Example 18-8 Resistors in Combination
Figure 18-12 shows two different combinations of three identical resistors,
each with resistance R. Find the equivalent resistance of the combination in
(a) Figure 18-12a and (b) Figure 18-12b.
(a)
V
(b)
V
R
Figure 18-12 ​Two combinations of three identical resistors What is the equivalent resistance
of each combination?
Set Up
Neither combination in Figure 18-12 is a ­simple
series or parallel arrangement of resistors. But
in Figure 18-12a resistors 1 and 2 are in parallel
with each other, and that combination is in series
with resistor 3. Similarly, in Figure 18-12b resistors 4 and 5 are in series with each other, and
that combination is in parallel with resistor 6. So
we can use Equations 18-11 and 18-20 together
to find the equivalent resistance of both arrangements of resistors.
Solve
1
R
R
3
2
(18-11)
Equivalent resistance of resistors in parallel:
1
R equiv
R
4
5
R
6
Equivalent resistance of resistors in series:
Requiv = R1 + R2 + R3 + . . . + RN
R
1
1
1
1
=
+
+
+ c +
(18-20)
R1
R2
R3
RN
resistors in series
R1
R2
resistors in parallel
R1
R2
(a) For the arrangement in Figure 18-12a,
first find the equivalent resistance of resistors 1 and 2.
Resistors 1 and 2 in Figure 18-12a are in parallel, so their equivalent
resistance R12 is given by Equation 18-20:
1
1
1
2
=
+
=
R 12
R
R
R
R
R 12 =
2
The combination of resistors 1 and 2 is in
series with resistor 3. This tells us the overall
equivalent resistance.
Equivalent resistor R12 is in series with resistor 3. The equivalent
resistance R123 of the entire combination is given by Equation 18-11:
R
3R
+ R =
R 123 = R 12 + R =
2
2
(b) For the arrangement in Figure 18-12b,
first find the equivalent resistance of resistors 4 and 5.
Resistors 4 and 5 in Figure 18-12b are in series, so their equivalent
resistance R45 is given by Equation 18-11:
R45 = R + R = 2R
The combination of resistors 4 and 5 is in
parallel with resistor 6. This tells us the
overall equivalent resistance.
Equivalent resistor R45 is in parallel with resistor 6. The equivalent
resistance R456 of the entire combination is given by Equation 18-20:
1
1
1
1
1
3
=
+
=
+
=
R 456
R 45
R
2R
R
2R
2R
R 456 =
3
Reflect
If we had more than three resistors, or if the resistors had different values, we could create a large number of combinations and equivalent resistances.