4.
Figure 4: Water pipe analogy
In this situation, the total flow of water through the main is equal to the sum
of the individual flows through the smaller branching pipes.
This is completely analogous to the flow of electrical current. The current
divides and flows down the individual branches, and the total circuit current
is exactly equal to the sum of the currents in each branch.
Example 2
Let’s calculate the total current from the previous example. We simply add
the individual branch currents.
I T  I1  I 2  I 3
864
 18 A
Kirchhoff’s current law
The idea that current behaves like water in pipes as described above is quite
intuitive, and it is expressed in Kirchoff’s current law.
Kirchhoff’s current law says:
“The sum of the currents entering a node (a junction) of an electric circuit
is equal to the sum of the currents leaving the node.”
8
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
Example 3
Two resistors R1 and R 2 are connected in parallel across a 180V voltage
source. The total current supplied by the voltage source is 5A. The current
through R1 is 2A.
(a) Draw the circuit diagram of the arrangement.
(b) Calculate the current in R 2 .
(c) Calculate the resistance values of R1 and R 2 .
Solution
(a)
Figure 5
I2




I1  I 2
I  I1
52
3A
R1

E
I1

180
2
90 
E
I2
(b) I
I2
(c )

R2



EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
180
3
60 
9
The current divider equation
Look at the parallel electric circuit in Figure 6. Note that IT divides at the
point C into I1 (the current through R1) and I2 (the current through R2).
These currents I1 and I2 can be calculated by Ohm’s law.
Figure 6: Parallel electrical circuit
I1 
V
R1
100
16.666
6A
V
I2 
R2

100
25
4A

Now notice that:

the ratio of R1 to R2 is 16.666 to 25 or 2:3 (found be dividing each
by 8.333).

the ratio of I1 to I2 is 6:4 = 3:2
From this we can deduce that the:

the largest current will flow through the smallest resistance, and

the ratio of the branch currents to each other (3:2) is the inverse of the
ratio of the branch resistances to each other (2:3).
This relationship is true in general, and can be expressed as an equation
known as the voltage divider equation.
I1 R2

I 2 R1
10
or
I 2 R1

I1 R2
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
Example 4 (using the current divider equation)
Determine the current flowing through R2 in the following circuit.
Figure 7
I 2 R1

I1 R2
I2 
I1 R1
R2
4 10
20
2A

EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
11
Equivalent Resistance
Now we return to the problem of calculating the equivalent resistance of two
or more resistors in parallel. Consider a parallel circuit with two resistors R1
and R2. The total current is the sum of the branch currents:
IT = I1 + I2
Now each of these resistors, and also the equivalent resistance will obey
Ohms Law, so:
V
V
V


RT R1 R2
Dividing through by V, we have:
1
1 1
 
RT R1 R2
Inverting both sides, we have:
RT 
1
1 1

R1 R2
This is the parallel resistance equation. To calculate the equivalent
resistance, you invert each resistance, add these values, and invert the result.
The equation can be extended to any number of resistances in parallel, so
that :
1
1
1
1




RT R1 R2 R3
Example 5
Let us take an example using parallel resistances of 12 Ω and 24 Ω.
12
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
1
1 1
 
RT 12 24
 0.0833  0.0417
 0.125 S
Note that the inverse of resistance is called conductance, which has the units
Siemens (S). Now to find RT we invert the result (using the 1/x button on the
calculator). This gives RT = 8Ω
Example 6
Let us take one further example by calculating the total resistance of a
circuit having three resistors of 5 Ω, 6 Ω and 10 Ω connected in parallel.
RT 
1
1
1
1


R1 R2 R3
1
1 1 1
 
5 6 10
1

0.2  0.167  0.1
1

0.467
 2.14 

Equal value resistors in parallel
If the resistors connected in parallel are of equal value then there is no need
to apply the given equation, as a much easier way exists.
All you do to determine the total resistance is take the resistance of one
resistor and divide it by the number of resistors in parallel. If we wish to
put this in equation form it would be:
R
n
where:
R = resistance value of one resistor
n = number of resistors in parallel
RT 
For example, if there are two equal 100 Ω resistors, their parallel
combination has a resistance of 100 ÷ 2 = 50 Ω.
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
13
Example 7
What is the equivalent resistance of a circuit having five 20 Ω resistors
connected in parallel?
R
n
20

5
4
RT 
Product over sum formula
Where there are only two resistors in parallel, there is slightly simpler way
of calculating RT by using product over sum equation, where:
Put in equation form this means:
RT 
R1  R2
R1  R2
Note that this only works with two resistors. With three or more in
parallel, the basic equation has to be used to find RT.
Example 8
Determine the total resistance of a circuit having resistors of 4 Ω and 20 Ω
in parallel.
RT 
R1  R2
R1  R2
4  20
4  20
80

24
 3.33 

We do not always know what the individual branch resistances are. It may
be that the total resistance and one branch resistance are known but that a
second branch resistance is not known. In this case, a transposition of the
basic equation is necessary.
14
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
Example 9
Two resistors are connected in parallel, one of 9 Ω the other of unknown
resistance. If their equivalent resistance is 6 Ω, determine the value of the
unknown resistor.
1
1 1
 
RT R1 R2
1
1
1
 
RT R1 R2
1 1 1
 
6 9 R2
0.16666  0.1111 
1
R2
0.0555 
1
R2
1
0.0555
 18 
R2 
Rules for calculating total resistance
To make some calculations easier for yourself, note and remember the
following basic rules applying to the total resistance of a parallel circuit.

The total resistance is always less than the value of any one of the
individual resistances.

Where only two resistors are connected in parallel the total resistance is
more easily found from product over sum.
RT 

R1  R2
R1  R2
If resistors of equal value are connected in parallel, the total resistance
can more easily be found from the equation:
RT 
resistance of one resistor
number of resistors in parallel
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
15
Power in a parallel circuit
As was the case with series circuits, power dissipated in parallel connected
resistors can be calculated from any of three power equations:
P  VI
V2
P
R
PI R
2
Remember that when calculating the power used by a particular resistor,
only values pertaining to that resistor may be used. If for example you were
determining the power used by resistor R2 the power equations would read:
P2  V2 I 2
P2  I 2 R2
2
V2 2
P2 
R2
The good news is that the individual powers dissipated by each resistor can
always be added. It does not matter how complex the circuit is, the total
power is always the sum of the individual powers.
Example 10
Two resistors of 20 Ω and 10 Ω are connected in parallel to a 20 volt dc
supply. Calculate the power dissipated by each resistor and the total power
dissipated by the circuit.
Figure 8
16
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
P1 
V12
R1
20 2
20
 20 W

P2 
V2 2
R2
20 2

10
 40 W
PT  P1  P2
 20  40
 60 W
The total power dissipated could also be found by finding total circuit
V2
resistance RT and then using PT  T .
RT
PT 
R1  R2
R1  R2
20 10
20  10
 6.666 

Then
PT 
RT 2
RT
202
6.66
 60 W

If you have Hampson, read the section ‘Parallel circuits’ on page 61 to page
63.
If you have Jenneson, efer to Section 4.4 in Jenneson on page 85 for
information and examples of parallel circuit analysis.
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
17
Activity 2
1
Find the equivalent or total resistance of this circuit, by using Ohm’s law.
Figure 9
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
2
Find the total resistance of the circuit in Figure 9 using individual resistor values.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
3
Using the product and sum equation, what is the equivalent resistance of a 4 Ω and a
12 Ω resistor connected in parallel?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
18
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
4
What is the equivalent resistance of three 18 Ω resistors in parallel?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
5
Two resistors in a parallel configuration dissipate 8 W and 6 W respectively. What is
the total power dissipated in the circuit?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
6
Two resistances of 10 Ω and 15 Ω are connected in parallel. What resistance must be
paralleled with them to produce a total resistance of 2 Ω?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
7
Three resistors of 4 Ω, 6 Ω and 12 Ω respectively are connected in parallel to a 16 V dc
supply. Determine:
(a) total circuit resistance
___________________________________________________________________
___________________________________________________________________
(b) total circuit current
___________________________________________________________________
___________________________________________________________________
(c) current through each resistor
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
19
(d) power used by each resistor
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
(e) total circuit power
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
Check your answers with those given at the end of the section.
20
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
Summary

When using Ohm’s law in parallel circuit branches, only use values
pertaining to that branch.

The voltage across all components connected in parallel is the same.

In a parallel circuit the sum of the currents through each branch equals
the total current (IT = I1 + I2…)

The branch currents in a parallel circuit divide in a ratio which is the
reverse of the ratio of the branch resistances.

In a parallel circuit:
RT 

1
1
1
1


R1 R2 R3
or
1
1
1
1
 

RT R1 R2 R3
If all resistors in a parallel circuit are of the same value then the equation
RT 
R
n
may be used, where:
R = resistance of an individual resistor
n = number of resistors in parallel.

If only two resistors are in parallel, then:
RT 
R1  R2
R1  R2

In a parallel circuit the equivalent or total resistance is always less than
the smallest branch resistance.

In practical wiring circuits, lamps and general purpose outlets are always
connected in parallel.
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
21

The total power in a parallel circuit can be found by:
 PT

VT 2
RT
 VT IT
 IT 2 RT
 PT
 P1  P2  P3 ...
Check your progress
In Questions 1–10 only one of the suggested answers is correct. Write the letter
corresponding to your answer in the brackets provided.
1
A parallel circuit is different to a series circuit in that it has:
(a) fewer current paths
(b) a single current path
(c) more than one current path
(d) no current paths
2
( )
Components that are connected in parallel form:
(a) several branches for current flow
(b) a single path for the current
(c) an open circuit
(d) a voltage divider
3
( )
The total resistance in a parallel circuit is:
(a) less than the smallest resistance
(b) equal to the average resistance
(c) equal to the sum of the resistors
(d) greater than the largest resistance
22
( )
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
4
The largest resistance in a parallel circuit will always have the:
(a) lowest current flowing through it
(b) highest current flowing through it
(c) highest voltage drop across it
(d) smallest voltage drop across it
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
( )
23
5
If an open circuit occurs in a parallel circuit, the total resistance will:
(a) increase
(b) remain the same
(c) decrease
(d) be unpredictable
6
( )
In a parallel circuit containing two lamps, if lamp 1 is open circuit:
(a) both lamps will be off
(b) lamp 1 will be on and lamp 2 off
(c) both lamps will be on
(d) lamp 1 will be off and lamp 2 on
7
( )
In a circuit containing two resistors connected in parallel, if resistor R2 is conducting
excessive current, resistor:
(a) Rl is open circuit
(b) R2 has a much reduced resistance
(c) R2 is open circuit
(d) R1 has a much reduced resistance
8
( )
The lowest value of individual resistance in a parallel combination of resistors
is always:
(a) equal to the equivalent resistance of the combination
(b) less than the equivalent resistance of the combination
(c) dependent on the voltage and current for its resistance
(d) greater than the equivalent resistance of the combination
9
( )
In a parallel circuit the supply current equals the:
(a) total power multiplied by the supply voltage
(b) sum of the branch currents
(c) supply voltage divided by the resistance of any one branch
(d) ratio of the branch currents
24
( )
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
10 Two resistors are connected in parallel, in which resistor Rl has twice the
resistance of resistor R2. The current taken by resistor R2 is:
(a) two thirds of the total supply current
(b) half that taken by resistor Rl
(c) one third of the total supply current
(d) one half of the total supply current
( )
11 Three resistors of 20 Ω, 40 Ω and 110 Ω or are connected in parallel across a 200 V dc
supply. Calculate the:
(a) equivalent resistance of the circuit
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
(b) current flowing in each resistor
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
(c) total current taken from the supply
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
(d) total power dissipated by the circuit.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
25
12 A 4 Ω and a 6 Ω resistor are connected in parallel across a 60 V dc supply. Calculate
the:
(a) equivalent resistance of the circuit
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
(b) total current taken from the supply
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
(c) current flowing in the 6 Ω resistor
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
(d) power dissipated by the 4 Ω resistor.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
26
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
13 If a 100 Ω resistor has 20 mA of current flowing through it, calculate the current that
will flow in a 60 Ω resistor connected in parallel with the 100 Ω resistor.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
14 A parallel circuit containing three resistors of 1 Ω, 2 Ω and 4 Ω has a total circuit
current of 5.6 A. Calculate the current flowing in each resistor.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
15 Calculate the value of the resistor that will give a total resistance of 4 Ω if it is
connected in parallel with a 12 Ω resistor.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Check your answers with those given at the end of the section.
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
27
Answers
Activity 1
1
A parallel circuit is one where the corresponding ends of the
components are joined together so that each component receives
the same voltage.
2
House lights, street lights, house power points.
3
Each light has its own control switch but lights are in parallel.
4
If one lamp filament breaks, the remaining lamps still have supply
and can be used. That is each lamp is independent of the others.
Activity 2
1
Total current IT  I1  I 2
 12  4
 16 A
VT  96 V
So RT 
VT
IT
96
16
6

28
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
2
R1

V1
I1
96
12
 8
V2

I2

R2
96
4
 24 
R1  R2

R1  R2

RT
8  24
8  24
192

32
 6

3
RT

R1  R2
R1  R2
4 12
4  12
48

16
 3

4
R
n
18

3
 6
RT

PT
 P1  P2
 86
 14 W
5
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
29
6
1 1
1 
  
RT  R1 R2 
1 1 1
  
2  10 15 

1
R3

1
R3
0.5  (0.1  0.067) 
1
R3
0.5  0.167

1
R3
0.333

1
R3
1
0.333
 3
R3 
7
(a) RT

1
1 1
1


R1 R2 R3

1
1 1 1
 
4 6 12
1
0.25  0.167  0.083
1
0.5
2
VT
RT



(b)
IT

16
2
 8A

30
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
(c )
I1

V1
R1
16
4
 4A
V2

R2

I2
16
6
 2.67 A
V3

R3

I3

(d ) P1
P2
P3
(e)
PT













16
12
1.33 A
V1 I1
16  4
64 W
V2 I 2
16  2.67
42.72 W
V3 I 3
16 1.33
21.28 W
VT IT
16  8
128 W
Check:
PT  P1  P2  P3
 64  42.72  21.28
 128 W
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666
31
Check your progress
1
(c)
2
(a)
3
(a)
4
(a)
5
(a)
6
(d)
7
(b)
8
(d)
9
(b)
10 (a)
11 (a) 11.9 
(b) 10 A, 5 A, 1.8 A
(c) 16.8 A
(d) 3.36 kW
12 (a) 2.4 
(b) 25 A
(c) 10 A
(d) 900 W
13 33.3 mA
14 3.2 A, 1.6 A, 0.8 A
15 6 
32
EEE042A: 11 Analyse parallel circuits
 NSW DET 2017 2006/060/04/2017 LRR 3666