From steady DC to signals that move
Almost everything in the earlier rungs was DC: voltages and currents that hold perfectly still while you measure them. But the two new parts from the last two guides care about nothing except change. A capacitor obeys I = C times dV/dt, so it pushes current only while its voltage is moving; an inductor obeys V = L times dI/dt, so it builds a voltage only while its current is moving. Hold a steady DC level across either one and, once it has settled, it just sits there. The interesting life of these parts begins when the signal refuses to sit still.
A signal that swings back and forth, over and over, is alternating current (AC) — the opposite of the one-way, steady direct current you have known so far. Mains power, music, and radio are all AC. Of all the shapes a swinging signal could take, one is special: the sine wave, the smooth up-and-down curve. It is the natural motion of anything that oscillates — a pendulum, a plucked string, a mass on a spring — and, as the next guide will show, it is the only waveform a network of resistors, capacitors, and inductors passes through with its shape intact, merely scaled and shifted. So we study the sine first, and everything else is built from it.
Notice how this changes the question we ask. In the last guide an RC circuit charging up was a one-time exponential approach to a final value, set by the time constant — drive it once, watch it settle, done. With AC we drive the same parts with a voltage that never settles, so the circuit never reaches a final value; it just keeps responding, cycle after cycle, forever. Same components, completely different question — and to answer it we first need a vocabulary for describing the sine itself.
Anatomy of a sine wave
Written out, a sine voltage is v(t) = Vpeak times sin(2 pi f t). Two numbers fully describe it. The first is its size. The amplitude, or peak value Vpeak, is the farthest it swings from the center line. Because it swings equally above and below, the full top-to-bottom height is twice that: the peak-to-peak value, Vpp = 2 times Vpeak. Picture a tide rising and falling around the calm sea level — the peak is high tide above mean, the peak-to-peak is the full reach from low tide to high.
The second number is how fast it swings. The frequency f counts how many complete cycles fit into one second, measured in hertz (Hz). Its partner is the period T, the time for one full cycle, and they are simple reciprocals: T = 1/f. Taiwan mains runs at 60 Hz, so one cycle lasts T = 1/60 ≈ 16.7 ms; a 1 kHz audio tone repeats every 1 ms; human hearing spans roughly 20 Hz to 20 kHz. The 2 pi inside the formula just converts whole cycles into radians, the angle a full turn sweeps — one cycle is 2 pi radians, or 360 degrees.
v(t)
^ .-"""-. .-"""-.
+Vpk| / \ / \ <- peak
| / \ / \
0 +-/-----------\-----------/-----------\--> t
| / \ / \
-Vpk|/ \ / \ <- trough
| '-...-'
|<------ T (one cycle) ------>| f = 1/T
peak-to-peak Vpp = 2 x Vpk (trough to peak)RMS: the honest size of an AC signal
Here is a real puzzle. If a sine swings from +156 V down to -156 V, what single number should we call its size? The peak (156 V) overstates it, because the signal only touches that value for an instant and spends most of the cycle smaller. The plain average over a whole cycle is worse: it is exactly zero, because every positive half is cancelled by an equal negative half. We need a fair number that captures how much actual work the signal does — how much heat it can deliver to a resistor.
That number is the RMS value — root mean square, and the name is the recipe read backwards. Square the waveform (squaring makes everything positive and, conveniently, power already goes as voltage squared), take the mean of that, then the square root to get back to volts. For a pure sine the answer is beautifully clean: Vrms = Vpeak / sqrt(2) ≈ 0.707 times Vpeak. Its meaning is the whole point: an AC voltage of X volts RMS heats a resistor exactly as much as a steady DC of X volts would. That is why mains is always quoted in RMS — Taiwan's 110 V socket means 110 V RMS, whose peak is 110 times sqrt(2) ≈ 156 V and whose peak-to-peak is about 311 V. And power into a resistive load uses the RMS value directly: P = Vrms^2 / R.
Phase: two waves out of step
Amplitude and frequency fully describe one lonely sine. But the moment you have two sines of the same frequency, a new question appears: are they in step, or does one lag behind the other? Phase is where a wave is within its cycle, measured as an angle, with one full cycle being 360 degrees (or 2 pi radians). The phase shift between two same-frequency waves is simply how far one is slid from the other: 90 degrees is a quarter-cycle slip, and 180 degrees is a perfect upside-down mirror, peak where the other has a trough.
We say a wave leads if it gets to a given point in the cycle earlier, and lags if it gets there later. One honest caveat: phase difference is only meaningful between two signals of the same frequency. If the frequencies differ, the gap between them drifts continuously rather than holding a fixed value, so a single phase number stops making sense.
Where does phase come from in a real circuit? Straight from the capacitor and inductor laws above. Because a capacitor's current follows the rate of change of its voltage (I = C times dV/dt), the current peaks when the voltage is changing fastest — which is as the voltage sweeps through zero, a quarter-cycle before the voltage itself peaks. So in a capacitor the current leads the voltage by 90 degrees; in an inductor it is the mirror image, the voltage leading the current (the current lags) by 90 degrees. The old hams' mnemonic is ELI the ICE man: in L, E leads I; in C, I leads E. This built-in quarter-cycle shift is exactly what makes reactance and impedance richer than plain resistance — the whole subject of the next guide.
The phasor picture, and seeing it for real
Juggling sines, cosines, and angle-addition by hand gets miserable fast. The escape is the phasor: picture each sine as a rotating arrow, a spinning clock hand. The arrow's length is the amplitude and the angle it starts at is the phase; as it spins at the frequency f, its shadow on the vertical axis traces out the sine in time. The magic is that two waves of the same frequency are two arrows turning together in lockstep, so they look frozen relative to each other. Phase difference becomes just the fixed angle between two arrows, and adding two waves becomes adding two arrows tip-to-tail. This single picture is what makes the impedance arithmetic of the next guide tractable instead of a trigonometric swamp.
All of this is something you can see with your own eyes on the bench. An oscilloscope draws voltage against time, so a sine appears as a wave you can measure directly: read the amplitude off the vertical grid, the period off the horizontal grid (then invert it for frequency), and with two channels you literally watch the phase shift between input and output as a sideways slide. You make the test sine with a function generator, and an AC multimeter reads its RMS for you.
- Feed the sine into one scope channel and adjust the timebase until two or three full cycles fill the screen.
- Read the peak-to-peak height off the vertical divisions, then halve it for the peak (and multiply the peak by 0.707 for the RMS of a sine).
- Measure the width of one cycle in time, then take its reciprocal: f = 1/T gives the frequency.
- For phase, show both signals at once, measure the sideways gap dt between matching zero-crossings, and convert: phase in degrees = 360 times dt / T.