Digital instruments formulas

N is the number of full digits, so for 3½ digits N = 3 and R = 0.001. R is the smallest step as a fraction of the decade range. The half digit adds overrange, not resolution.

3½ digit: 2000 counts, maximum display 1999. 4½ digit: 20000 counts, maximum 19999. A plain N-digit display has $10^N$ counts.

This is the place value of the last displayed digit. Use the first form when ranges are named by decade (1 V, 10 V, ...). Use the second when they are named by full count (2 V, 20 V, ... on a 2000-count meter).

$(\text{f.s.}{)}_{\min }$ is the full scale of the lowest range, taken at its decade value. For example, a range that reads up to 199.9 mV counts as 100 mV. S is the smallest change in input the meter can display at all, and it equals the place value of the last digit on the lowest range.

$V_{\text{count}}$ is the value of one count on the range in use. The two terms are added because this is a worst-case bound.

The count term is fixed in volts, so the percentage error rises as the reading falls.

$S_r$ is the ramp slope in V/s and $f_c$ is the clock frequency. Both must be accurate and stable, because nothing cancels them and nothing averages out noise.

Use magnitudes; $V_{in}$ and $V_{ref}$ have opposite polarity. For an input that varies during run-up, $V_{in}$ here is its average over $T_1$. R, C and $f_c$ cancel as long as they stay constant for one conversion, so the only component value the reading depends on is $V_{ref}$. The integrator, switches and comparator must still behave ideally (linear and free of offset), and the result is quantised to ±1 count.

Assumes a dc input and an integrator that starts from 0 V. At full-scale input this must stay inside the op-amp's output swing, which sets the smallest usable RC.

For a dc input. Add any auto-zero time on top. Integrating DVMs are slow compared with successive-approximation and flash converters, because $T_1$ is usually one or more periods of the power-line frequency.

Assumes a sinusoidal interference of frequency f added to the input, and a reading equal to the average of the input over $T_1$. NMRR is the ratio of the peak interference to the worst-case error it causes in that average. It is infinite (ideal rejection) when $f{T}_1$ is a whole number, that is, when $T_1$ is an integer multiple of the interference period.

$V_{FS}$ is the full-scale input range. Some treatments use $V_{FS}/(2^n-1)$ because they define the top code as equal to $V_{FS}$; use whichever definition the question gives. An ideal rounding converter has quantisation error ±½ LSB.

This is fixed and does not depend on $V_{in}$. Some designs add start or end-of-conversion clock cycles on top.

The conversion time varies with $V_{in}$. A full-scale input takes the longest.

All comparisons happen in parallel, so a conversion takes one clock period. An 8-bit flash converter needs 255 comparators.

The source is modelled as its Thevenin equivalent ($V_{Th}$, $R_{Th}$), and $R_{in}$ is the DMM's input resistance. The error is taken relative to the true (unloaded) voltage $V_{Th}$. It is negative: the meter always reads low.

$\overline{|v|}$ is the full-wave rectified average of the actual waveform. The indication equals the true rms only for a sinusoid, because 1.11 is the sine form factor. A half-wave average-responding meter instead multiplies the half-wave average by $\pi /\sqrt{2}\approx 2.22$; it too is correct only for a sine.

$V_{\text{rms}}$ is the true rms of the actual waveform, so this error is relative to the true value. $k_f$ is 1.11 for a sine, 1 for a square wave and $2/\sqrt{3}\approx 1.155$ for a triangle wave. So a square wave reads 11.1% high and a triangle wave reads about 3.8% low.

An ac-coupled true-rms meter shows only $V_{ac,\text{rms}}$. To get the total rms, combine it with the dc reading as shown.

The $N_s$ samples must span a whole number of cycles; otherwise the reading ripples. It stays correct when harmonics are present, provided the sampling rate satisfies $f_s>2f_{\max }$.

$T_s=1/f_s$ is the sampling interval. Energy is the running sum of instantaneous power × time.

C is the meter constant in impulses per kWh (imp/kWh). For example, C = 3200 imp/kWh gives 0.3125 Wh (1125 J) per pulse.

P is the active power in kW and C is in imp/kWh.

Error is taken relative to the true energy. It is positive when the meter over-registers. $n_{\text{expected}}=C\times E_{\text{true}}$ in kWh.