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Series Combination Rules

Module by: Don Johnson. E-mail the author

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Summary: A discussion of combination rules for circuit elements connected in series and of the voltage divider rule.

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This module has been superceded; please check out this module for more up-to-date material.

Figure 1: The circuit shown is perhaps the simplest circuit that performs a signal processing function. The input is provided by the voltage source v in v in and the output is the voltage v out v out across the resistor labelled R 2 R 2 .
(a) (b)
Figure 1(a) (circuit4.png)Figure 1(b) (circuit4a.png)

The results shown in other modules (circuit elements,KVL and KCL, interconnection laws) with regard to the circuit above (Figure 1), and the values of other currents and voltages in this circuit as well, have profound implications.

Resistors connected in such a way that current from one must flow only into another—currents in all resistors connected this way have the same magnitude—are said to be connected in series. For the two series-connected resistors in the example, the voltage across one resistor equals the ratio of that resistor's value and the sum of resistances times the voltage across the series combination. This concept is so pervasive it has a name: voltage divider.

The input-output relationship for this system, found in this particular case by voltage divider, takes the form of a ratio of the output voltage to the input voltage.

v out v in = R 2 R 1 + R 2 v out v in R 2 R 1 R 2 (1)
In this way, we express how the components used to build the system affect the input-output relationship. Because this analysis was made with ideal circuit elements, we might expect this relation to break down if the input amplitude is too high (Will the circuit survive if the input changes from 1 volt to one million volts?) or if the source's frequency becomes too high. In any case, this important way of expressing input-output relationships—as a ratio of output to input—pervades circuit and system theory.

The current i 1 i 1 is the current flowing out of the voltage source. Because it equals i 2 i 2 , we have that v in i 1 = R 1 + R 2 v in i 1 R 1 R 2 :

Resistors in series:

The series combination of two resistors acts, as far as the voltage source is concerned, as a single resistor having a value equal to the sum of the two resistances.
This result is the first of several equivalent circuit ideas: In many cases, a complicated circuit when viewed from its terminals (the two places to which you might attach a source) appears to be a single circuit element (at best) or a simple combination of elements at worst. Thus, the equivalent circuit for a series combination of resistors is a single resistor having a resistance equal to the sum of its component resistances.

Figure 2: The resistor (on the right) is equivalent to the two resistors (on the left) and has a resistance equal to the sum of the resistances of the other two resistors.
Figure 2 (circuit4b.png)

Thus, the circuit the voltage source "feels" (through the current drawn from it) is a single resistor having resistance R 1 + R 2 R 1 R 2 . Note that in making this equivalent circuit, the output voltage can no longer be defined: The output resistor labeled R 2 R 2 no longer appears. Thus, this equivalence is made strictly from the voltage source's viewpoint.

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