Signals in circuit theory
An ordinary multimeter measures almost any signal other than a pure sinusoid wrongly, and most signals in power electronics are not pure sinusoids. The difference between average value and RMS value explains why.
01Classifying a signal before analyzing it
Before solving any circuit you need to know what kind of signal is exciting it, because each kind allows different tools. Four classifications organize almost everything that follows in this course:
| Classification | Distinguishes | Example |
|---|---|---|
| Continuous-time / discrete-time | Whether it is defined for every instant of time, or only at isolated instants | The voltage of a generator (continuous-time) versus the samples of an A/D converter (discrete-time) |
| Periodic / aperiodic | Whether it repeats exactly every \( T \) seconds, forever | A mains sinusoid (periodic) versus a single turn-on pulse (aperiodic) |
| Deterministic / random | Whether its future value can be predicted exactly, or only on average | A generated sinusoid (deterministic) versus the thermal noise of a resistor (random) |
| Causal / non-causal | Whether it is zero for all \( t<0 \) | The response of a real circuit to a switch that closes at \( t=0 \) is always causal |
Unless stated otherwise, this course works with deterministic signals. The treatment of random signals —noise, in particular— belongs to later courses, but it is worth having the distinction clear from now on: no real circuit is completely free of the second class.
02The elementary signals that build everything else
A handful of signals appear over and over, because any more complicated signal can be built by combining them —an idea that is exploited fully in the transient response unit—.
| Signal | Definition | What it is for in this course |
|---|---|---|
| Step \( u(t) \) | 0 for \( t<0 \), 1 for \( t \geq 0 \) | Models a switch that closes: “the source is connected at \( t=0 \)” |
| Impulse \( \delta(t) \) | Unit area concentrated at one instant, zero at every other | Models an instantaneous discharge of energy; it is the derivative of the step |
| Ramp \( r(t) = t\,u(t) \) | Grows linearly from \( t=0 \) | Models a load that increases progressively; it is the integral of the step |
| Exponential \( e^{-t/\tau}u(t) \) | Decays from 1 with time constant \( \tau \) | The form of every transient response of a first-order RC or RL circuit |
| Sinusoid \( A\cos(\omega t+\varphi) \) | Periodic, with amplitude \( A \), angular frequency \( \omega \) and phase \( \varphi \) | The sinusoidal steady-state signal, the central topic of several of the following units |
03Average value and RMS value: what an instrument actually measures
For a periodic signal, the average value can be misleading: that of a pure sinusoid is zero, even though the signal clearly “does something.” What matters for calculating the power delivered is the effective value or RMS (root mean square): the DC voltage that would deliver the same average power to the same resistor.
| Waveform | RMS value | Rectified average value |
|---|---|---|
| Sinusoid, amplitude \( V_p \) | \( V_p/\sqrt{2} \approx 0.707\,V_p \) | \( 2V_p/\pi \approx 0.637\,V_p \) |
| Symmetric square wave, amplitude \( V_p \) | \( V_p \) | \( V_p \) |
| Triangle / sawtooth, amplitude \( V_p \) | \( V_p/\sqrt{3} \approx 0.577\,V_p \) | \( V_p/2 \) |
| Pulse with duty cycle \( D \), amplitude \( V_p \) | \( V_p\sqrt{D} \) | \( V_p D \) |
You choose the shape —and the duty cycle, where applicable— and the panel computes the average value, the RMS value and the form factor using exactly the formulas in the table.
04Energy signals and power signals
An energy signal has finite total energy —\( \int_{-\infty}^{\infty}|x(t)|^2\,dt < \infty \)— typically because it dies out with time, like a single pulse or a decaying transient response. A power signal has infinite energy but finite average power, typically because it is periodic and lasts forever, like any steady-state sinusoid.
No real circuit produces a sinusoid that has always existed and will always exist; it is a useful idealization of “steady state,” after any transient has died out. The transient response unit of this course deals with energy signals; the sinusoidal steady-state and spectra unit, with power signals. They are two different tools —Laplace for one, Fourier series and integrals for the other— and choosing the wrong one does not give an approximate result: it does not converge.
05Where it shows up in electronics
An ordinary multimeter does not measure RMS: it rectifies the signal, measures its average value and multiplies by a calibrated constant so that the result is correct if the signal is sinusoidal (that constant is \( \pi/(2\sqrt2) \approx 1.11 \), the ratio between the RMS and the average value of a sinusoid). With any other waveform —the output of a motor drive, a switching charger, a PWM signal— that multimeter gives a wrong number, sometimes by more than 10 %. A true RMS multimeter computes the integral for real, whatever the waveform, and that is why it costs more and is essential for measuring power electronics.
06In the lab
Using this page’s lab, calculate the true RMS value of a square wave of 10 V amplitude and compare it with what an “average-responding, RMS-calibrated” multimeter would show, which multiplies the rectified average value by 1.11. Express the error as a percentage.
Calculate the RMS value of a PWM signal of 12 V amplitude at duty cycles of 10 %, 50 % and 90 %, and explain why that value —not the average value— is the one that determines the power delivered to a resistor.
For a battery charger with a rectifier input stage and a switching output, identify at least three different signals in the circuit and classify each one according to the four categories of the first section.
07Common mistakes
- Confusing the average value with the RMS value. The average value of a symmetric sinusoid is zero; its RMS value is not.
- Using an ordinary multimeter to measure non-sinusoidal signals. Without the “true RMS” label, the instrument assumes a waveform that may not be the real one.
- Thinking that “energy signal” and “power signal” are the same idea with a different name. They are mutually exclusive categories with different analysis tools.
- Applying the sinusoid’s RMS formula to any other waveform. Each waveform has its own factor between RMS and peak value.
- Forgetting the duty cycle when calculating the RMS of a pulse. A 10 % pulse is not the same as a 90 % one, even when the amplitude is the same.
08Self-assessment
Why is the average value of a symmetric sinusoid no use for calculating the power it delivers?
Because it is zero —the positive and negative half-cycles cancel—, while power depends on the square of the signal, which is never negative.
What does the effective or RMS value of a signal represent?
The value of a DC voltage that would deliver the same average power to the same resistor.
What is the RMS value of a symmetric square wave of amplitude \( V_p \)?
\( V_p \), because the square of the signal equals \( V_p^2 \) all the time.
What is the difference between an energy signal and a power signal?
An energy signal has finite total energy, typically because it dies out with time; a power signal has finite average power but infinite energy, typically because it is periodic and lasts forever.
Why can a multimeter without “true RMS” give a wrong value on a PWM signal?
Because it measures the rectified average value and multiplies it by a constant that is only correct if the signal is sinusoidal.
Which elementary signal is the derivative of the step?
The impulse \( \delta(t) \).
09Further reading
- Charles K. Alexander and Matthew N. O. Sadiku. Fundamentals of Electric Circuits. 7th ed., McGraw-Hill, 2021. The chapter on effective values and elementary signals, with the same treatment as this page.
- Alan V. Oppenheim, Alan S. Willsky and S. Hamid Nawab. Signals and Systems. 2nd ed., Prentice Hall, 1996. The complete classification of signals —continuous-time, periodic, energy and power— with the mathematical rigor that this page simplifies.
- Fluke Corporation. ABCs of DMMs: A Guide to Selecting and Using Digital Multimeters. Fluke Corporation. The manufacturer’s technical explanation of the difference between an “average-responding, RMS-calibrated” multimeter and a true RMS one.