Series and Parallel Circuits

The results shown in other modules (circuit elements, KVL and KCL, interconnection laws) with regard to this circuit, 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.

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

is the current flowing out of the voltage source. Because it equals =

, we have that

:

This result is the first of several equivalent circuit idea: 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.

Thus, the circuit the voltage source "feels" (through the current drawn from it) is a single resistor having resistance

. Note that in making this equivalent circuit, the output voltage can no longer be defined: The output resistor labeled

no longer appears. Thus, this equivalence is made strictly from the voltage source's viewpoint.

One interesting simple circuit (Figure 3) has two resistors connected side-by-side, what we will term a parallel connection, rather than in series. Here, applying KVL reveals that all the voltages are identical:

and

. This result typifies parallel connections. To write the KCL equation, note that the top node consists of the entire upper interconnection section. The KCL equation is

. Using the v-i relations, we find that

This circuit highlights some important properties of parallel circuits. You can easily show that the parallel combination of

and

has the v-i relation of a resistor having resistance

. A shorthand notation for this quantity is

. As the reciprocal of resistance is conductance, we can say that for a parallel combination of resistors, the equivalent conductance is the sum of the conductances.

Similar to voltage divider for series resistances, we have current divider for parallel resistances. The current through a resistor in parallel with another is the ratio of the conductance of the first to the sum of the conductances. Thus, for the depicted circuit,

. Expressed in terms of resistances, current divider takes the form of the resistance of the other resistor divided by the sum of resistances:

.

Let

denote the equivalent resistance of the parallel combination of

and

. Using

's v-i relation, the voltage across it is

. The KVL equation written around the leftmost loop has

; substituting for

, we find

or

Thus, we have the input-output relationship for our entire system having the form of voltage divider, but it does not equal the input-output relation of the circuit without the voltage measurement device. We can not measure voltages reliably unless the measurement device has little effect on what we are trying to measure. We should look more carefully to determine if any values for the load resistance would lessen its impact on the circuit. Comparing the input-output relations before and after, what we need is

. As

, the approximation would apply if

or

. This is the condition we seek:

Figure 8 summarizes the series and parallel combination results. These results are easy to remember and very useful. Keep in mind that for series combinations, voltage and resistance are the key quantities, while for parallel combinations current and conductance are more important. In series combinations, the currents through each element are the same; in parallel ones, the voltages are the same.