Phasors and Impedance

A phasor is simply one complex number that records a wave's size and its timing, and impedance is the AC version of resistance. The name is short for "phase vector": picture an arrow whose length is the wave's size and whose angle is its phase. Phasors are what let you solve AC circuits with algebra instead of differential equations. They work because every voltage and current in an AC circuit is a wave with the same rhythm, so each one only needs those two numbers: in $\tilde V = V_0e^{j\phi}$, that's $V_0$ (the size of the wave) and $\phi$ (its head start). Then $\tilde V = Z\tilde I$ is just Ohm's law again, with $Z$ (the impedance) playing the part of resistance, except it can also shift the timing. The complex numbers make it look scary, but it's the DC rules you already know with a little rotation added. Bear with it.

Why bother? In differential equations you found an AC circuit's steady state by guessing $A\cos\omega t + B\sin\omega t$, substituting and grinding out two equations for $A$ and $B$. That's fine for one loop and miserable for a whole network. Phasors turn the calculus into complex arithmetic, and every DC technique comes back: series and parallel combinations, voltage dividers and Kirchhoff's laws. One change of notation first: circuits write $j$ for $\sqrt{-1}$, the $i$ of math class, because $i$ already means current.

Sinusoids as shadows. Euler's formula, $e^{j\theta} = \cos\theta + j\sin\theta$, gives

Picture $V_0e^{j(\omega t + \phi)}$ as an arrow of length $V_0$ spinning counterclockwise at angular speed $\omega$. The real voltage is its shadow on the real axis. Now here's the key observation. In a linear circuit driven at $\omega$, every voltage and current spins at that same $\omega$, so the factor $e^{j\omega t}$ is common to all of them and tells you nothing new. What tells them apart is the phasor $\tilde V = V_0e^{j\phi}$: its length is the amplitude and its angle is the phase.

The payoff: derivatives become multiplication.

So differentiating a sinusoid just multiplies its phasor by $j\omega$: scale by $\omega$ and rotate a quarter turn forward. That makes sense if you think about it, since the derivative of $\cos$ is $-\sin$, which is the same wave shifted a quarter cycle ahead. Taking real parts commutes with adding, with multiplying by real constants and with differentiating, so we can work entirely with complex exponentials and take the real part at the very end.

Each element's impedance, derived.

• Resistor: $v = Ri$, so $\tilde V = R\tilde I$. Voltage and current are in step.
• Inductor: $v = L\,di/dt$, so $\tilde V = j\omega L\,\tilde I$. The voltage is the current turned $90^\circ$ forward: voltage leads current.
• Capacitor: $i = C\,dv/dt$, so $\tilde I = j\omega C\,\tilde V$ and $\tilde V = \frac{1}{j\omega C}\tilde I = -\frac{j}{\omega C}\tilde I$, since $1/j = -j$. The voltage lags the current by $90^\circ$.

Notice all three read $\tilde V = Z\tilde I$. That's one Ohm's law for everything, with a complex impedance $Z$. Its magnitude $|Z|$ is the ratio of amplitudes $V_0/I_0$, and its angle is how far the voltage leads the current.

Frequency is built in. An inductor's reactance $\omega L$ grows with frequency and a capacitor's $1/(\omega C)$ shrinks. Think about what that means at the extremes. At DC ($\omega \to 0$) the inductor is a wire and the capacitor an open circuit, exactly the steady states you found in the RL and RC lessons. At very high frequency they swap roles. That one fact is the basis of every filter, and the second picture further down shows it.

Circuit laws carry over. KVL and KCL hold at every instant, and turning sinusoids into phasors is linear, so the laws hold for phasors too. Impedances in series add, in parallel they combine like resistors, and dividers work as before. Only the arithmetic is complex.

One loop, both ways. For a series RL circuit driven by $V_0\cos\omega t$, the phasor equation is $(R + j\omega L)\tilde I = V_0$, so

The current has amplitude $V_0/\sqrt{R^2 + \omega^2L^2}$ and lags the voltage by $\arctan(\omega L/R)$. Undetermined coefficients gives exactly the same thing, after a page of algebra.

A common mistake. In series RL, it's tempting to think the source voltage is just the resistor's voltage plus the inductor's, $V_0 = V_R + V_L$, because that's how series voltages worked in DC. It isn't. The two voltages peak at different moments, so they add as phasors, like perpendicular arrows: $V_0^2 = V_R^2 + V_L^2$. The first picture further down shows this. A few smaller slips: using $f$ where $\omega = 2\pi f$ belongs; taking $\arctan$ in the wrong quadrant (checking the sign of the real part prevents that); and forgetting that phasors describe only the steady state. The transient from switching on is a separate, decaying solution.

Every AC calculation an electrical engineer does runs on phasors: sizing the cables and breakers for a building's motors, designing the matching networks that couple a phone's radio to its antenna, analyzing the power grid as one giant network of complex impedances and specifying the loudspeaker crossovers and tone controls whose behavior shifts with frequency. Body-composition scales estimate body fat from the impedance of tissue at a few frequencies, since cell membranes act like tiny capacitors.