Electricity & Electrical Circuits

Chief Noter

July 30, 2021
4 min read
Circuits Electronics Kirchhoff's Laws Ohm's Law Resistance

This notes sheet looks at electricity & electrical circuits, including basic electrical quantities such as potential (voltage), resistance & current, and key electrical laws: Ohm’s law & Kirchhoff’s Laws.

Electrical Potential & Kirchhoff’s Laws
Resistance & Resistors
Electrical Circuits
Summary

Electrical Potential & Kirchhoff’s Laws

Electrical Potential (Voltage)

Just like all potential quantities, there is no measurable zero voltage. Instead, we define a zero point in a circuit and measure the difference from this ‘earth’ or ‘ground’. This is defined with the symbol:

electrical ground symbol, electrical earth symbol

The passive sign convention states that:

Sources of voltage (emf) have negative power
Loads on voltage (pd) have positive power

A load is anything in the circuit that uses electrical power, like a resistor or a diode.

Kirchhoff’s Laws

Kirchhoff’s First Law states that:

Charge & current are conserved around a circuit.

This means that when current reaches a branch it splits so that the total current after the junction is the same as the total current before branching.

The second law states that:

Potential Energy around a closed circuit is conserved.

This means that the sum of all the load voltages (potential differences) must equal the total source voltage (electromotive force). Therefore, the voltage in parallel branches is the same.

Resistance& Resistors

Electrical resitance occurs wherever there is a current, and is given by Ohm’s Law:

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

$V=IR$

Resistor Series

Resistors do not come in any value, instead there are a few series of standard resistance values. These series have a set number of resistances between 1 and 10, and then repeat for every power of ten.

Series	Original Values	Then…
E3 Series	1.0, 2.2, 4.7	10, 22, 47, 100, 220, 470…
E6 Series	1.0, 1.5, 2.2, 3.3, 4.7, 6.8	10, 15, 22, 33, 47, 68, 100, 150…
E12 Series	1.0, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2	10, 12, 15, 18, 22, 27, 33, 39, 47, 56…

Internal Resistance

All sources of emf have internal resistance, caused by electrons colliding with atoms inside the power supply itself. This needs to be accounted for, as the output voltage of the source (the p.d.) will be less than what it produces (the emf):

$\varepsilon=V+Ir$

$\varepsilon=I(R+r)$

$\varepsilon$ is the emf produced by the source
$V$ is the terminal p.d.
$I$ is the current
$r$ is the internal resistance
$R$ is the total load resistance in the circuit

Non-Ideal Voltmeter

An ideal voltmeter would give a totally accurate value for voltage. However, an actual voltmeter has an internal resistance. This must be accounted for in parallel:

Non-Ideal Ammeter

Similarly, an ammeter has an internal resistance. Unlike the voltmeter, however, this is modelled in series:

Electrical Circuits

Resistors in Series & Parallel

From Kirchhoff’s Laws, we know that:

$R=R_1+R_2+...$ in series

$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+...$ in parallel

Potential Dividers

electricity & electrical circuits, potential divider, potential divider diagram, potential divider equation, potentiometer

A potential divider is a circuit in which two or more resistors are connected in series, and the voltage is split across them. This allows a voltage output to be controlled in the ratio:

$\frac{V_1}{V_2}=\frac{R_1}{R_2}$

From Kirchhoff’s and Ohm’s Laws, the potential divider equation is quickly derived:

$V_{out}=V_{in}\frac{R_2}{R_1+R_2}$

A potentiometer is a variable potential divider.

Thevenin’s Theorem

Thevenin’s theorem is the idea that any network with two terminals that contains only voltage & current sources and resistors can be modelled by a simplified equivalent network.

For example, the Thevenin Theorem is useful to find the current in the 8Ωresistor in the circuit below:

electricity & electrical circuits, thevenin, thevenise

Remove the resistor we are trying to find the current in
Redraw the circuit as two potential dividers

Model the potential dividers as Thevenin sources

Left Side

$\frac{1}{R_L}=\frac{1}{4}+\frac{1}{4}$

$R_L=2\Omega$

$12\frac{4}{4+4}=6V$

Right Side

$\frac{1}{R_R}=\frac{1}{10}+\frac{1}{15}$

$R_R=6\Omega$

$25\frac{10}{10+15}=10V$

Redraw the circuit and calculate the net voltage

$V_{net}=6-10=-4V$

Solve to find the current

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

$I=\frac{-4}{2+8+6}=-0.25 A$

Ohm’s Law: $V=IR$
Kirchhoff’s Current law states that charge is conserved around a circuit
- The current in series loads is the same
- The sum of the currents in all parallel branches must equal the initial series current
Kirchhoff’s Voltage Law states that potential energy around a closed circuit is conserved
- The sum of all series voltages must equal the source p.d.
- The voltage in parallel branches is the same
Internal resistance is given by $\varepsilon=V+Ir=I(R+r)$
Resistors is series: $R=R_1+R+2+...$
Resistors in parallel: $\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+...$

Electricity & Electrical Circuits

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