Electronics
Parallel
Resistive
Circuits
Part 1
1
Copyright © Texas Education Agency, 2014. All rights reserved.
What is a Parallel Circuit?

A parallel circuit is a circuit with more than
one path for current flow

This type of circuit is very common
This is the type of circuit that is used to deliver
power to an outlet in your home
Circuit analysis in a parallel circuit starts the
same way as a series circuit—with Kirchhoff’s
Laws


2
Copyright © Texas Education Agency, 2014. All rights reserved.
Review of Kirchhoff’s Law’s

Voltage law- the sum of all voltages in a closed
loop is equal to zero



The sum of the voltage drops equals the sum of the
voltage sources
All of the voltage is always used in a loop
Current law- the sum of the currents into a node
is equal to the sum of the currents leaving the
node

The current into a conductor is the same as the
current out of the conductor
Copyright © Texas Education Agency, 2014. All rights reserved.
The Simplest Parallel Circuit

Here is an example of the simplest parallel circuit

This circuit has a power supply and two paths for
current flow
4
Copyright © Texas Education Agency, 2014. All rights reserved.
The Simplest Parallel Circuit

The two resistors are different loads
VS

R1
R2
Load one is labeled R1 and load two is labeled R2
5
Copyright © Texas Education Agency, 2014. All rights reserved.
Paths for Current Flow

Path One
VS
R1
R2
6
Copyright © Texas Education Agency, 2014. All rights reserved.
Paths for Current Flow

Path Two
VS
R1
R2
7
Copyright © Texas Education Agency, 2014. All rights reserved.
Paths for Current Flow

Path Two
VS

R1
R2
Now let’s apply Kirchhoff’s Voltage Law to
each path
8
Copyright © Texas Education Agency, 2014. All rights reserved.
Voltage in Parallel Circuits

Path One- place polarities for the two
components
VS
R1
R2
9
Copyright © Texas Education Agency, 2014. All rights reserved.
Kirchhoff’s Law in Parallel Circuits

Path One- place polarities for the two
components
VS

R1
R2
In a path for current flow from one side of the
battery to the other, the sum of the voltage in
a closed loop equals zero
10
Copyright © Texas Education Agency, 2014. All rights reserved.
Kirchhoff’s Law in Parallel Circuits

Path One- start from the top of the battery, and
read polarities going into each component
VS


R1
R2
+ VS – VR1 = 0 or
VS = VR1
11
Copyright © Texas Education Agency, 2014. All rights reserved.
Kirchhoff’s Law in Parallel Circuits

Path Two
VS


R1
R2
+ VS – VR2 = 0 or
VS = VR2
12
Copyright © Texas Education Agency, 2014. All rights reserved.
Voltage in Parallel Circuits
VS = VR1 = VR2

This is the first equation for a parallel circuit
VS

R1
R2
This equation says that the voltage in each
parallel path is the same
13
Copyright © Texas Education Agency, 2014. All rights reserved.
Current in a Parallel Circuit


Both paths exist at the same time
The current that flows through R1 does not
flow through R2
VS

R1
R2
The current that flows through R2 does not
flow through R1
14
Copyright © Texas Education Agency, 2014. All rights reserved.
Current in a Parallel Circuit


Each current is separate and independent
To calculate each current flow, use Ohm’s Law
R1
VS
V1
I1 =
R1
R2
V2
I2 =
R2
15
Copyright © Texas Education Agency, 2014. All rights reserved.
Current in a Parallel Circuit
V1

I1 =

Apply Kirchhoff’s Current Law to this circuit
VS

R1 ,
I2 =
V2
R2
R1
R2
Current law- the sum of the currents into a
node is equal to the sum of the currents
leaving the node
Copyright © Texas Education Agency, 2014. All rights reserved.
16
Current in a Parallel Circuit


A node is where current splits or combines
It is a junction or branching point for current
VS
R1
R2
17
Copyright © Texas Education Agency, 2014. All rights reserved.
Current in a Parallel Circuit


A node is where current splits or combines
It is a junction or branching point for current
R1
VS

R2
Here are the nodes
18
Copyright © Texas Education Agency, 2014. All rights reserved.
Current in a Parallel Circuit

Current combines or comes back together
here
VS

R1
R2
Current splits apart here
19
Copyright © Texas Education Agency, 2014. All rights reserved.
Water Flow Equivalent

Here is a picture showing the same effect
using water flow in a pipe

Water flow here is the same as water flow
here
20
Copyright © Texas Education Agency, 2014. All rights reserved.
Water Flow Equivalent

Here is a picture showing the same effect
using water flow in a pipe

Flow splits into two parts here
21
Copyright © Texas Education Agency, 2014. All rights reserved.
Water Flow Equivalent

Here is a picture showing the same effect
using water flow in a pipe

These two points are the equivalent of an
electrical node or junction

Where flow splits and then comes back together
22
Copyright © Texas Education Agency, 2014. All rights reserved.
Current in a Parallel Circuit

There are actually three different currents
VS
R1
R2
23
Copyright © Texas Education Agency, 2014. All rights reserved.
Current in a Parallel Circuit


There are actually three different currents
Here is I1
VS
R1
R2
24
Copyright © Texas Education Agency, 2014. All rights reserved.
Current in a Parallel Circuit


There are actually three different currents
Here is I1
VS

R1
R2
Here is I2
25
Copyright © Texas Education Agency, 2014. All rights reserved.
Current in a Parallel Circuit

Here is IT (total current)
VS

R1
R2
IT is the current leaving and entering the
battery
26
Copyright © Texas Education Agency, 2014. All rights reserved.
Water Flow Equivalent

Here is the picture using current flow symbols
IT
IT
I2
27
Copyright © Texas Education Agency, 2014. All rights reserved.
Current in a Parallel Circuit

From Kirchhoff’s Current Law
IT = I1 + I2
IT
VS
R1
R2
28
Copyright © Texas Education Agency, 2014. All rights reserved.
Current in a Parallel Circuit

From Kirchhoff’s Current Law
IT = I1 + I2
IT
VS

R1
R2
This is the second parallel circuit equation
29
Copyright © Texas Education Agency, 2014. All rights reserved.
Resistance in a Parallel Circuit


Start with the equation for parallel circuit
current IT = I1 + I2
Using Ohm’s Law, substitute for current I = V
so
IT =
VT
VT

RT
, I1 =
V1
=
RT
V1
R1
, I2 =
V2
+
R1
V2
R
R2
R2
Recall the voltage rule in a parallel circuit
VS = VR1 = VR2

Substitute this rule into the previous equation
30
Copyright © Texas Education Agency, 2014. All rights reserved.
Resistance in a Parallel Circuit

After substitution
VS

RT
=
R1 +
VS
R2
VS is the same in each term so it divides out,
giving us the following formula for resistance in a
parallel circuit
1
RT

VS
=
1
R1
+
1
R2
This is the third parallel circuit equation
31
Copyright © Texas Education Agency, 2014. All rights reserved.
Parallel Circuit Equations
For two resistors
I T = I1 + I2
VS = VR1 = VR2
1
RT
=
1
R1
+
1
R2
32
Copyright © Texas Education Agency, 2014. All rights reserved.
Parallel Circuit Equations
I T = I1 + I2
VS = VR1 = VR2
1
RT
=
1
R1
+
1
R2
(current adds)
(voltage is the same)
(resistance is more complex,
but it basically divides)
33
Copyright © Texas Education Agency, 2014. All rights reserved.
Parallel Circuit Equations
I T = I1 + I2
VS = VR1 = VR2
1
RT
=
1
R1
+
1
R2
(current adds)
(voltage is the same)
(resistance is more complex,
but it basically divides)
These three formulas (plus Ohm’s Law)
form a “tool kit” to analyze parallel circuits.
34
Copyright © Texas Education Agency, 2014. All rights reserved.
Understanding Resistance in a
Parallel Circuit


Resistance looks a little more complicated, so
let’s examine it more closely
Consider the following circuit
S1
S2
S3
VS
L1

L2
L3
Each switch is open; each light is off
35
Copyright © Texas Education Agency, 2014. All rights reserved.
Understanding Resistance in a
Parallel Circuit



Close S1 and L1 comes on
We get current I1 from the battery
Each light is identical
S1
S2
S3
VS
L1

L2
L3
Total current = I1 , total resistance = R1
36
Copyright © Texas Education Agency, 2014. All rights reserved.
Understanding Resistance in a
Parallel Circuit



Next close S2 and L2 comes on
We get additional current I2 from the battery
Total current = I1 + I2, double the current
S1
S2
S3
VS
L1

L2
L3
This means total resistance must be cut in half
compared to the previous circuit
37
Copyright © Texas Education Agency, 2014. All rights reserved.
Do the Math

Use the following formula
1
RT

=
1
R1
+
1
R2
Assume R1 = R2 = 30 Ω
1
RT
=
1
RT
1
R1
+
1
R2
1
1
= +
= .0333 + .0333
30
30
= .0667 or RT =
1
= 15 Ω
.0667
38
Copyright © Texas Education Agency, 2014. All rights reserved.
Example Problem 1

For the following circuit, calculate RT and IT
VS =
15 V


R1 =
300 Ω
R2 =
200 Ω
Begin by writing down the equations we need
Start with the formula for RT. Once we calculate
that, we can solve for IT
39
Copyright © Texas Education Agency, 2014. All rights reserved.
Example Problem 1

For the following circuit, calculate RT and IT
R1 =
300 Ω
VS =
15 V

R2 =
200 Ω
Begin by writing down the equations we need
1
RT
=
1
R1
+
1
R2
and
IT =
VT
RT
40
Copyright © Texas Education Agency, 2014. All rights reserved.
Example Problem 1
1
RT
=
1
R1
1
+
R2
1
1
=
+
300
200
41
Copyright © Texas Education Agency, 2014. All rights reserved.
Example Problem 1
1
RT
1
R1
1
+
R2
1
1
=
+
300
200
=
1
RT
= 0.00333 + 0.005 = 0.00833
1
RT =
0.00833
42
Copyright © Texas Education Agency, 2014. All rights reserved.
Example Problem 1
1
RT
1
R1
1
+
R2
1
1
=
+
300
200
=
1
RT
= 0.00333 + 0.005 = 0.00833
1
RT =
0.00833
RT = 120 Ω
43
Copyright © Texas Education Agency, 2014. All rights reserved.
Example Problem 1
1
RT
1
R1
1
+
R2
1
1
=
+
300
200
=
1
RT
= 0.00333 + 0.005 = 0.00833
1
RT =
0.00833
RT = 120 Ω
VT
IT =
RT
15 V
=
120 Ω
44
Copyright © Texas Education Agency, 2014. All rights reserved.
Example Problem 1
1
RT
1
R1
1
+
R2
1
1
=
+
300
200
=
1
RT
= 0.00333 + 0.005 = 0.00833
1
RT =
0.00833
RT = 120 Ω
VT
IT =
RT
15 V
=
= .125
120 Ω
A
45
Copyright © Texas Education Agency, 2014. All rights reserved.