Series Resistors
Equivalent Resistance
Series resistors:
Equivalent Resistance
Introduction
When resistors are connected in series, a simplification of the circuit is possible. Consider the resistive circuit shown in Fig. 1(a). Since the resistors are in series, they both carry the same current. Ohm's law gives:
(1)
\[{v_1} = {R_1}i\] \[{v_2} = {R_2}i\]Applying KVL around the loop:
(2)
\[ - v + {v_1} + {v_2} = 0 \Rightarrow v = {v_1} + {v_2}\]Substituting equations (1) into equation (2) and solving for the current i results in:
(3)
\[i = \frac{v}{{{R_1} + {R_2}}}\]- Now consider the circuit of Fig. 1(b). Application of Ohm's law to this circuit and solution for the current, i, gives:
- Comparing equation (3) with equation (4), we can see that the circuits of Figs. 1(a) and 1(b) have the voltage-current characteristics at their terminals if we select:
(4)
\[i = \frac{v}{{{R_{eq}}}}\]
(5)
\[{R_{eq}} = {R_1} + {R_2}\]Figures 1(a) and 1(b) are called equivalent circuits if the equivalent resistance of Fig. 1(b) is chosen as shown in equation (5). Req of equation (5) is called the equivalent resistance of the series combination of resistors R1 and R2.
Important Point:
This result can be generalized to a series combination of N resistances as follows:
- A series combination of N resistors R1, R2, ... , RN can be replaced with a single equivalent resistance \({R_{eq}} = {R_1} + {R_2} + \cdots + {R_N}\). The equivalent circuit can be analyzed to determine the current through the series combination of resistors.
Example 1:
Replace any series resistors in the circuit below with their equivalent resistance.
Solution:
All four resistors in the circuit are in series. We can add all four resistances to obtain the equivalent resistance, so the equivalent resistance is:
And the equivalent circuit is:
Example 2:
Replace any series resistors in the circuit below with their equivalent resistance.
Solution:
The two resistors to the far right—the 1.5 kΩ and 2.2 kΩ resistors—are not in series. However, recall that KCL can be applied over any enclosed surface. If we define a surface which encloses the 1.5 kΩ and 2.2 kΩ resistors, as shown below, we see that the 4.7k and 6.8k resistors are in series and can be combined into a ingle 11.5 kΩ resistor.
The 1k and 1.5k resistors to the left of the source are more obviously in series, and can be combined into a single 2.5k resistor. The final circuit is shown below.
Test Your Knowledge!
- Where possible in the circuits below, replace series resistances with their equivalent resistance.
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- Using only the fixed resistors from the Digilent® Analog Parts
Kit, create resistors with resistances within 5% of the following values. (You
may assume that the resistors in the Analog Parts Kit have resistances which are
exactly their nominal values.)
- 2.5 kΩ
- 3 kΩ
- 2 kΩ
- 12 kΩ
Selected Answers:
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- Here are some possible solutions. (There are other combinations which will also meet the criteria.)
- 1 kΩ and 1.5 kΩ resistors in series
- Two, 1.5 kΩ resistors in series
- 1.5 kΩ and 470Ω resistors in series
- 4.7 kΩ, 6.8 kΩ, and 1.5 kΩ resistors in series. (Could also meet the criteria with a 10 kΩ resistor and a 2.2 kΩ resistor in series.)
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