After series circuits, which you studied in Unit 7, the next simplest
type of circuit is the parallel circuit, which we’ll take up next.
Again, we’ll restrict our attention to parallel resistive circuits, which
contain only resistors in addition to voltage sources.
Unit 7 Review
This unit will build on material that you studied in Unit
7. So let’s begin by taking this self-test to review what you
learned in that unit.
Recall from Unit 2 that two components are connected in series if
they are connected to each other at exactly one point and if no other
component is connected to that point.
Example: In the circuit shown below, R2 and R3 are connected in
series, and R3 and R4 are also connected in series.
On the other hand, two components are connected in parallel if
they are connected to each other at two points.
Example: In the circuit shown above, the voltage source and R1
are connected in parallel.
Parallel-Connected Components Have the Same Voltage
The most important property of parallel connections is that the
voltage is the same across every parallel-connected component.
Example: In the circuit shown below, the voltage source and R1 are
connected in parallel, so we know that the voltage across the source
must be the same as the voltage across R1. But R1 and R3 are not connected
in parallel, so we cannot assume that the voltage across R1 is equal
to the voltage across R3.
A parallel circuit is one in which all of the components are
connected in parallel with each other. Here’s an example:
Voltage in a Parallel Circuit
As noted above, parallel-connected components have the same voltage.
Therefore, all of the components in a parallel circuit must have
the same voltage as each other.
Kirchhoff’s Current Law
Kirchhoff’s Current Law says that the sum of all currents entering
a point is equal to the sum of all currents leaving that point.
We use the abbreviation KCL as a shorthand way of
referring to Kirchhoff’s Current Law.
KCL in Parallel Resistive Circuits
When applied to a parallel resistive circuit with a single voltage
source, KCL says that if you add the currents through all of the resistors,
the sum must be equal to the value of the total current leaving the
Here’s why. Consider the parallel circuit shown below, which shows
the directions of the currents flowing out of the source and through
Looking at the point labeled A, we see that there is
one current flowing into that point, namely IT,
the circuit’s total current.
There are two currents leaving point A, namely I1 and I2.
Since KCL tells us that the sum of the currents entering a point
is equal to the sum of the currents leaving that point, we can
IT = I1 + I2
Applying the same kind of reasoning to a parallel circuit with
more resistors, we’ll always reach the same conclusion: the sum
of the resistor currents is equal to the current flowing out of
the voltage source.
KCL in Other Circuits
KCL is a general rule that applies in all circuits,
not just parallel circuits and not just circuits containing resistors.
In more complicated circuits, it can get tricky to apply KCL correctly,
but when applied correctly it is a powerful tool. We’ll see this in
Total Parallel Resistance
Suppose you have n resistors connected in parallel, where n is
any number. The total conductance of these resistors is equal to the
sum of the n individual conductances. In symbols:
GT = G1 + G2 +
… + Gn
From this we can derive an expression for the total equivalent resistance
of n resistors connected in parallel:
RT = 1 ÷ (1÷R1 +
1÷R2 + … + 1÷Rn)
This formula is often called the reciprocal formula,
since it involves taking the reciprocal of the sum of the reciprocals
of the resistors. (Remember, the reciprocal of a number just means
1 divided by that number.)
An important fact about parallel-connected resistances is that the
total equivalent resistance is always less than each of the
individual resistances, including the smallest one. For example, looking
at the circuit shown below, we can say immediately that the total
resistance of the three resistors must be less than 680 Ω.
If you get a value greater than 680 Ω when you apply the
reciprocal formula to this circuit, you know that you’ve made a mistake.
Below are three formulas that you can use to find total parallel
resistance in certain special cases. However, the
reciprocal formula is the general
formula that works for all cases of resistors in parallel, including
these special cases, so you can always use it if you wish.
Special Case #1: Two Parallel Resistors
In many cases, we wish to figure out the total resistance of two resistors
connected in parallel. We could use the
reciprocal formula to find this total
resistance, or we could use the following special-case formula:
RT = (R1 × R2) ÷ (R1 +
In words, the total resistance of two parallel resistors is equal
to their product divided by their sum.
For obvious reasons, this rule is often called the product-over-sum
Special Case #2: Parallel Resistors of
Another special case arises when you have two or more resistors in
parallel, and all of the resistors have the same individual resistance.
(For example, pehaps you have three 100-Ω resistors in parallel
with each other.) Again, we could use the
reciprocal formula in such cases, or
we could use the following special-case rule:
For n parallel resistors, each having resistance R,
RT = R ÷ n
In words, if you have several resistors of the same value connected
in parallel, the total resistance is equal to the individual resistance
value divided by the number of resistors.
For obvious reasons, this rule is often called the value-over-number
We’ve still got one more special case to cover, but this animated
lesson summarizes the cases that we’ve covered so far.
Special Case #3: Resistor in Parallel with a Much Smaller Resistor
When one resistance is much greater than another one connected in
parallel with it, the total resistance of the combination is very nearly
equal to the smaller of the two. In symbols:
If R1 >> R2,
then R1||R2 ≈ R2
Here we have used the symbol || for the parallel combination of two
resistors, and we have also used two standard mathematical symbols: >> means "much
greater than," and ≈ means "approximately equal to."
The Effect of Adding More Branches to a Parallel Circuit
If you add another parallel resistor to a parallel circuit, the circuit’s
total resistance decreases. This can be a difficult
concept for students to understand, and the following animated lesson
does a nice job of explaining it.
Since adding another parallel resistor decreases the circuit’s total
resistance, it also increases the circuit’s total
From a practical standpoint, adding too many additional parallel
branches can cause the circuit’s total current to grow so large that
it causes problems, as shown in this animated lesson.
Analyzing Parallel Resistive Circuits
We noted above that all of the components in a parallel circuit must
have the same voltage as each other.
Of course, once we know the voltage across any resistor, we can use
Ohm’s law to find the current through that resistor.
So we now know enough to be able to find currents and voltage drops
in a parallel resistive circuit. There are four basic steps.
Recall that in a parallel circuit, every component has the same
voltage. Therefore, each resistor’s voltage is equal to the source
voltage. In symbols,
VS = V1 = V2 =
… = Vn
Use Ohm’s law in the form I = V ÷ R to
find the current through each resistor. In symbols,
I1 = V1 ÷ R1 I2 = V2 ÷ R2
and so on for each of the resistors.
Use the reciprocal formula (or one of the special-case formulas
given above) to find the circuit’s total resistance:
RT = 1 ÷ (1÷R1 +
1÷R2 + … + 1÷Rn)
Use one of the following methods to find the circuit’s total
Either add together all of the individual
IT = I1 + I2 +
… + In
Or apply Ohm’s law in the form I = V ÷ R to
the entire circuit. In words, the total current produced by
the voltage source is equal to the source voltage divided by
the total resistance. In symbols,
IT = VS ÷ RT
Voltage Sources Connected in Parallel?
In general, you should not connect different-valued voltage sources
in parallel with each other.
An exception to this is the case of rechargeable batteries. For
instance, suppose you’ve got a "dead" car battery whose
voltage is close to 0 V. You can recharge the battery by connecting
it in parallel with a good car battery or in parallel with a battery
charger that produces a voltage of about 12 V.
Though we generally don’t connect different-valued voltage sources
in parallel with each other, we do sometimes connect equal-valued voltage sources
in parallel with each other. Why would we want to do this? The following animated lesson explains.
Current Sources Connected in Parallel
A current source is a device that supplies the same current
to any resistance connected across its terminals.
The schematic symbol for a current source is shown below.
Current sources can be connected in parallel.
Current sources connected in parallel can be replaced by a single
equivalent current source that produces a current equal to the algebraic
sum of the individual sources.
A group of resistors connected in parallel is often called a current
divider because the total current entering the group is divided
among the various resistors in inverse proportion to the resistance
of each one.
For example, if you have two resistors in parallel and one resistor
is twice as large as the other one (for example,
suppose that one is 20 kΩ and the other is 10 kΩ),
then there will be twice as much current through
the smaller resistor as there is through the larger one.
On the other hand, if one of the parallel resistors is three
times as large as the other one (say, 30 kΩ and
10 kΩ), then there will be three times as
much current through the smaller resistor as there is through
the larger one.
Remember that, as in these examples, if two resistors of different
size are in parallel with each other, the smaller resistor gets more
current than the larger resistor.
The Current-Divider Rule
For branches in parallel, the current Ix through
any branch equals the ratio of the total parallel resistance RT to
the branch’s resistance Rx, multiplied by the total
current IT entering
the parallel combination. In equation form:
Ix = (RT ÷ Rx) × IT
Here x is a variable representing the number of the resistor
that you’re interested in.
For instance, if you’re trying to find the current through resistor
R1, you would replace x with 1 to get:
I1 = (RT ÷ R1) × IT
On the other hand, applying the rule to resistor R4 in a parallel
circuit gives us:
I4 = (RT ÷ R4) × IT
Note that RT in this formula means the equivalent
resistance (given by the reciprocal formula), not the sum of
The current-divider rule given above applies whenever
you have any
resistors in parallel. There’s another form of the current-divider
rule that applies only to cases of two resistors in parallel.
However, I’ve found that students usually get confused if they
try to remember these special-case formulas in addition to the general
formula. Therefore, I recommend that you just remember
the general formula and use it for all cases.
Power in a Parallel Circuit
To find the power dissipated in a resistor in a parallel circuit,
use any of the same formulas that you used for series circuits:
P = V × I
P = I2 × R
P = V2 ÷ R
Recall that in each of these equations, R is the resistor’s
resistance, V is the voltage across the resistor, and I is
the current through the resistor.
Total Circuit Power
Just as with series resistive circuits, there are two
ways to compute total power dissipated in a parallel resistive circuit.
You’ll get the same answer either way:
Either find the power for each resistor, and
then add these powers:
PT = P1 + P2 + P3 +
… + Pn
Or apply any one of the power formulas to the
PT = VS × IT
PT = IT2 × RT
PT = VS2 ÷ RT
These are the same power formulas from above, except that now we’re
applying them to the entire circuit, instead of to a single resistor.
Troubleshooting Parallel Circuits
Recall from the previous Unit that the two most common types of circuit
problems are opens (breaks) and shorts (paths
of zero resistance connecting points that should not be connected).
Recall also that the current through an open
is zero, and that
the voltage across a short is zero.
In a parallel circuit, an open resistor has no
effect on the current passing through the other resistors. But it
does increase the circuit’s total resistance and therefore decreases
the circuit’s total current.
A shorted resistor in a parallel
circuit is basically the same thing as connecting a wire directly
from the power supply’s positive terminal to its negative terminal.
This is a very bad thing to do, and will cause the circuit’s total
current to increase to an excessive value.
If the circuit is
properly protected by a fuse or circuit breaker, the fuse will blow
or the breaker will trip, cutting off all current to the circuit.
If the circuit is not properly protected, the excessive current caused
by a short can start a fire or damage the circuit’s power supply.
Unit 8 Review
This e-Lesson has covered several important topics, including:
parallel connections and parallel circuits
Kirchhoff’s Current Law (KCL)
total resistance of resistors in parallel
power in parallel circuits
shorts and opens in parallel circuits.
To finish the e-Lesson, take this self-test to check your understanding
of these topics.
Congratulations! You’ve completed the e-Lesson for this unit.