Electrical engineering

Unit summary table

	Unit		Concept	Relationships
J	joule	E	Energy	E = QV
C	coulomb	Q	Electric charge	Q = E / V
A	ampere	I	Electric current	I = Q / t
V	volt	V	Voltage	V = E / Q
W	watt	P	Power	P = VI = E / t
\Omega	ohms	R	Resistance	R = V / I
F	farad	C	Capacitance
s	second	t	Time

Definitions

Voltage

The voltage between two points is the cost in energy required to move a unit of positive charge (coulombs) from the negative point to the positive point. A joule of work is done in moving a coulomb of charge through a potential difference of one volt (V = E / Q).

Voltages are generated by doing work on charges.

Current

Current is the rate of flow of electric charge through a point. A current of one ampere equals a flow of one coulumb per second (I = Q / t). Current is conventionally considered to flow from a positive point to a negative point, though the actual electron flow is the opposite.

We get currents by placing a voltage across two points.

Power

Power is the amount of energy dissipated by a circuit per second. One watt of power is dissipated while a current of one ampere passes through a circuit element with a voltage across it of one volt (P = V I).

An open circuit will dissipate no power. An open circuit is one where R \to \infty, so from \lim_{R \to \infty} I = V / R we can see that I = 0, and from P = VI we can see that P = 0.

A short circuit will also dissipate no power. A short circuit is one where R = 0, and from P = I^2 R we can see that P = 0.

Dynamic resistance

Dynamic resistance (also called incremental resistance or small-signal resistance) is resistance as a function of applied voltage, and is denoted as \frac{\Delta V}{\Delta I}, \frac{dV}{dI}, or R_{dyn}.

Laws

Kirchhoff’s voltage law

The sum of the voltage drops from point A to B via one path through a circuit equals the sum by any other route, and is simply the voltage drop between A and B. Therefore, components connected in parallel have the same voltage across them.

Kirchhoff’s current law

The sum of currents flowing into a node (a junction) in a circuit is equal to the sum of currents flowing out of that node. This is because, as per the principle of conservation of charge, the total electric charge in an isolated system never changes.

Ohm’s law

The relationship is I = V / R, where I is the current through the conductor, V is the voltage measured across the conductor, and R is the resistance of the conductor. R is constant, independent of the current.

Thévenin’s theorem

The Thévenin-equivalent voltage is denoted V_{th}, and the Thévenin-equivalent resistance is denoted R_{th}.

V_{th} is equal to the open-circuit voltage of the original circuit. The short-circuit current I_{th} of the Thévenin-equivalent circuit is I_{th} = V_{th} / R_{th}, so from this we can see that R_{th} = V_{th} / I_{th}.

Resistors

For resistors connected in series, the currents passing through each resistor are equal, but the voltage drop across each resistor is proportional to resistance. The total resistance R of n resistors in series is the sum of the individual resistances:

R_{total} = \sum_{k=1}^{n} R_k

For resistors connected in parallel, the voltage drops across each resistor are equal, but the current passing through each resistor is proportional to resistance. The total conductance \frac{1}{R} of n resistors in parallel is the sum of the individual conductances:

\frac{1}{R_{total}} = \sum_{k=1}^{n} \frac{1}{R_k}

For two resistors in parallel, the total resistance is:

R_1 \parallel R_2 = (\frac{1}{R_1} + \frac{1}{R_2})^{-1} = \frac{R_1 R_2}{R_1 + R_2}

For three resistors in parallel, the total resistance is:

R_1 \parallel R_2 \parallel R_3 = (\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3})^{-1} = \frac{R_1 R_2 R_3}{R_1 R_3 + R_2 R_3 + R_1 R_3}

Voltage dividers

A voltage divider is a passive linear circuit that produces an output voltage (V_{out}) that is a fraction of the input voltage (V_{in}). The circuit comprises two resistors (R_1 and R_2) in series, with V_{in} placed across both R_1 and R_2, and V_{out} being measured across R_2. When a load is connected to the voltage divider, we represent it as a third resistor R_3 placed in parallel to R_2 (the voltages across R_2 and R_3 are equal).

Transfer function

The transfer function of a circuit describes the relationship between V_{in} and V_{out}. The transfer function H of a voltage divider with no load connected, or with a load R_3 connected, is:

H = \frac{V_{out}}{V_{in}} = \frac{R_2}{R_1 + R_2} = \frac{R_2 R_3}{R_1 R_2 + R_2 R_3 + R_1 R_3}

Open-circuit voltage

To derive the open-circuit voltage of a voltage divider:

The total resistance R of the voltage divider is R_1 + R_2 (via resistors in series)
The voltage V across the voltage divider is V_{in} (by definition)
The current I through the voltage divider is \frac{V_{in}}{R_1 + R_2} (via Ohm’s law, I = \frac{V}{R})
The currents through resistors in series are equal (via Kirchhoff’s current law)
The voltage V_{R_2} across R_2 is V_{in} \frac{R_2}{R_1 + R_2} (via Ohm’s law, V = IR)

The output voltage V_{out} of a voltage divider with no load connected is measured across R_2, therefore the open-circuit voltage of a voltage divider is:

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

Voltage under load

To derive the output voltage V_{out} of a voltage divider when a load resistor R_3 is connected:

The total resistance R of the circuit is R_1 + (R_2 \parallel R_3)
This expands to R = R_1 + \frac{R_2 R_3}{R_2 + R_3} (via resistors in series and parallel)
The voltage V across the voltage divider is V_{in} (by definition)
The current I through the circuit is V_{in} (R_1 + \frac{R_2 R_3}{R_2 + R_3})^{-1} (via Ohm’s law, I = VR^{-1})
The voltage V_{R_3} across R_3 is V_{in} (\frac{R_2 R_3}{R_2 + R_3}) (R_1 + \frac{R_2 R_3}{R_2 + R_3})^{-1} (via Ohm’s law, V = IR)
This simplifies down to V_{in} \frac{R_2 R_3}{R_1 R_2 + R_2 R_3 + R_1 R_3}

The output voltage V_{out} of a voltage divider with a load resistance R_3 connected is measured across that load resistance, therefore the voltage of a voltage divider under load is:

V_{out} = V_{in} \frac{R_2 R_3}{R_1 R_2 + R_2 R_3 + R_1 R_3}

Power dissipation

To derive the power dissipation of a voltage divider:

The total resistance R of the voltage divider is R_1 + R_2 (via resistors in series)
The voltage V across the voltage divider is V_{in} (by definition)
The current I through the voltage divider is \frac{V_{in}}{R_1 + R_2} (via Ohm’s law, I = \frac{V}{R})
The power P dissipated by the voltage divider is \frac{V_{in}^2}{R_1 + R_2} (via def. of power, P = IV)

The power P dissipated by a voltage divider is:

P = \frac{V_{in}^2}{R_1 + R_2}

From this, we can see that the power dissipated by a voltage divider is inversely proportional to the total resistance. The greater the resistances of the two resistors, the lesser the power that will be dissipated.

Thévenin-equivalent circuit

The Thévenin-equivalent voltage of a voltage divider is the same as the open-circuit voltage, so:

V_{th} = V_{in} \frac{R_2}{R_1 + R_2}

The short-circuit current I of a voltage divider (with ground being connected between R_1 and R_2) is I = V_{in} / R_1, so the Thévenin-equivalent resistance is:

R_{th} = \frac{V_{th}}{I} = V_{in} \frac{R_2}{R_1 + R_2} \cdot (\frac{V_{in}}{R_1})^{-1} = \frac{R_1 R_2}{R_1 + R_2}

The Thévenin-equivalent voltage divider circuit with a load comprises two resistors (R_{th} and R_{load}) in series, with V_{th} placed across both R_{th} and R_{load}, and V_{out} measured across R_{load} (R_{load} is identical to R_3 from the original circuit).

From this, we can see that V_{out} will drop significantly when the load resistance (R_{load}) is not significantly greater than the internal resistance (R_{th}) of the voltage divider. The lesser the internal resistance of the voltage divider, the stiffer V_{out} will be.

Diodes

A diode has polarity. The lead adjacent to the contrasting stripe is the cathode, and the opposite lead is the anode.

A diode is said to be forward biased if a positive voltage is placed across it such that the anode is positive and the cathode is negative. A diode is said to be reverse biased if a negative voltage is placed across it such that the cathode is positive and the anode is negative.

An ideal diode will act as a perfect conductor when a positive voltage is placed across it, and will act as a perfect insulator when a negative voltage is placed across it.

In actuality, a diode will not begin to conduct until a small forward voltage is placed across it (around 0.6V). The voltage where this occurs is called the threshold voltage of the diode.

When a large enough reverse voltage is placed across a diode (around -50V), the diode will stop acting as an insulator and will act instead as a conductor. The voltage where this occurs is called the breakdown voltage (V_{br}) of the diode.

The voltage drop across a diode will not exceed its threshold or breakdown voltages.

Zener diodes

A Zener diode is a diode that is designed to operate around its breakdown voltage. Zener diodes can be used to provide a constant voltage source, as the voltage drop across the diode won’t exceed the breakdown voltage.

Light-emitting diodes

A light-emitting diode can be used in place of a zener diode. Light-emitting diodes have a forward voltage drop of between 1V and 2V.

The lead adjacent to the flat edge on the circumference is the cathode, and the opposite lead is the anode.