Electrical Circuits: Ohm's Law, Kirchhoff's Laws & Circuit Analysis

V = IR

P = VI = I^2 R = \dfrac{V^2}{R}

Volt (V), ampere (A), ohm (Ω), watt (W)

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R_{eq} = R_1 + R_2 + \cdots + R_n

\dfrac{1}{R_{eq}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots For two resistors: R_{eq} = \dfrac{R_1 R_2}{R_1 + R_2}

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A favorite exam trap.

The sum of currents entering a node equals the sum leaving it: \sum I_{in} = \sum I_{out} .

The sum of voltages around any closed loop is zero: \sum V = 0 .

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

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I_1 = I_{total}\dfrac{R_2}{R_1 + R_2}

R = \dfrac{\rho L}{A} : longer wires resist more, thicker wires resist less

Ammeter in series (it has very low resistance). Voltmeter in parallel across the component (it has very high resistance).

Conventional current flows from + to − (high to low potential). Electrons actually move the opposite way.

Q = CV , stored energy E = \tfrac{1}{2}CV^2 . In parallel, capacitances add; in series, they combine by reciprocals (the opposite of resistors).

V = L\dfrac{di}{dt} , stored energy E = \tfrac{1}{2}LI^2 . Inductors combine like resistors (add in series).

A capacitor acts as an open circuit (no current). An inductor acts as a short circuit (a plain wire).

\tau = RC . After one \tau a charging capacitor reaches about 63% of the final voltage; after about 5\tau it is considered fully charged.

Any linear two-terminal circuit can be replaced by a voltage source V_{Th} in series with a resistance R_{Th} . V_{Th} : open-circuit voltage. R_{Th} : resistance seen from the terminals with independent sources turned off (voltage sources shorted, current sources opened).

A current source I_N in parallel with R_N , where I_N is the short-circuit current and R_N = R_{Th} = \dfrac{V_{oc}}{I_{sc}} .

A load receives maximum power when R_L = R_{Th} ; that power is P_{max} = \dfrac{V_{Th}^2}{4R_{Th}} .

Z_R = R , Z_L = j\omega L , Z_C = \dfrac{1}{j\omega C}

V_{rms} = \dfrac{V_{peak}}{\sqrt{2}} \approx 0.707\,V_{peak}

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