# 3.5 Series and parallel circuits

A brief look at series and parallel circuits. Also defines voltage and current divider.

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 samemagnitude—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 resistancestimes 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 formof a ratio of the output voltage to the input voltage. $\frac{{v}_{\mathrm{out}}}{{v}_{\mathrm{in}}}=\frac{{R}_{2}}{{R}_{1}+{R}_{2}}$ In this way, we express how the components used to build the system affect the input-output relationship. Because thisanalysis 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 toohigh. In any case, this important way of expressing input-output relationships—as a ratio of output toinput—pervades circuit and system theory.

The current ${i}_{1}$ is the current flowing out of the voltage source. Because it equals ${i}_{2}$ , we have that $\frac{{v}_{\mathrm{in}}}{{i}_{1}}={R}_{1}+{R}_{2}$ :

The series combination of two resistors acts, as far as the voltage source is concerned, as a single resistor having avalue 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 twoplaces to which you might attach a source) appears to be a single circuit element (at best) or a simple combination ofelements at worst. Thus, the equivalent circuit for a series combination of resistors is a single resistor having aresistance 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 ${R}_{1}+{R}_{2}$ . Note that in making this equivalent circuit, the output voltagecan no longer be defined: The output resistor labeled ${R}_{2}$ no longer appears. Thus, this equivalence is made strictly from the voltage source's viewpoint.

#### Questions & Answers

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RAW Reply
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Damian
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Brian Reply
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Rafiq
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Damian
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LITNING Reply
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LITNING Reply
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LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Rafiq
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Rafiq
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Bob Reply
write examples of Nano molecule?
Bob
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brayan
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Damian
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Damian Reply
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Renato
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Adin Reply
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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Anassong Reply
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Bharti
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Damian Reply
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How can I make nanorobot?
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s. Reply
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NANO
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Lily
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Answers please
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Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
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