NASA-Threads
Electricity and Magnetism
Lesson 23: Combining Resistors
Lesson 23: Combining Resistors
Equivalent Resistance
Today, we are going to learn about collapsing the circuit so that the resistors in series and
parallel are represented with one equivalent resistor.
Hands On
In a previous class, we built circuits containing more than one resistor and measured the
voltage and current of the circuits. The circuits were built so that the resistors were
configured to be in series and in parallel. In series, it was noted that the voltage drop over
each resistor was different. In contrast, the current that was measured was the same
through each resistor. On the same lines, resistors in parallel received the same voltage
drop, but had different current values passing through. These findings all correspond to
what you would expect from KVL and KCL.
The same principles will be applied to this hands-on experiment. We will create circuits
with resistors in series and parallel, but we will see how we can reduce the circuit by
replacing multiple resistor configurations with single resistors.
1. To begin, we will start off with a simple circuit containing resistors in series:
a. Create a circuit using two 220  resistors and one 120  resistor in series. (Use
a Vdd as voltage source.)
5V
220Ω
220Ω
120Ω
b.
c.
d.
e.
Measure the overall voltage for the circuit. (~5 V)
Measure the overall current passing through the circuit. (~8.9 mA)
Using Ohm’s Law calculate the overall resistance of the circuit. (~560 )
Now take out the two resistors and replace them with a single 560  resistor.
Repeat steps a – c. You should get the same results.
2. Now we will create a circuit with resistors in parallel
a. Create a circuit using one 220  resistor in parallel with one 470  resistor. (Use
Vdd as voltage source.)
5V
220kΩ
470Ω
b. Measure the overall voltage of the circuit. (~5 V)
c. Measure the overall current passing through the circuit. (~33 mA)
Hint: Measure the current for Resistor 1 and Resistor 2 then add them together.
d. Using Ohm’s Law, calculate the overall resistance of the circuit. (~150 )
NASA-Threads
Electricity and Magnetism
Lesson 23: Combining Resistors
e. Now take out the two resistors and replace them with a single 150  resistor.
Repeat steps a – c. You should get the same results.
3. Now let’s make it a little more interesting and create a circuit using resistors in series
and parallel.
a. Create a circuit where two parallel 1 k resistors are in series with a 10 
resistor.
5V
10Ω
1kΩ
1kΩ
b. Measure the overall voltage of the circuit. (~5 V)
c. Measure the overall current passing through the circuit. (~9.8 mA)
Hint: Measure the current for Resistor 1 and Resistor 2, then add them together.
d. Using Ohm’s Law, calculate the overall resistance of the circuit. (~510 )
e. Now take out the two 1 k resistors and the 10  resistor and replace them with
a single 510  resistor. Repeat steps a – c. You should get the same results.
Discussion
How does this work? How can we replace multiple resistors and still obtain the same
voltage and current through the whole circuit.
We use the idea of equivalent resistances.
When applied to resistors in series, the equivalent resistance is the sum of the resistors,
expressed as Equation 1 where 𝑅𝑒𝑞 is the equivalent resistance, 𝑅𝑖 is a resistor in the
circuit, 𝑛 is the number of resistors.
n
R eq = R1 + R 2 + ⋯ + R n = ∑ R i
i=1
When applied to resistors in parallel, the equivalent resistance is the inverse of the sum of
the inverses of the resistances, expressed as Equation 2.
1
1
R eq =
=
1
1
1
1
∑ni=1
+
+
⋯
+
R1 R 2
Rn
Ri
On paper, rework the circuits we created by calculating their equivalent resistances using
the equations. See if they are the same as the resistances you calculated using Ohm’s Law
when you measured the values using a multimeter.
Homework
If you do not finish calculating the circuits by hand, complete the calculations for
homework.