Circuit Analysis

A. KCL, KVL

Kirchhoff's Laws

KCL, KVL, Ohm's law, and the passive sign convention — the bookkeeping that turns any resistive circuit into a solvable set of linear equations.

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B. Series/parallel equivalent circuits

Series-Parallel Equivalents and Dividers

Collapse resistor networks with series/parallel rules, read off voltages and currents with the divider formulas, and convert sources and delta-wye to finish stubborn topologies.

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C. Thevenin and Norton theorems

Thevenin and Norton Equivalents

Reduce any linear two-terminal network to a single source and resistance, swap between Thevenin and Norton forms, and apply superposition and maximum power transfer.

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D. Node and loop analysis

Nodal and Mesh Analysis

Two systematic methods that turn any resistive or AC circuit into a small linear system you can solve fast under exam pressure.

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E. Waveform analysis

Waveform Analysis: RMS and Average

Period, frequency, average, and RMS for the common waveforms — plus first-order RC/RL transients — so you can read any signal the exam throws at you.

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F. Phasors and impedance

Phasors, Impedance, and AC Power

Phasor analysis, R/L/C impedance, series-parallel combination, and the P/Q/S power triangle — the core AC toolkit for the FE-ECE exam.

Phasors turn calculus into algebra: a sinusoid at a fixed frequency becomes a single complex number, and every derivative in a circuit's differential equations becomes a multiplication by

j\omega

. That is the whole reason AC analysis is tractable, and the FE Reference Handbook's AC Circuits section gives you the phasor transforms, the impedance table, and the complex-power definitions. The points are lost in the bookkeeping — degrees versus radians, peak versus RMS, the sign of reactive power — so this concept builds the fluency to move between time domain, phasor domain, and the power triangle without slipping.

Phasor representation

A sinusoid

x(t)=X_{\max}\cos(\omega t+\phi)

maps to the phasor

\mathbf{X}=X\angle\phi

, a complex number carrying magnitude and phase at the single frequency

\omega

. The FE handbook lets the magnitude be either peak or RMS depending on context — for power work, use RMS magnitudes. Convert freely between polar

X\angle\phi

and rectangular

X\cos\phi+jX\sin\phi

: addition is easy in rectangular form, multiplication and division easy in polar form.

x(t)=X_{\max}\cos(\omega t+\phi)\;\longleftrightarrow\;\mathbf{X}=X\angle\phi

Impedance of R, L, and C

Impedance

Z=V/I

is the phasor generalization of resistance, a complex number

Z=R+jX

with resistance

R

and reactance

X

in ohms. A resistor is purely real,

Z_R=R

. An inductor has positive reactance

Z_L=j\omega L

— current lags voltage by

90^\circ

. A capacitor has negative reactance

Z_C=1/(j\omega C)=-j/(\omega C)

— current leads voltage by

90^\circ

. The mnemonic ELI the ICE man captures it: in an inductor (L) voltage E leads current I; in a capacitor (C) current I leads voltage E.

Z_R=R,\qquad Z_L=j\omega L,\qquad Z_C=\frac{1}{j\omega C}=-\frac{j}{\omega C}

Series and parallel impedance

Impedances combine exactly like resistances. In series they add; in parallel the reciprocals add (admittances add). Admittance

Y=1/Z=G+jB

has conductance

G

and susceptance

B

in siemens, and is the natural quantity for parallel branches and nodal analysis. For two impedances in parallel, the product-over-sum shortcut applies, just as with resistors.

Z_{\text{series}}=\sum_k Z_k, \qquad \frac{1}{Z_{\text{par}}}=\sum_k \frac{1}{Z_k}=\sum_k Y_k

AC power: real, reactive, apparent

With RMS phasors, real power

P

(watts) is the average power actually consumed, reactive power

Q

(vars) is the power that sloshes in and out of inductors and capacitors, and apparent power

S

(volt-amperes) is their magnitude. The angle

\theta

is measured from voltage to current and equals the impedance angle. Inductive (lagging) loads absorb positive

Q

; capacitive (leading) loads supply it, giving negative

Q

.

P=V_{\text{rms}}I_{\text{rms}}\cos\theta, \quad Q=V_{\text{rms}}I_{\text{rms}}\sin\theta, \quad S=V_{\text{rms}}I_{\text{rms}}

Complex power and the power triangle

Complex power packages everything:

\mathbf{S}=\mathbf{V}\mathbf{I}^* = P+jQ

, where

\mathbf{I}^*

is the conjugate of the current phasor. Its magnitude is

S=\sqrt{P^2+Q^2}

and its angle is the impedance angle

\theta

, forming the right-triangle relationship below. The power factor is

\text{pf}=\cos\theta=P/S

: lagging for inductive loads (current lags voltage), leading for capacitive. Equivalent forms

S=I_{\text{rms}}^2 Z = V_{\text{rms}}^2/Z^*

let you compute power directly from impedance.

\mathbf{S}=\mathbf{V}\mathbf{I}^*=P+jQ, \qquad S=\sqrt{P^2+Q^2}, \qquad \text{pf}=\cos\theta=\frac{P}{S}

Power-factor correction

A poor (low) power factor means large current for the same real power, so utilities penalize it and engineers correct it. Adding a capacitor in parallel supplies leading reactive power that cancels part of an inductive load's lagging

Q

, raising the power factor toward unity without changing

P

. The required capacitor reactive power is the difference between the original and target

Q

, both computed at the same real power.

Q_C=P\left(\tan\theta_1-\tan\theta_2\right)

Exam strategy

Fix your magnitude convention first: use RMS phasors so power formulas need no factor of

\tfrac12

. Build

Z=R+jX

in rectangular form, convert to polar to get

I=V/Z

, and read

\theta

straight off the impedance angle. For power, anchor on the triangle

S=\sqrt{P^2+Q^2}

and remember the sign: inductive/lagging gives

+Q

, capacitive/leading gives

-Q

. Compute

P

from

I^2R

as an independent check on

S\cos\theta

. For correction problems, work entirely with

Q

at constant

P

. Sanity-check that

S\ge P

always, and that pf lies between 0 and 1.

Key equations

Phasor transform

X_{\max}\cos(\omega t+\phi)\;\leftrightarrow\;X\angle\phi

Maps a sinusoid at frequency $\omega$ to a complex phasor. Magnitude is peak or RMS by context; use RMS for power.

Impedance

Z=\frac{\mathbf{V}}{\mathbf{I}}=R+jX

Ratio of voltage to current phasors. $R$ = resistance, $X$ = reactance, both in ohms.

Inductor / capacitor reactance

X_L=\omega L,\qquad X_C=\frac{1}{\omega C}

$Z_L=+jX_L$ (current lags $90^\circ$ ); $Z_C=-jX_C$ (current leads $90^\circ$ ). $\omega=2\pi f$ in rad/s.

Admittance

Y=\frac{1}{Z}=G+jB

Reciprocal of impedance. $G$ = conductance, $B$ = susceptance, in siemens. Adds in parallel.

Series / parallel impedance

Z_s=\sum Z_k,\qquad Z_p=\frac{Z_1 Z_2}{Z_1+Z_2}

Impedances add in series; product-over-sum for two in parallel (or sum admittances).

Real power

P=V_{\text{rms}}I_{\text{rms}}\cos\theta=I_{\text{rms}}^2 R

Average power dissipated (W). $\theta$ = angle from voltage to current = impedance angle.

Reactive power

Q=V_{\text{rms}}I_{\text{rms}}\sin\theta=I_{\text{rms}}^2 X

Reactive power (var). Positive for inductive (lagging), negative for capacitive (leading) loads.

Apparent power

S=V_{\text{rms}}I_{\text{rms}}=\sqrt{P^2+Q^2}

Magnitude of complex power (VA). Always $S\ge P$ .

Complex power

\mathbf{S}=\mathbf{V}\mathbf{I}^*=P+jQ

$\mathbf{I}^*$ = conjugate of current phasor. Real part is $P$ , imaginary part is $Q$ .

Power factor

\text{pf}=\cos\theta=\frac{P}{S}

Lagging (inductive) when current lags voltage; leading (capacitive) when it leads. Dimensionless, 0 to 1.

Power from impedance

\mathbf{S}=I_{\text{rms}}^2\,Z=\frac{V_{\text{rms}}^2}{Z^*}

Direct complex power from load impedance; convenient when only $Z$ and one of $V$ or $I$ is known.

Power-factor correction

Q_C=P\left(\tan\theta_1-\tan\theta_2\right)

Capacitor reactive power (kvar) to move from $\theta_1$ to a better $\theta_2$ at fixed real power $P$ . The FE Reference Handbook 10.6 prints the power triangle ( $P$ , $Q$ , pf) but not this correction formula.

Worked examples

Series RL load: current and power

Problem. A series circuit of $R=30\,\Omega$ and an inductor with reactance $X_L=40\,\Omega$ is driven by $120\,\text{V}$ RMS at $0^\circ$ . Find the current, power factor, and the real, reactive, and apparent power.

Solution. Impedance:

Z=30+j40=50\angle53.1^\circ\,\Omega

. Current:

I=120/50=2.40\,\text{A}

at

-53.1^\circ

(lagging). Power factor:

\cos53.1^\circ=0.600

lagging.

S=VI=120(2.40)=288\,\text{VA}

;

P=S\cos\theta=288(0.6)=173\,\text{W}

;

Q=S\sin\theta=288(0.8)=230\,\text{var}

(lagging). Sanity:

P=I^2R=2.40^2(30)=173\,\text{W}

and

Q=I^2X=2.40^2(40)=230\,\text{var}

— both match the triangle results.

Z=30+j40=50\angle53.1^\circ\,\Omega

Parallel R and C

Problem. A $50\,\Omega$ resistor is in parallel with a capacitor whose reactance is $50\,\Omega$ , driven by $100\,\text{V}$ RMS. Find the equivalent impedance, the total current, and the real and reactive power.

Solution. Admittances:

Y_R=1/50=0.0200\,\text{S}

,

Y_C=+j/50=j0.0200\,\text{S}

(capacitor susceptance is positive).

Y=0.0200+j0.0200=0.02828\angle45^\circ\,\text{S}

, so

Z=1/Y=35.4\angle{-45^\circ}\,\Omega

. Current:

I=VY=100(0.02828)=2.83\,\text{A}

at

+45^\circ

(leading).

S=VI=100(2.83)=283\,\text{VA}

;

P=283\cos45^\circ=200\,\text{W}

;

Q=-283\sin45^\circ=-200\,\text{var}

(capacitive, supplied). Sanity:

P=V^2/R=100^2/50=200\,\text{W}

and

|Q|=V^2/X_C=100^2/50=200\,\text{var}

— both confirm.

Z=\frac{1}{Y}=35.4\angle{-45^\circ}\,\Omega

Power-factor correction

Problem. A single-phase load draws $10\,\text{kW}$ at $0.70$ power factor lagging from a $480\,\text{V}$ source. Find the apparent power and reactive power, then the capacitor kvar needed to correct the power factor to $0.95$ lagging.

Solution.

S_1=P/\text{pf}=10/0.70=14.3\,\text{kVA}

.

\theta_1=\cos^{-1}0.70=45.6^\circ

, so

Q_1=P\tan\theta_1=10\tan45.6^\circ=10.2\,\text{kvar}

(lagging). Target:

\theta_2=\cos^{-1}0.95=18.2^\circ

,

Q_2=10\tan18.2^\circ=3.29\,\text{kvar}

. Capacitor:

Q_C=Q_1-Q_2=10.2-3.29=6.92\,\text{kvar}

. Sanity: real power is unchanged at

10\,\text{kW}

; new apparent power is

10/0.95=10.5\,\text{kVA}

, below the original

14.3\,\text{kVA}

, so current (and losses) drop as expected.

Q_C=P(\tan\theta_1-\tan\theta_2)

Common pitfalls

•Mixing peak and RMS phasor magnitudes. With peak magnitudes, power formulas need a factor of $\tfrac12$ ( $P=\tfrac12 V_{\max}I_{\max}\cos\theta$ ); with RMS they do not. Choose one convention and stay in it.
•Capacitor reactance sign: $Z_C=-j/(\omega C)$ is negative imaginary. Writing $+jX_C$ flips current from leading to lagging and breaks the power factor.
•Calling the power factor leading or lagging backwards. Lagging = current lags voltage = inductive = $+Q$ ; leading = capacitive = $-Q$ . Tie it to the sign of $Q$ , not guesswork.
•Using $\theta$ as the source-voltage angle instead of the angle from voltage to current. $\theta$ in the power formulas is the impedance angle, $\angle Z$ .
•Adding impedances in parallel directly. Impedances add in series; in parallel add the admittances (or use product-over-sum), then invert.
•Forgetting $\omega=2\pi f$ when computing reactance. $X_L=\omega L=2\pi f L$ , not $fL$ ; a factor of $2\pi$ error scales every reactance.
•In power-factor correction, subtracting apparent powers. Work with reactive power $Q$ at constant $P$ — $S$ values do not subtract because their angles differ.

References

NCEES FE Reference Handbook — Electrical and Computer Engineering: AC Circuits (Phasor Transforms, Impedance, Complex Power) — Source of the impedance table, phasor transforms, and the P/Q/S complex-power definitions.
NCEES FE Reference Handbook — Mathematics: Complex Numbers / Euler's Identity — Polar-rectangular conversion and complex arithmetic underlying phasor analysis.
Hayt, Kemmerly & Durbin, Engineering Circuit Analysis — Sinusoidal steady-state analysis, impedance, and AC power.