*Individual resistors can be connected together in either a series connection, a parallel connection or combinations of both series and parallel, to produce more complex resistor networks whose equivalent resistance is the mathematical combination of the individual resistors connected together. Circuits consisting of just one battery and one load resistance are very simple to analyze, but they are not often found in practical applications. Usually, we find circuits where more than two components are connected together.*

**There are two basic ways in which to connect more than two circuit components: Series& Parallel:**

# SERIES RESISTORS AND VOLTAGE DIVISION

A series circuit is a circuit in which resistors are arranged in a chain, so the current has only one path to take. The current is the same through each resistor. The total resistance of the circuit is found by simply adding up the resistance values of the individual resistors: equivalent resistance of resistors in series : R = R_{1} + R_{2} + R_{3} + …

A series circuit is shown in the diagram above. The current flows through each resistor in turn. In this circuit the electrons flow in a counter-clockwise direction, from point 4 to point 3 to point 2 to point 1 and back around to 4.

If the values of the three resistors are:

With a 10 V battery, by V = I R the total current in the circuit is:

I = V / R = 10 / 20 = 0.5 A. The current through each resistor would be 0.5 A.

## Voltage Division

- v1= iR1 & v2 = iR2

KVL:

- v-v1-v2=0
- v= i(R1+R2)
- i = v/(R1+R2 ) =v/Req
- or v= i(R1+R2 ) =iReq
- iReq = R1+R2

## VOLTAGE DIVISION FORMULA

v1 = iR1 & v2 = iR2

i = v/(R1+R2 )

Thus:

### v1=vR1/(R1+R2)

### v2=vR2/(R1+R2)

*To solve for Req:*

*Req = R1+R2 …*

**Parallel Resistors & Current Division**

*A parallel circuit has more than one resistor (anything that uses electricity to do work) and gets its name from having multiple (parallel) paths to move along . Charges can move through any of several paths. If one of the items in the circuit is broken then no charge will move through that path, but other paths will continue to have charges flow through them.*

*A parallel circuit is a circuit in which the resistors are arranged with their heads connected together, and their tails connected together. The current in a parallel circuit breaks up, with some flowing along each parallel branch and re-combining when the branches meet again. The voltage across each resistor in parallel is the same. *

*The total resistance of a set of resistors in parallel is found by adding up the reciprocals of the resistance values, and then taking the reciprocal of the total: *

*equivalent resistance of resistors in parallel: *

*1 / R = 1 / R _{1} + 1 / R_{2} + 1 / R_{3} +… *

A parallel circuit is shown in the diagram above. In this case the current supplied by the battery splits up, and the amount going through each resistor depends on the resistance. If the values of the three resistors are:

= v/R1+ v/R2

= v(1/R1+1/R2)

=v/Req

- v =iReq
- 1/Req = 1/R1+1/R2
- Req = R1R2 / (R1+R2 )

## CURRENT DIVISION FORMULA

v = i1R1 = i2R2

v=iReq = iR1R2 / (R1+R2 )

and i1 = v /R1 & i2 =v/ R2

Thus:

### i1= iR2/(R1+R2)

### i2= iR1/(R1+R2 )

*To solve for Req:*

*Req = 1/R1+ 1/R2 … 1/Rn*

*Example:*

- Total resistor value:

2. The total current can be calculated as

3. The current in each branch

verify that,

4. The power dissipated by each resistor

or

or

or

# CONDUCTANCE

*Conductance is an expression of the ease with which electric current flows through a substance. In equations, conductance is symbolized by the uppercase letter G. The standard unit of conductance is the siemens (abbreviated S), formerly known as the mho.*

#### Series conductance:

1/Geq = 1/G1 +1/G2+…

#### Parallel conductance:

Geq = G1 +G2+…

**LEARNINGS:**

- In a series circuit, all components are connected end-to-end, forming a single path for electrons to flow.
- In a parallel circuit, all components are connected across each other, forming exactly two sets of electrically common points.
- When two resistors R1 = (1/G1) and R2 = (1/G2) are in series, their equivalent resistance Req and equivalent conductance Geq are:
- When two resistors R1 = (1/G1) and R2 = (1/G2) are in parallel, their equivalent resistance Req and equivalent conductance Geq are:
- The voltage division principle for two resistors in series is:
- The current division principle for two resistors in parallel is: