Analog Electronics I · 120 h · Topic 8 of 8

AC measurements

In DC, a single number is enough. In AC the voltage changes from instant to instant, so you have to define what you are measuring: the amplitude, the average value or the RMS value. Mixing them up is the most common source of error in the whole lab.

Measurement and instrumentation AC RMS value Function generator Frequency counter

01The parameters of an AC signal

Figure 1. The rotating phasor generates the sine wave (animated). Below, the four values used for the same wave: peak, peak-to-peak, RMS and average.

Parameter	Symbol	Definition
Amplitude or peak value	V_p	Maximum value the signal reaches relative to zero.
Peak-to-peak value	V_pp	From minimum to maximum. In a symmetrical wave, 2·V_p.
Period	T	Duration of one complete cycle, in seconds.
Frequency	f	Cycles per second, in hertz. f = 1/T.
Angular frequency	ω	ω = 2πf, in radians per second. It appears in the expression v(t) = V_p·sin(ωt + φ).
Phase	φ	Offset relative to a reference, in degrees or radians.
Average value	V_avg	Mean over one cycle. For a pure sine wave it is zero; it is calculated over the half cycle or over the rectified wave.
RMS value	V_RMS	The DC value that would produce the same heat in a resistor.

02The RMS value

This is the central concept of the whole topic. A sine wave with a 10 V peak does not heat a resistor the way 10 V DC does: it spends most of the time below that value. The RMS value is the equivalent DC in terms of power.

⚡ Vector study of alternating current Where the sine wave comes from and what each characteristic value means, with the rotating vector on screen and controls for amplitude, frequency and phase. ›

Operational definition

It is called RMS for Root Mean Square, which describes the procedure in reverse order: the signal is squared, the mean is taken over one cycle, and then the root is extracted. It is squared because power is proportional to v², so the average makes energetic sense even when the signal changes sign.

V_{RM S} = \frac{V_{p}}{\sqrt{2}} = 0.707 \cdot V_{p}

V_{a v g} = \frac{2 V_{p}}{π} = 0.637 \cdot V_{p}

Sine wave only. The average value is taken over the full-wave rectified wave.

220 V is RMS

The household mains voltage is 220 V RMS. Its peak value is 220 × 1.414 = 311 V, and the peak-to-peak value reaches 622 V. That is the value the insulation of every component connected to the mains has to withstand, and the reason a filter capacitor for direct mains rectification must be rated 400 V, not 250 V.

Form factor and crest factor

k_{f} = \frac{V_{RM S}}{V_{a v g}} k_{c} = \frac{V_{p}}{V_{RM S}}

The form factor explains the error of cheap instruments; the crest factor shows how much headroom an amplifier or a power supply needs for a signal with peaks.

Waveform	V_RMS	V_avg	k_f	k_c
Sine	V_p/√2 = 0.707 V_p	0.637 V_p	1.111	1.414
Symmetrical square	V_p	V_p	1.000	1.000
Triangle	V_p/√3 = 0.577 V_p	0.5 V_p	1.155	1.732
Sawtooth	0.577 V_p	0.5 V_p	1.155	1.732
Half-wave rectified sine	0.5 V_p	0.318 V_p	1.571	2.000

Example · Same amplitude, different power

Three signals with a 10 V peak across a 100 Ω resistor:

Square: V_RMS = 10 V → P = 100/100 = 1.00 W
Sine: V_RMS = 7.07 V → P = 50/100 = 0.50 W
Triangle: V_RMS = 5.77 V → P = 33.3/100 = 0.33 W

The same height on the oscilloscope screen, three different powers. That is why you cannot talk about “the voltage” of an AC signal without saying which one.

03How each instrument measures AC

This is the point that decides whether a measurement is useful or not. There are two families of instruments, and they do not give the same result.

Average-responding (RMS-calibrated)

It rectifies the signal, measures its average value and multiplies it by 1.111 to display it as if it were RMS.

Correct only with a pure sine wave.
This is what inexpensive multimeters and analog meters with a rectifier do.
With a square wave it reads 11 % too high; with a triangle wave, 4 % too low; with the clipped output of a dimmer or a switching power supply, errors of up to 40 %.

True RMS

It actually calculates the root of the mean of the square, whatever the waveform.

Correct with any waveform, within its allowable crest factor (typically 3).
It says so explicitly on the front of the instrument.
Note: many are “AC-coupled”, meaning they ignore the DC component. Those that include it say True RMS AC+DC.

How to tell which instrument you have

Use the multimeter to measure a square wave of 10 V_pp from the function generator (V_p = 5 V). A True RMS meter will read 5.00 V; an average-responding one will read about 5.55 V. It is a two-minute test that is worth doing with the shop instruments and writing the result on the instrument itself.

There is one more limit: bandwidth. Almost all multimeters measure AC well up to 400 Hz or 1 kHz, and beyond that the reading starts to fall. To measure the output of an audio amplifier at 10 kHz you need an instrument that specifies that bandwidth, or you can use the oscilloscope directly.

04The function generator

It is the counterpart of the oscilloscope: one measures signals, the other produces them. Between the two you can test any circuit without depending on a real signal being available.

🔬 Instrument simulator The generator and the oscilloscope on the bench, joined by a BNC cable that you connect by dragging. The frequency dial, the range multiplier, the amplitude and the DC offset, and on the other side the division count for reading what came out. The bench also includes a spectrum analyzer and the modules for building modulations. ›

Control	What it does	Practical detail
Waveform	Sine, square, triangle; sometimes sawtooth and pulse.	The sine wave is for frequency response; the square wave, for seeing response times and distortion.
Frequency	Coarse range (decades) + fine adjustment.	Always check with the oscilloscope or the frequency counter: the generator dial often has quite a large error.
Amplitude	Output level.	Many instruments show it in V_pp, assuming a 50 Ω load. See the warning below.
Offset (DC)	Adds a DC component.	Essential for driving single-supply circuits, which do not accept negative voltages.
Duty cycle / symmetry	Changes the ratio between the high time and the low time.	Turns the triangle into a sawtooth and the square wave into a pulse train.
Attenuator	Fixed steps of −20 dB / −40 dB.	Needed to go down to millivolts without the signal filling up with noise.
Sweep	Automatically runs through a range of frequencies.	With the oscilloscope in X-Y mode it draws the response curve of a filter in a single pass.

The 50 Ω output impedance

Almost all generators have a 50 Ω internal impedance. If the instrument is set up assuming a 50 Ω load but is connected to a high-impedance circuit (the usual case in electronics), the actual output voltage is twice what the front panel shows. Many generators have a “load: 50 Ω / high Z” setting to correct this. In any case, the rule is to always verify the amplitude with the oscilloscope, not to trust the display.

Never short the output

And never connect the generator output to a point that has its own voltage (for example, the collector of a powered transistor). The generator delivers signal, it does not absorb it: its output stage gets damaged. The signal is always injected through the circuit's coupling capacitor.

05Frequency counters

A digital frequency counter does something very simple: it counts the cycles that enter during a known time interval.

Figure 2. Digital frequency counter (animated): the pulses that enter while the gate is open are counted. A longer gate gives more resolution and fewer readings per second.

f = \frac{N}{t_{g a t e}}

With a 1 s gate, the counter shows hertz directly. With 0.1 s, the reading is multiplied by 10 and the measurement is 10 times faster, but with 10 times less resolution.

The ± 1 count error

The signal and the gate are not synchronized, so the last cycle may or may not get in: there is always an uncertainty of ± 1 count. With a 1 s gate that is ± 1 Hz, which on 1 MHz is negligible (0.0001 %) but on 10 Hz is a 10 % error.

That is why, for low frequencies, instruments measure the period instead of the frequency: they count pulses of a fast clock during one cycle of the signal, and then invert. It is exactly the same ± 1 count error, but applied to a much larger number. Many frequency counters do this on their own below a certain frequency.

Method	Range where it is best	What it counts
Frequency counting	Above ~1 kHz	Cycles of the signal in a standard time.
Period measurement	Below ~1 kHz	Pulses of the standard clock during one cycle of the signal.
With prescaler	Above the counter speed (VHF/UHF)	Divides the signal by 10, 64 or 256 before counting it.

The input conditioning block is a Schmitt trigger: it converts the signal (sinusoidal, noisy, of any shape) into clean pulses. Its hysteresis is what keeps superimposed noise from generating false counts, and that is why a very small or very dirty signal can give erratic readings or nothing at all.

06Other ways to measure frequency

With the oscilloscope

You measure the period by counting divisions and then invert it. Less accurate than a frequency counter (the screen resolution is the limit) but it has an enormous advantage: you can see the shape of the signal and detect whether something is wrong. Digital oscilloscopes include automatic frequency measurement.

Lissajous figures

In X-Y mode, with the unknown signal on one axis and a reference on the other, the figure stays still only if the frequency ratio is a simple whole number. By counting how many times the figure touches the horizontal edge and the vertical edge you obtain that ratio. It is a historical method, older than the digital frequency counter, but it is still useful for comparing two signals.

07In the lab

Lab 1 · Measured RMS and calculated RMS

Generator at 100 Hz, sine wave, 10 V_pp verified with the oscilloscope.

Calculate V_p = 5 V and V_RMS = 3.54 V.
Measure with the multimeter on AC. It should agree within 2 %.
Switch to a square wave with the same amplitude. The correct RMS value is now 5.00 V. Note what the multimeter reads: if it reads about 5.55 V, it is average-responding (RMS-calibrated).
Switch to a triangle wave: the correct RMS value is 2.89 V.
Build a table with the three cases and the error of each one.

Lab 2 · Checking the generator

Set the generator to 1 kHz according to its dial and measure the actual frequency with the frequency counter and with the oscilloscope. Note the dial error. Repeat at 10 Hz, 10 kHz and 100 kHz. The usual conclusion is that the generator dial is not a measuring instrument.

Lab 3 · The ± 1 count error

Measure a 10 Hz signal with the frequency counter on a 1 s gate and note how much the reading fluctuates. Switch to a 10 s gate and check that it stabilizes. Then measure 1 MHz with both gates: the relative difference is imperceptible. It is the practical demonstration of why low frequencies are measured by period.

Lab 4 · Offset and single-supply circuits

Drive the input of a common-emitter amplifier with the generator. First without offset: the negative half cycle enters the base-emitter junction in reverse bias and the output is distorted. Then add a positive offset (or couple with a capacitor, which is the correct solution) and compare. It is the experimental justification for the coupling capacitors seen in small-signal amplifiers.

08Common errors

Symptom	Usual cause
The multimeter and the oscilloscope do not agree	RMS is being compared with peak-to-peak. You have to convert: V_RMS = V_pp/(2√2) for a sine wave.
The AC reading differs from the calculated value by 4 % to 11 %	Non-sinusoidal wave measured with an average-responding instrument: it reads too high with the square wave and too low with the triangle.
The actual amplitude is double what the generator shows	The generator is set for a 50 Ω load and the circuit is high impedance.
The multimeter reads less and less as the frequency goes up	The instrument bandwidth has been exceeded (typically 400 Hz to 1 kHz).
The frequency counter jumps around or does not read	Signal amplitude too small for the input conditioner, or noise that causes false counts.
The multimeter on AC reads something with a DC signal	True RMS AC+DC instrument, which includes the DC component. Check the mode.
A 250 V capacitor exploded when connected to the mains	The mains is 220 V RMS: the peak is 311 V. A 400 V one was needed.

09Self-assessment

What exactly does it mean for an AC voltage to be 220 V RMS?

That it produces in a resistor the same heat as 220 V of direct current. Its peak value is 311 V and the peak-to-peak value, 622 V.

A 24 V_pp sine wave: what are its peak, RMS and average values?

V_p = 12 V. V_RMS = 12 × 0.707 = 8.49 V. V_avg (full-wave rectified) = 12 × 0.637 = 7.64 V.

Why is the RMS value of a symmetrical square wave equal to its peak value?

Because the signal is always at +V_p or at −V_p: its square is always V_p², the mean of the square is V_p² and the root gives V_p. It spends no time at intermediate values, unlike the sine wave.

An average-responding multimeter measures a square wave with a 5 V peak. What does it read?

The instrument measures the average value of the rectified wave and multiplies it by 1.111 (the form factor of the sine wave, with which it was calibrated). For a square wave the average value equals the peak: 5 V. So it reads 5 × 1.111 = 5.55 V, when the true RMS value is 5.00 V: an error of +11 %. With a triangle wave it is the other way around: it reads 2.78 V against the actual 2.89 V, a −4 % error.

What is the offset control of the function generator for?

To add a DC component to the signal. It is essential when the circuit under test is powered from a single supply and does not accept negative voltages at its input.

A frequency counter with a 1 s gate measures 50 Hz. What is the relative error due to the counting uncertainty?

The error is ± 1 count out of 50: ± 2 %. To improve it you have to lengthen the gate (10 s gives ± 0.2 %) or measure the period instead of the frequency.

Why does the frequency counter need a Schmitt trigger at the input?

To convert any waveform into clean digital pulses, and above all because its hysteresis keeps superimposed noise from crossing the threshold several times and generating false counts.

What crest factor does a triangle wave have, and what is the practical implication?

k_c = √3 = 1.73. It means that for the same RMS value, its peaks are 22 % higher than those of a sine wave: the power supply and the components have to withstand that amplitude even though the instrument shows the same value.

You measure the output of a dimmer with an ordinary multimeter. Is it reliable?

No. The dimmer clips the sine wave, so its form factor is no longer 1.111 and the average-responding instrument gives large errors. You need a True RMS meter, or else measure with the oscilloscope and calculate.