2.2 Complex Numbers & Laplace Transforms
Key Takeaways
- AC circuit analysis lives in the complex plane: convert to phasors, do arithmetic in rectangular for add/subtract and in polar for multiply/divide, then convert back.
- Euler's identity e^(jθ) = cosθ + j sinθ links rectangular and polar forms and underlies every phasor and Laplace/Fourier relationship on the exam.
- Impedances combine like resistances: Z_L = jωL and Z_C = 1/(jωC) = -j/(ωC), so capacitor current leads and inductor current lags by 90 degrees.
- The Laplace transform turns differential equations into algebra; s = jω connects the s-domain transfer function to steady-state frequency response.
- FE controls and linear-systems questions reward recognizing standard Laplace pairs (step 1/s, exponential 1/(s+a), and the s-domain forms of derivative and integral) straight from the Handbook table.
Complex numbers and phasors
Alternating-current (AC) analysis is complex-number arithmetic. A sinusoid v(t) = V_m cos(ωt + φ) is represented by the phasor V = V_m∠φ, a single complex number that drops the time dependence at a fixed frequency. NCEES phasors typically use the amplitude (peak) convention in the Handbook, though some problems use RMS magnitude; read the stem carefully.
A complex number has two interchangeable forms:
- Rectangular: z = a + jb (electrical engineering uses j, not i, to avoid clashing with current).
- Polar: z = r∠θ, where r = sqrt(a^2 + b^2) and θ = atan2(b, a).
Euler's identity ties them together: e^(jθ) = cosθ + j sinθ, so z = r e^(jθ) = r(cosθ + j sinθ).
| Operation | Use which form | Rule |
|---|---|---|
| Add / subtract | Rectangular | Add real parts and imaginary parts separately |
| Multiply | Polar | Multiply magnitudes, add angles |
| Divide | Polar | Divide magnitudes, subtract angles |
| Magnitude / angle | Convert to polar | r = sqrt(a^2+b^2), θ = atan2(b,a) |
The biggest time sink is converting back and forth by hand. Drill the rectangular↔polar keys on your NCEES-approved calculator until they are automatic.
Complex impedance
In the phasor domain, Ohm's law becomes V = I Z, where impedance Z is complex and combines in series and parallel exactly like resistance:
- Resistor: Z_R = R (angle 0 degrees).
- Inductor: Z_L = jωL (impedance angle +90 degrees, so current lags voltage by 90 degrees).
- Capacitor: Z_C = 1/(jωC) = -j/(ωC) (angle -90 degrees, so current leads voltage by 90 degrees).
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. At resonance in a series RLC circuit, Z_L and Z_C cancel and the impedance is purely resistive at ω_0 = 1/sqrt(LC).
Laplace transforms
The Laplace transform F(s) = ∫ f(t) e^(-st) dt converts a time-domain differential equation into an algebraic equation in s, solves it, and inverts back. This is the backbone of linear systems and control systems questions. Two properties do most of the work:
- Derivative: L{f'(t)} = s F(s) - f(0).
- Integral: L{∫ f dt} = F(s)/s.
For zero initial conditions, differentiation becomes multiplication by s and integration becomes division by s, which is why a transfer function H(s) = Output(s)/Input(s) captures system behavior. Substituting s = jω gives the steady-state frequency response, connecting Laplace directly to Bode plots and phasors.
| f(t) | F(s) |
|---|---|
| δ(t) (unit impulse) | 1 |
| u(t) (unit step) | 1/s |
| e^(-at) u(t) | 1/(s + a) |
| t u(t) | 1/s^2 |
| sin(ωt) u(t) | ω/(s^2 + ω^2) |
| cos(ωt) u(t) | s/(s^2 + ω^2) |
These pairs are in the FE Reference Handbook. The exam skill is recognizing which row matches the stem, not memorizing the integral definition.
Express the impedance of a 0.1 H inductor at angular frequency omega = 100 rad/s in polar form.
What is the Laplace transform of f(t) = e^(-3t) u(t)?