AC Waveforms Review
Mains power, the audio from a microphone and the clock of a microcontroller are all alternating signals — voltages that swing back and forth tens, thousands or millions of times every second. Before you can analyze any AC circuit, you need a firm grasp of how AC signals are described: what parameters define a waveform, how frequency and period relate, what amplitude means, and how phase describes the timing relationship between two signals. This lesson reviews all of those fundamentals and makes sure the vocabulary is solid before the mathematics begins in earnest.
A complete sine wave with all key parameters labeled. Every AC signal can be fully described by its amplitude, frequency (or period), and phase.
View LargerWhat Is Alternating Current?
In the What Is Electricity module you learned the distinction between direct current (DC) and alternating current (AC). DC is electron flow in one consistent direction — the kind produced by a battery. AC is electron flow that continuously reverses direction, first flowing one way, then the other, in a repeating cycle. The wall outlet in your home supplies AC; the power supply in your equipment converts it to DC for the internal circuitry, but the audio, clock and wireless signals inside that equipment are also AC, often at vastly higher frequencies.
The reason AC is used for power transmission is that transformers can only work with AC. A transformer exploits the changing magnetic field produced by a changing current. DC produces a static magnetic field that cannot transfer energy through a transformer core. By using AC, power companies can step voltages up for long-distance transmission (reducing current losses) and step them back down for home use — all without mechanical moving parts.
AC is also how signals carry information. An audio signal is an AC voltage whose shape follows the sound; a Wi-Fi signal is a high-frequency AC current in an antenna that radiates electromagnetic energy; a switching power supply chops DC into high-frequency AC so that a small transformer can handle it. In each case the frequency of the AC oscillation is one of its defining properties: a 2.4 GHz Wi-Fi signal is a current alternating 2,400,000,000 times per second.
Unlike DC, where "voltage" and "current" are simple constant values, AC quantities vary continuously with time. This is what makes AC analysis more complex — and more interesting — than DC analysis. The tools you learned in the DC Circuit Theory module (Ohm's Law, KVL, KCL) all still apply to AC circuits, but they must be applied to instantaneous values or to special AC-specific quantities like impedance and reactance that you will learn in the following lessons.
The Sine Wave
Among all the possible waveform shapes, the sine wave holds a special place in electronics and physics. It is the only waveform shape that, when passed through any linear circuit (resistors, capacitors, inductors), produces output signals of exactly the same shape — only the amplitude and phase change. Square waves, triangular waves, and all other periodic waveforms are made up of sine waves added together at different frequencies (a result called Fourier analysis). For these reasons, the sine wave is the fundamental unit of AC analysis.
A sine wave is the projection of circular motion onto a straight line. Imagine a point moving at constant speed around a circle. If you plot the height of that point against time, you get a perfect sine wave. The mathematical description is:
v(t) = Vpeak × sin(2πft + φ)
where Vpeak is the peak amplitude, f is the frequency in hertz, t is time in seconds, and φ (phi) is the initial phase angle in radians.
This equation tells you the exact voltage at any instant in time. At t = 0, with zero phase shift, the voltage starts at zero and rises sinusoidally. After one quarter of a cycle the voltage reaches its positive peak. After half a cycle it returns to zero. After three quarters it reaches its negative peak. After one full cycle it returns to zero and the pattern repeats exactly.
The sine wave's smooth, continuous variation — with no sharp corners or sudden jumps — is why it is the natural shape produced by oscillating LC circuits. When a capacitor and inductor exchange energy back and forth, the voltage and current in the circuit vary sinusoidally. This is the foundation of every LC oscillator and every tuned circuit.
Frequency and Period
Two parameters describe the time behavior of a repeating waveform: frequency and period. They describe exactly the same thing from opposite perspectives.
Period (T) is the time taken to complete one full cycle. It is measured in seconds. One cycle means the waveform starts at some value, goes through its full range of positive and negative values, and returns to exactly the same starting point ready to repeat. The symbol is T, and the unit is the second (s) or milliseconds (ms) for audio frequencies.
Frequency (f) is the number of complete cycles that occur in one second. It is measured in hertz (Hz), where 1 Hz = 1 cycle per second. The symbol is f. For audio, frequencies range from about 20 Hz to 20,000 Hz. For RF, frequencies range from kHz through GHz.
Frequency and period are mathematical reciprocals of each other:
f = 1 / T and T = 1 / f
If T is in seconds, f is in hertz. If f is in hertz, T is in seconds.
The US electrical grid operates at 60 Hz. What is the period of one cycle?
T = 1 / f = 1 / 60 = 0.01667 seconds = 16.67 milliseconds
This is why AC line voltage ripple on a half-wave rectified power supply appears at 60 Hz (one pulse per line voltage cycle) or 120 Hz on a full-wave rectified supply (two pulses per line voltage cycle).
A microcontroller runs from a 16 MHz crystal. What is the period of one clock cycle?
T = 1 / f = 1 / 16,000,000 = 6.25 × 10-8 seconds = 62.5 nanoseconds
In the time it takes to blink your eye (about 0.1 s), 1,600,000 clock cycles have occurred.
Angular frequency (ω, omega) is a closely related quantity used extensively in AC circuit mathematics. Where f is in cycles per second, ω is in radians per second. Since there are 2π radians in one complete cycle:
ω = 2πf (radians per second)
You will see 2πf appear in every reactance formula in this module. It always represents angular frequency.
Amplitude
Amplitude describes how large a waveform is — the maximum displacement of the signal from zero. However, there are several ways to express amplitude, and confusing them leads to errors in circuit calculations. The full treatment of RMS, peak, and peak-to-peak values is in the next lesson (the RMS, Peak and Peak-to-Peak Values lesson), but a brief introduction here is needed to complete the waveform picture.
Peak amplitude (Vpeak) is the maximum voltage reached by the waveform, measured from the zero reference level to the top of the waveform. For a symmetric sine wave, the negative peak has the same magnitude.
Peak-to-peak amplitude (Vpp) is the total voltage swing from the most negative point to the most positive point. For a symmetric sine wave: Vpp = 2 × Vpeak. An oscilloscope measures peak-to-peak directly from the screen.
RMS amplitude (Vrms) is the root-mean-square value, which is the DC-equivalent voltage that would deliver the same power to a resistive load. Your multimeter reads RMS. For a sine wave: Vrms = Vpeak / √2 ≈ Vpeak × 0.707.
US line voltage is rated at 120 V RMS. What is the peak voltage?
Vpeak = Vrms × √2 = 120 × 1.414 = 169.7 V
This is why capacitors in utility power supplies must be rated for at least 200 V — the peak voltage reaches nearly 170 V even though the "line voltage" is called 120 V.
Phase
Phase describes where a waveform is in its cycle at a given moment. Two waveforms with the same frequency can be offset in time — one may reach its peak slightly before or after the other. This offset is expressed as a phase angle in degrees or radians.
A single waveform in isolation has no meaningful phase — phase only has meaning when comparing two signals of the same frequency. If one signal reaches its positive peak 90 degrees before another signal, we say the first signal leads by 90°, or equivalently that the second signal lags by 90°.
Phase is critical in AC circuit analysis because capacitors and inductors introduce phase shifts between voltage and current:
- In a purely resistive circuit: voltage and current are in phase (0° phase difference)
- In a purely capacitive circuit: current leads voltage by 90°
- In a purely inductive circuit: voltage leads current by 90°
- In a mixed RLC circuit: the phase angle is somewhere between −90° and +90° depending on the relative magnitudes of resistance and reactance
Two signals of the same frequency with a 90° phase difference. The blue signal leads the red signal — it reaches its peak one quarter of a cycle earlier. Phase is always relative: neither signal is inherently "ahead."
View LargerThe phase difference between voltage and current in a circuit determines the power factor — the fraction of apparent power (V × I) that is actually delivered as real power to the load. In a purely resistive circuit, all apparent power is real power (power factor = 1). In a purely reactive circuit, no real power is delivered at all (power factor = 0). Most practical loads, such as motors and transformers, fall between these extremes.
Other Waveform Shapes
Mains power and audio tones are sine waves, but you will encounter other waveform shapes in digital electronics, power supplies, and test equipment. Understanding these is important for interpreting oscilloscope displays and troubleshooting.
| Waveform | Description | Where you see it |
|---|---|---|
| Sine wave | Smooth, continuous, mathematically pure oscillation. Contains only a single frequency. | Mains power, audio tones, oscillator outputs, signal generator test signals |
| Square wave | Alternates abruptly between two fixed levels. Rich in odd harmonics (3f, 5f, 7f…). | Digital logic signals, PWM motor and LED control, clock signals in microcontrollers |
| Triangle wave | Linear ramps up and down symmetrically. Odd harmonics with lower amplitude than square wave. | Some VFO control signals, test signals for checking amplifier linearity |
| Sawtooth wave | Ramps linearly in one direction, then resets abruptly. Contains both odd and even harmonics. | Oscilloscope sweep waveform, some PWM control applications |
| Pulse | A square wave with unequal on/off times, described by its duty cycle. | Radar pulses, switching power supply waveforms, PWM motor control |
The fact that square and sawtooth waves contain many harmonics matters in practice. A digital signal or a switching power supply with very fast edges generates strong harmonics that can interfere with nearby equipment. This is why designers deliberately slow edges (with series resistors on fast signals, or snubbers in power supplies), reducing harmonic content dramatically.
AC Waveforms in Practice
Most equipment deals with AC signals at several different frequencies and amplitudes at once. Understanding the waveform parameters helps you interpret what you see on a test instrument and diagnose problems.
The oscillator or clock generates a precise signal at a fixed frequency. A 16 MHz crystal oscillator produces a waveform at exactly that frequency. The purity and stability of this signal determine how accurately the equipment keeps time and how much energy spills into other frequencies as interference.
The audio chain handles sine waves in the audio frequency range (about 20 Hz to 20 kHz). A microphone produces a complex waveform that is the sum of many audio sine waves at different frequencies, amplitudes, and phases. Fourier analysis tells us that any periodic waveform is equivalent to a sum of sine waves — a fact the entire field of audio and signal processing relies on.
The power supply rectifies the 60 Hz line voltage sine wave, producing a pulsating DC with significant ripple at 120 Hz (for full-wave rectification). Filter capacitors smooth this into an approximately constant DC voltage, but the residual ripple is a low-amplitude AC signal that, if it reaches the audio stages, causes audible hum.
The oscilloscope on your bench displays voltage waveforms in the time domain — voltage on the vertical axis, time on the horizontal axis. From an oscilloscope display you can directly read the period (time for one complete cycle) and peak-to-peak amplitude, then calculate frequency and RMS. This is the most direct way to characterize an AC signal.
You are looking at an oscilloscope screen. The waveform takes up exactly 4 divisions across the screen. The time/division setting is 50 microseconds per division. The waveform height is 6 divisions peak-to-peak, and the volts/division setting is 0.5 V/division.
Period: T = 4 divisions × 50 µs/division = 200 µs = 0.0002 s
Frequency: f = 1 / T = 1 / 0.0002 = 5,000 Hz = 5 kHz
Peak-to-peak voltage: Vpp = 6 × 0.5 = 3.0 V
Peak voltage: Vpeak = 3.0 / 2 = 1.5 V
RMS voltage: Vrms = 1.5 × 0.707 = 1.06 V
This signal could be an audio tone, a clock output, or a test tone used to check an amplifier.
Frequently Asked Questions
Is the frequency the same everywhere in a series AC circuit?
Yes. In a linear circuit, frequency does not change. The voltages across individual components may have different amplitudes and different phase angles, but all signals in a series circuit oscillate at the same frequency as the source. This is a fundamental property of linear systems. Non-linear circuits (like amplifiers driven into clipping, or mixers) can generate signals at new frequencies, but passive linear circuits — resistors, capacitors, inductors — never change the frequency of a signal passing through them.
Why does the US use 60 Hz for utility power instead of something higher?
60 Hz was chosen when power systems were first standardized in the 1890s as a compromise between competing considerations. Higher frequencies reduce transformer core size and weight, which is advantageous. But higher frequencies increase energy losses in long-distance transmission lines. 60 Hz (and Europe's 50 Hz) represent historical compromises. The human eye starts to perceive flicker in incandescent bulbs below about 50 Hz, which also influenced the choice. For electronics work, 60 Hz matters primarily because it is the frequency of the hum you will hear if power supply ripple reaches your audio circuits.
What is the difference between frequency and angular frequency?
Frequency (f) measures cycles per second in hertz. Angular frequency (ω = 2πf) measures radians per second. Since one complete cycle corresponds to 2π radians of the sine function's argument, multiplying f by 2π converts cycles to radians. Angular frequency appears in the reactance formulas (XC = 1/ωC, XL = ωL) because the mathematics of sinusoidal functions is most natural in radians. When you see 2πf in a formula, you can always substitute ω — they are the same thing.
Can I apply Ohm's Law directly to an AC circuit?
Yes, but with important caveats. For a purely resistive AC circuit, Ohm's Law works exactly as in DC: V = IR using RMS values throughout. For circuits with capacitors or inductors, you must use impedance (Z) instead of simple resistance: V = I × Z. Impedance is a complex quantity that accounts for both the magnitude of opposition to current and the phase angle. The full treatment is in the Impedance lesson (Impedance). For now: use V = IR for resistors only, and V = IZ for any circuit containing reactive components.
Test Your Knowledge
Answer the questions below to check your understanding. Every answer can be found in the lesson above.