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Sinusoidal Signals: Amplitude, Frequency, and Phase

A sinusoidal signal is the reference shape for AC analysis. Real signals can be distorted, noisy, or switched, but sine waves are the foundation for power, filters, impedance, phasors, and frequency response.

Learning Objectives

By the end of this lesson, you should be able to:

  • identify amplitude, peak-to-peak value, period, frequency, and phase;
  • use the relationship f = 1 / T;
  • write a sine wave equation in time-domain form;
  • explain why phase matters when comparing two signals;
  • avoid confusing peak, peak-to-peak, and RMS values.

The Basic Sine Wave

Basic Sine Wave

A common voltage sine wave can be written as:

$$
v(t)=V_\text{peak}\sin(2\pi ft+\phi)
$$

where:

  • V_peak is the peak amplitude;
  • f is frequency in hertz;
  • t is time in seconds;
  • phi is phase angle.

Amplitude

Amplitude is the distance from the center line to the positive or negative peak.

Sine Wave Amplitude

Related terms:

Term Meaning
peak maximum magnitude from zero
peak-to-peak positive peak to negative peak
RMS equivalent heating value for a periodic waveform

For a pure sine wave:

$$
V_\text{RMS}=\frac{V_\text{peak}}{\sqrt{2}}
$$

That is why 120 V RMS mains has about 170 V peak, and 230 V RMS mains has about 325 V peak.

Frequency and Period

Frequency is cycles per second. Period is the time for one cycle.

$$
f=\frac{1}{T}
$$

Frequency Comparison

Worked Example

For a 50 Hz signal:

$$
T=\frac{1}{50}=0.02\ \text{s}=20\ \text{ms}
$$

For a 60 Hz signal:

$$
T=\frac{1}{60}=16.67\ \text{ms}
$$

Phase

Phase describes where a waveform is within its cycle relative to a reference.

Phase Relationship

Two signals may have the same amplitude and frequency but still not reach their peaks at the same time. That timing offset is phase.

title "Two sine waves with a phase offset"
time start=0 end=20 unit=ms divisions=5

REF: sine label="reference phase" amplitude=1 cycles=1 unit=norm color=#2563eb
LAG: sine label="90 degree lag" amplitude=1 cycles=1 phase=-90 unit=norm color=#dc2626

This waveform is illustrative and uses normalized amplitude.

Why Phase Matters

Phase affects:

  • power factor in AC power systems;
  • filter response;
  • motor torque in multi-phase systems;
  • timing margins in communication and clocked systems;
  • cancellation and reinforcement when signals combine.

Three-Phase Power

Industrial power often uses three sine waves separated by 120 degrees.

Three-Phase Power

That spacing provides smoother power transfer and efficient motor operation.

Rotating Generators

When a coil rotates in a magnetic field, induced voltage varies approximately sinusoidally.

Rotating Coil Generates Sine Wave

This is one reason sine waves are natural in AC power generation.

Common Phase Angles

Phase Interpretation for a sine reference
0 degrees starts at zero and rises positive
90 degrees starts at positive peak
180 degrees inverted relative to reference
270 degrees or -90 degrees starts at negative peak

Common Mistakes

  • Calling peak voltage and RMS voltage the same value.
  • Forgetting that frequency and period are reciprocals.
  • Comparing phase between signals of different frequency without defining the reference.
  • Assuming every AC waveform is a perfect sine wave.
  • Ignoring phase when calculating AC power.

Summary

A sinusoidal signal is described by amplitude, frequency, and phase. Frequency sets how fast it repeats, amplitude sets its size, and phase sets its timing relative to a reference. These ideas are the language of AC circuits, filters, motors, communications, and power systems.

Further Reading