Unit summary

	Unit		Quantity	Relationship
J	joule	U	energy	U = QV
C	coulomb	Q	charge	Q = U / V
A	ampere	I	current	I = Q / t
V	volt	V	voltage	V = U / Q
W	watt	P	power	P = U / t
Ω	ohm	R	resistance	R = V / I
F	farad	C	capacitance	C = Q / V

Energy

In a circuit, work is done by moving charge through a component, transferring energy in the process.

This can be restated in terms of current, charge over time. Current affects the continuous operation of a circuit.

When designing circuits, we can create a potential difference across a circuit using a voltage source, such as a battery. The amount of current that will flow through the circuit for each unit of voltage placed across it is inversely proportional to the resistance of the circuit.

Laws

Kirchoff’s voltage law

Kirchoff’s current law

This is also known as conservation of current.

Thévenin’s theorem

The Thévenin-equivalent voltage is denoted V_{th}, and is equal to the open-circuit voltage of the original circuit. This is because the circuits are equivalent, and so the voltage drops across the circuits must be identical.

The Thévenin-equivalent resistance is denoted R_{th} and can be calculated from the Thévenin-equivalent voltage and current via Ohm’s law. The Thévenin-equivalent current I_{th} is equal to the short-circuit current of the original circuit.

Components

Resistors

A resistor is an imperfect conductor.

The current I that will flow through a circuit is proportional to the voltage V across the circuit, and inversely proportional to the resistance R of the circuit. This relationship is called Ohm’s Law.

I = \frac{V}{R}

Resistors in series

The total resistance R_{total} of n resistors connected in series is equal to the sum of the individual resistances. Adding a resistor will increase the resistance.

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

The current through each resistor is equal, but the voltage drop across each resistor is proportional to the individual resistance.

Resistors in parallel

The total resistance R_{total} of n resistors connected in parallel is equal to the reciprocal of the sum of the reciprocals of the individual resistances. Adding a resistor will decrease the resistance.

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

The name for the reciprocal of a resistance is conductance. The equation for the total resistance of parallel resistors is more intuitive in terms of conductances. The total conductance G_{total} of n resistors connected in parallel is equal to the sum of the individual conductances. Adding a resistor will increase the conductance.

G_{total} = \sum_{k=1}^{n} G_k

The voltage drop across each resistor is equal, but the current through each resistor is inversely proportional to the individual resistance.

The voltage drop across two parallel resistors is \frac{R_1 R_2}{R_1 + R_2}, and across three parallel resistors is \frac{R_1 R_2 R_3}{R_1 R_3 + R_2 R_3 + R_1 R_3}.

Diodes

A diode has polarity. The lead adjacent to the contrasting stripe is the cathode (positive end), and the other lead is the anode (negative end).

A diode is forward biased if the voltage across the terminals is positive (positive end is positive) and reverse biased if negative (positive end is negative).

An ideal diode will act as a perfect conductor when forward-biased (R \to \infty), and a perfect insulator when reverse-biased (R=0).

Capacitors

A capacitor is a device with two charge-storing plates separated by an insulating material. A capacitor of capacitance C and with voltage V across the terminals has charge of Q accumulated on one plate and -Q accumulated on the other. The stored charge is proportional to both the voltage and the capacitance.

Q = CV

Capacitance is proportional to the area A of each plate in square centimeters, the distance d between the two plates in millimeters, and the dielectric constant \epsilon of the insulating material.

C = 8.85 \x 10^{-14} \epsilon A / d

The current through a capacitor is proportional to the rate of change of the voltage.

I = C \frac{dV}{dt}

The energy U_c stored in a capacitor is proportional to the voltage across the capacitor.

U_c = \frac{1}{2}CV^2

The power flowing through a capacitor isn’t lost as heat like it is in a resistor. Instead, it’s stored as energy (voltage and charge) and is returned when discharged.

Capacitors in series

The total capacitance C_{total} of n capacitors connected in series is equal to the reciprocal of the sum of the reciprocals of the individual capacitances. Adding a resistor will increase the resistance.

\frac{1}{C_{total}} = \sum_{k=1}^{n} \frac{1}{C_k}

Because there is no external connection to the connection between each pair of capacitors in series, the stored charge in each capacitor is equal.

Capacitors in parallel

The total capacitance C_{total} of n capacitors connected in parallel is equal to the sum of the individual capacitances. Adding a capacitor will increase the capacitance.

C_{total} = \sum_{k=1}^{n} C_k

Circuits

RC circuits

A charged capacitor placed across a resistor discharges according to an inverse exponential curve.

V = Ae^{-t/RC}