Circuit Analysis DC and AC Cheat Sheet

Key Facts

• Ohm's law relates voltage, current, and resistance by $V = IR$ for an ideal resistor.
• Kirchhoff's current law says the algebraic sum of currents at a junction is zero, so $\sum I = 0$.
• Kirchhoff's voltage law says the algebraic sum of potential differences around a closed loop is zero, so $\sum \Delta V = 0$.
• Series resistors add directly as $R_{\text{eq}} = R_1 + R_2 + \cdots$, while parallel resistors satisfy $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$.
• Capacitors combine oppositely to resistors: series capacitors satisfy $\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$, while parallel capacitors add as $C_{\text{eq}} = C_1 + C_2 + \cdots$.
• The time constant for an RC circuit is $\tau = RC$, and the time constant for an RL circuit is $\tau = \frac{L}{R}$.
• AC impedance is $Z_R = R$, $Z_L = i\omega L$, and $Z_C = \frac{1}{i\omega C}$, where $\omega = 2\pi f$.
• Average AC power is $P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos \phi$, where $\phi$ is the phase angle between voltage and current.

Vocabulary

Node

A node is a connection point in a circuit where two or more elements meet and share the same electric potential.

Loop

A loop is any closed path through a circuit that returns to its starting point.

Impedance

Impedance is the complex opposition to AC current, written as $Z$, and measured in ohms.

Phasor

A phasor is a rotating complex representation of a sinusoidal voltage or current with a fixed amplitude and phase.

Reactance

Reactance is the frequency-dependent part of impedance caused by inductors or capacitors.

Resonance

Resonance occurs in an RLC circuit when inductive and capacitive reactances cancel, so $\omega_0 = \frac{1}{\sqrt{LC}}$.