4- Waveforms

What is an analogue signal?

A continuously varying physical quantity that changes smoothly with time

What is a waveform?

A graph showing how a signal changes with time

What is the general sine equation?

v = V_msin(ωt)

What is the alternative sine equation form?

v = V_msin(2πt/T)

What is the cosine equivalent of a sine wave?

v = V_mcos(ωt) = V_mcos(2πt/T)

What is Vm?

The peak (maximum) value of the waveform

What is ω?

Angular frequency in rad/s (ω = 2πf)

What is T?

The period, the time for one complete cycle

What is f?

The frequency, number of cycles per second (f = 1/T)

What is a phase shift?

The phase difference between two waves of the same frequency

What identity links sine and cosine?

sinθ = cos(θ − π/2)

How does sine relate to cosine in time?

A sine wave lags a cosine wave by π/2

What is the general phase-shifted cosine form?

vb = Vm cos(ωt + φ)

What does a positive phase shift mean?

Waveform leads (reaches peak earlier)

What does a negative phase shift mean?

Waveform lags (reaches peak later)

What phase shift corresponds to vb = −sin(ωt)?

φ = π/2

What phase shift corresponds to vb = −cos(ωt)?

φ = π

Why do we use RMS for AC waveforms?

To find the equivalent DC voltage that delivers the same power to a resistor

What does RMS voltage represent physically?

The DC-equivalent voltage that gives the same energy to a resistor over a cycle as the AC waveform

What is instantaneous power for a resistor?

v² / R

What is the general RMS formula?

Vrms = √(1/T ∫₀ᵀ v² dt)

What is the RMS of a sine wave?

Vrms = Vm / √2 ≈ 0.707Vm

What assumption is made for the derivation of sine RMS?

The sine wave has zero DC offset

What waveform amplitude does a quoted AC mains value represent?

RMS value by convention (e.g. UK mains 220–250V AC) unless otherwise stated

What is a periodic waveform?

A signal that returns to its initial value after a fixed period and repeats indefinitely

What are common periodic waveforms?

Square wave, sawtooth wave, triangular wave, sinusoidal wave

What does the variable x in waveforms represent?

A physical quantity such as voltage current or another variable that changes with time.

What is the average or offset value of a periodic waveform?

The time-average value over one cycle Xav = (1/T) ∫₀ᵀ x(t) dt

What does unipolar mean in waveforms?

A waveform that never changes sign (always above zero)

What happens if the average of a waveform is reduced below zero?

It becomes alternating and changes sign during the cycle

What is a digital signal?

A signal that takes only discrete values rather than varying continuously

What is sampling?

Converting a continuous analogue signal into discrete values at specific time intervals

What is a piecewise-constant waveform?

A digitally-generated signal made of constant-voltage or current steps, each lasting a fixed time interval Ts

What does Ns represent?

The number of time steps in one full period of the waveform

What is the average value of a piecewise-constant waveform?

The simple average of all sample values over one period

What is the equation for average voltage of a piecewise signal?

Vav = (1/Ns) Σ Vn across all samples

What is the RMS value formula for piecewise-constant signals?

Vrms = √(1/Ns Σ Vn²)

What is a PWM waveform?

A square-wave signal where the fraction of time spent 'on' is varied to control average power

What does PWM stand for?

Pulse Width Modulation

What is the duty cycle α?

The proportion of each period for which the voltage is at its maximum value

How is duty cycle defined mathematically?

α = time on / period where 0 ≤ α ≤ 1

What does the waveform output equal during the on time?

v = Vmax for 0 < t ≤ αT

What does the waveform output equal during the off time?

v = 0 for αT ≤ t ≤ T

What is the average voltage of a PWM signal?

Vav = αVmax

What is the RMS voltage of a PWM signal?

Vrms = √(αVmax²) = Vmax√α

How does power relate to duty cycle?

Average power is α times the power delivered by a constant Vmax

What is PWM used for in mechatronics?

To switch power electronically and control power delivered to loads like motors and heaters