# Fundamental Physics/Electronics/Resistors

## Resistor

A resistor ( ${\displaystyle R}$ ) is an electronic component that resists, restricts, or opposes the flow of electrical current.

Symbol of resistor

Construction of resistor

${\displaystyle R=\rho {\frac {l}{A}}}$

where ${\displaystyle l}$ is the length of the conductor, measured in metres [m], A is the cross-section area of the conductor measured in square metres [m²] and ρ (rho) is the electrical resistivity (also called specific electrical resistance) of the material, measured in ohm-metres (Ω·m).

In This formula ${\displaystyle \rho ,{l},{A}}$ are constant,therefore ${\displaystyle R}$ already is constant.

## Resistance

Resistance is the resistor's characteristic to resist the current flow and reduces the voltage. Resistance has symbol R measured in ohm Ω can be calculated from Ohm's law

Ohm's law

${\displaystyle V=IR}$

Therefore,

${\displaystyle R={\frac {V}{I}}}$

## Electricity response

 passive electrical component Picture Electric response Resitor DC response ${\displaystyle V=IR}$${\displaystyle I={\frac {V}{R}}}$${\displaystyle P_{V}=IV}$${\displaystyle R={\frac {V}{I}}={\frac {1}{G}}=\rho {\frac {l}{A}}}$${\displaystyle G={\frac {I}{V}}={\frac {1}{R}}=\sigma {\frac {A}{l}}}$${\displaystyle B=IL=I{\frac {\mu }{2\pi r}}}$${\displaystyle P_{R}=I^{2}R(T)}$${\displaystyle R(T)=R_{o}+nT}$${\displaystyle R(T)=R_{o}e^{nT}}$${\displaystyle P=P_{V}-P_{R}}$ AC response ${\displaystyle v_{R}(t)=i(t)Z_{R}}$${\displaystyle i_{R}(t)={\frac {v(t)}{Z_{R}}}}$ ${\displaystyle P_{R}(t)=i(t)^{2}Z_{R}={\frac {v^{2}(t)}{Z_{R}}}}$${\displaystyle Z_{R}={\frac {v_{R}(t)}{i_{R}(t)}}=R+X_{R}=R\angle 0=R}$${\displaystyle X_{R}=0}$

## Resistor Configuration

### Resistors in Series

For n resitors connected adjacent to each other as shown

The total resistance

${\displaystyle R_{1}+R_{2}+...+R_{n}}$

For 2 series resistor of same value

${\displaystyle R_{t}=R_{1}+R_{2}=R+R=2R}$

### Resistors in parallel

For n resitors connected facing each other as shown

The total resistance

${\displaystyle {\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+...+{\frac {1}{R_{n}}}}$

For 2 parallel resistor of same value

${\displaystyle {\frac {1}{R_{t}}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}={\frac {R_{1}+R_{2}}{R_{1}R_{2}}}={\frac {R+R}{RR}}={\frac {2R}{RR}}={\frac {2}{R}}}$
${\displaystyle R_{t}={\frac {1}{2}}R}$

### Resistors in 2 port network

2 resistor to form 2 port network as shown

${\displaystyle v_{o}={\frac {v_{i}}{R_{1}+R_{2}}}R_{2}}$
${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {R_{1}+R_{2}}{R}}_{2}}$

3 resistor connected in T configuration

${\displaystyle v=v_{i}{\frac {R_{2}}{R_{1}+R_{2}}}}$
${\displaystyle v=v_{o}{\frac {R_{2}}{R_{3}+R_{2}}}}$
${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {R_{3}+R_{2}}{R_{1}+R_{2}}}}$

3 resistor connected in Π configuration

${\displaystyle i_{1}={\frac {v_{i}}{R_{1}}}}$
${\displaystyle i_{3}={\frac {v_{o}-v_{i}}{R_{2}}}}$
${\displaystyle i_{3}={\frac {v_{o}}{R_{3}}}}$
${\displaystyle i_{1}=i_{2}+i_{3}}$
${\displaystyle {\frac {v_{i}}{R_{1}}}={\frac {v_{o}-v_{i}}{R_{2}}}+{\frac {v_{o}}{R_{3}}}}$
${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {Y_{3}+Y_{2}}{Y_{1}+Y_{2}}}}$

## Resistor color code

Color Significant
figures
Multiplier Tolerance Temp. Coefficient (ppm/K)
Black 0 ×100 250 U
Brown 1 ×101 ±1% F 100 S
Red 2 ×102 ±2% G 50 R
Orange 3 ×103 15 P
Yellow 4 ×104 (±5%) 25 Q
Green 5 ×105 ±0.5% D 20 Z
Blue 6 ×106 ±0.25% C 10 Z
Violet 7 ×107 ±0.1% B 5 M
Gray 8 ×108 ±0.05% (±10%) A 1 K
White 9 ×109
Gold ×10-1 ±5% J
Silver ×10-2 ±10% K
None ±20% M
1. Any temperature coefficent not assigned its own letter shall be marked "Z", and the coefficient found in other documentation.