The whole Electrical and Electronic Measurements formula sheet, to print

Sign convention used throughout these notes: error = measured minus true, so a positive error means the instrument reads high. The correction is what must be added to the reading to get the true value. Some texts define the error with the opposite sign; check the convention a question uses.

Taken relative to the true value unless the question says otherwise. When the error is small, dividing by $A_m$ instead changes the result only in the second order. An accuracy quoted as a percentage of full scale is a different quantity (see limiting errors).

Accuracy is the complement of the relative error: a reading 2% in error is 98% accurate.

Change in steady output $q_o$ per change in input $q_i$; for a linear instrument, the slope of its calibration line. The unit combines output and input units.

Voltmeter of resistance $R_V$ across a source of open-circuit (Thevenin) voltage $V_s$ and internal (Thevenin) resistance $R_s$. The error is always negative (the meter reads low) and shrinks as $R_V/R_s$ grows. For a PMMC voltmeter, $R_V$ = (ohms-per-volt sensitivity) $\times$ (range in use), where ohms-per-volt $=1/I_{\text{fs}}$, the reciprocal of the full-scale current.

Ammeter of resistance $R_A$ inserted in a branch; $I_t$ is the current before insertion and $R_{\text{th}}$ the Thevenin resistance of the circuit seen from the break. The ammeter always reads low; the error is small when $R_A\ll R_{\text{th}}$.

$A_s$ = nominal or specified value, $\delta A$ = guaranteed bound, $A_a$ = actual value, which can lie anywhere in the band.

For an accuracy quoted as a fraction ${\epsilon }_{\text{FS}}$ of the full-scale value $A_{\text{FS}}$. At half scale the relative limiting error is twice the quoted figure; at one-tenth scale, ten times. Hence the range is chosen so that the reading falls in the upper part of the scale. A specification as a percentage of span uses the span (upper minus lower range value) in place of $A_{\text{FS}}$; the two differ only for a range that does not start at zero.

Absolute limiting errors add, for a difference as well as a sum. For a difference, $X$ is small when $x_1\approx x_2$, so the relative error can be very large.

Relative limiting errors add, for a quotient as well as a product.

Each relative error is weighted by the magnitude of its exponent, whether positive, negative or fractional ($k$ is an exact constant). $P=V^2/R$ gives $2{\epsilon }_V+{\epsilon }_R$; $Z_0=\sqrt{L/C}$ gives $\frac{1}{2}{\epsilon }_L+\frac{1}{2}{\epsilon }_C$.

Linearised, so valid for small errors. Use it whenever a variable appears more than once, as in $R_1{R}_2/(R_1+R_2)$: applying the product and sum rules separately treats the two appearances as independent, which can only overstate the error, never understate it.

The deviations from the mean always sum to zero, a useful arithmetic check.

Magnitudes are needed because the signed deviations cancel. For a large normally distributed set, $\bar{D}\approx 0.80\sigma$.

$\sigma$ (divide by $n$) describes a whole population or a very large set. $s$ (divide by $n-1$) is the estimate from a finite sample, because one degree of freedom is used up in computing $\bar{x}$; the difference matters only when $n$ is small. Use the one the question asks for.

Half of a large set of normally distributed readings lie within $\bar{x}\pm r$. Use the same $\sigma$ or $s$ that the question specifies.

Scatter of the mean of $n$ independent readings, each of standard deviation $\sigma$. Quadrupling the number of readings halves it; it does nothing to systematic error.

$s$ is the sample standard deviation (with $n-1$) of the $n$ readings.

$\pm a$ are the limits. Rectangular when any value within the limits is equally likely (a tolerance with no further information); a display of resolution $\Delta$ gives $a=\Delta /2$, so $u=\Delta /(2\sqrt{3})$. A certificate quoting $U$ at coverage factor $k$ gives $u=U/k$.

For $y=f(x_1,x_2,\dots )$ with uncorrelated inputs. $c_i$ is the sensitivity coefficient of input $x_i$. The same root-sum-square rule propagates the standard deviations, or the probable errors, of independent quantities: ${\sigma }_X=\sqrt{{\sum }_i(\partial X/\partial x_i{)}^2{\sigma }_i^2}$ and $r_X=\sqrt{{\sum }_i(\partial X/\partial x_i{)}^2{r}_i^2}$.

The statistical counterpart of the power rule for limiting errors: relative uncertainties weighted by the exponents, squared, added, square-rooted.

$k=2$ gives about 95% coverage and $k=3$ about 99.7% when the combined distribution is close to normal with many degrees of freedom. State $k$ with the result.

A junction irradiated at microwave frequency $f$ shows constant-voltage steps at integer $n$, each $f/K_J$ high: about $2.07\mu$V per GHz. Since 2019 $K_J$ is exact. $N$ junctions in series, each on step $n$, give $V=Nnf/K_J$; arrays of thousands of junctions give 1 V and 10 V.

In a two-dimensional electron gas at low temperature and high magnetic field, the Hall resistance takes plateau values for integer $i$, independent of the sample's dimensions and material. The $i=2$ plateau, $R_K/2\approx 12906.4\Omega$, is commonly used.

Every point equally reliable. The line passes through the centroid $(\bar{x},\bar{y})$ and the residuals sum to zero.

Same slope as above with smaller numbers. $S_{xy}=\sum (x_i-\bar{x})(y_i-\bar{y})$ and $S_{xx}=\sum (x_i-\bar{x}{)}^2$.

Only when the model is known to have zero intercept. This line does not in general pass through the centroid.

The line passes through the weighted centroid $\left(\sum w_i{x}_i/\sum w_i,\sum w_i{y}_i/\sum w_i\right)$.

Combines independent measurements of the same quantity made with different precision. The combined uncertainty is smaller than the smallest individual one; with equal ${\sigma }_i=\sigma$ it reduces to $\bar{x}$ and $\sigma /\sqrt{n}$.

Products of opposite arms are equal. The condition does not involve $E$, $G$ or the source resistance, and it is unchanged if the supply and the galvanometer are interchanged (only the sensitivity changes).

Limiting error is a guaranteed (worst-case) bound: for a product or quotient, the relative limiting errors of the factors add, whether a factor multiplies or divides. For example, arms each known to $\pm 0.05\%$ give $R$ to within $\pm 0.15\%$. Any lack of detector sensitivity (the smallest unbalance that can be seen) adds to this.

Open-circuit voltage of B with respect to D, and the resistance seen from B–D. This simple $R_{\text{th}}$ holds for a supply of negligible internal resistance; with a source resistance, find $R_{\text{th}}$ by a delta–star conversion instead.

Current in a galvanometer of resistance $G$ connected across an unbalanced bridge. $S_i$ is the galvanometer's current sensitivity (e.g. mm/μA).

Change in the open-circuit output when the unknown moves from its balance value $R$ to $R+\Delta R$, with $\Delta R\ll R$ for the approximate forms. For a given $R$, $S/(R+S{)}^2$ is largest at $S=R$, so equal arms give the largest open-circuit output per ohm of unbalance. The middle form is exact and shows the output is linear only for small $\Delta R$.

Open-circuit output with one, two or four active arms (e.g. strain gauges), each changing by $\pm \Delta R$. The active arms must be placed so their effects add: two adjacent arms must change in opposite senses, two opposite arms in the same sense. The full bridge, and the half bridge with two adjacent arms changing in opposite senses, are exactly linear. The quarter bridge, and a half bridge with two opposite arms changing in the same sense (exact output $E\Delta R/(2R+\Delta R)$), are linear only for small $\Delta R$.

$P,Q$ = outer ratio arms; $p,q$ = inner ratio arms ($P$ and $p$ on the side of the unknown $R$, $Q$ and $q$ on the side of the standard $S$); $r$ = resistance of the link between $R$ and $S$. The second term is the error caused by the link, and it has the sign of $(P/Q-p/q)$.

In practice the two ratio pairs are ganged so they stay equal, the link is made short and thick so any residual mismatch multiplies a small $r$, and every ratio arm is made much larger than the lead and contact resistances in series with it. Used for resistances below about 1 Ω, such as shunts and busbar sections.

A capacitor $C$ charged to $V_0$ discharges through the unknown $R$; $V$ is the voltage after time $t$, read on a voltmeter that draws no current (an electrostatic voltmeter). It suits very high resistances, where the decay is slow enough to time accurately; with a small $R$ the voltage collapses too quickly.

$R_1$ is the leakage resistance of the capacitor itself (together with that of the voltmeter), found by a second test with $R$ disconnected. It is always in parallel with $R$, so the first test gives $R^{\prime }<R$.

Products of the impedances of opposite arms are equal. The admittance form is convenient when arms are parallel combinations.

Both must hold at once. $\theta$ is the impedance angle: positive for an inductive arm, negative for a capacitive arm, zero for a pure resistance.

Subscript $s$ = series equivalent circuit, $p$ = parallel equivalent circuit; in the parallel model the coil's $Q=R_p/(\omega L_p)$. $\delta$ is the loss angle: $90^{\circ }$ minus the angle by which the capacitor's current leads its voltage. For small $\delta$ the capacitor's power factor $\cos \varphi =\sin \delta \approx \tan \delta =D$. For a coil, $D=1/Q$. From here on $Q$ denotes a quality factor, not a ratio arm of the DC bridges.

Converts between the series model (natural for an arm like $R+j\omega L$) and the parallel model. For $Q\ge 10$, $L_p$ and $L_s$ agree within 1%; for small $D$, $C_p\approx C_s$.

Arm 2 is a variable standard inductor $L_2$ with total series resistance $R_2$ (its own resistance plus any added resistor); arms 3 and 4 are resistors, $R_4$ opposite the unknown and $R_3$ opposite the standard. From $(R_1+j\omega L_1)R_4=(R_2+j\omega L_2)R_3$. Frequency-independent; its drawback is the need for a variable standard inductor.

Arm 4 = $R_4$ in parallel with a standard capacitor $C_4$. From $Z_1=R_2{R}_3{Y}_4=R_2{R}_3(1/R_4+j\omega C_4)$. The balance equations are free of frequency, and if $R_4$ and $C_4$ are the adjusted elements the two conditions are independent. Suited to medium-Q coils (roughly $1<Q<10$).

Arm 4 = $R_4$ in series with $C_4$. From $Z_1=R_2{R}_3/(R_4-j/(\omega C_4))$. Both $L_1$ and $R_1$ depend on $\omega$, so the frequency must be known. Suited to high-Q coils ($Q>10$).

Because ${\omega }^2{C}_4^2{R}_4^2=1/Q^2$. The approximation overstates $L_1$ by the factor $1+1/Q^2$: 1% at $Q=10$, 0.25% at $Q=20$. Use the exact form when a question gives $\omega$ and all the element values.

Topology as described above: the series $r$–$C$ branch is in parallel with $R_4$, and the detector taps its midpoint. $R_1$ is the total resistance of the coil's arm: if a variable resistor $r_1$ is added in series with a coil of resistance $R_c$ to obtain balance, $R_c+r_1=R_2{R}_3/R_4$. The balance is frequency-independent, and $r$ does not appear in the resistance condition.

From $Z_1=Z_2{Z}_3{Y}_4=\frac{R_3}{j\omega C_2}\left(\frac{1}{R_4}+j\omega C_4\right)$. Both equations are free of frequency. $R_4$, the resistor in the arm opposite the unknown, is in the numerator for $C_1$.

Series-model dissipation factor of the unknown; unlike $C_1$ and $r_1$, it contains $\omega$. With $R_4$ fixed, the $C_4$ dial can be calibrated directly in $D$ at one frequency, and $R_3$ is varied for the $C_1$ balance.

From the imaginary part of $Z_1{Y}_2=R_3/R_4$: $\omega C_2{R}_1=1/(\omega C_1{R}_2)$.

Real part of the same equation. $R_3$ is the resistor opposite the parallel-RC arm, $R_4$ the one opposite the series-RC arm.

With ganged $R_1=R_2$ and equal fixed capacitors, one dial reads frequency directly. At $f_0$ the series-RC/parallel-RC divider passes exactly one third of the voltage with zero phase shift, which is why a Wien-bridge oscillator's amplifier needs a gain of 3.

At resonance the loop's reactances cancel, so the current is $E/R$ ($R$ = coil resistance) and the capacitor voltage is $Q$ times the injected voltage. Assumes $R_{\text{sh}}$ and the capacitor's losses are negligible compared with $R$, the voltmeter does not load the circuit, and the self-capacitance is neglected.

$C_d$ = the coil's self (distributed) capacitance, which adds to the tuning capacitance as far as the resonant frequency is concerned. Neglect it only when $C\gg C_d$.

The shunt is part of the resonant loop, so the measured $Q_m$ is slightly low. The correction is negligible when $R_{\text{sh}}\ll R$ and matters for coils of very low resistance.

$C_1$ and $C_2$ are the tuning-capacitor settings for resonance at $f_1$ and at the higher frequency $f_2$. From ${\omega }^{2}L(C+C_d)=1$ at both frequencies: $C_1+C_d=n^2(C_2+C_d)$.

$Q_{\text{ind}}$ is the reading taken at tuning setting $C$. The self-capacitance acts across the coil's terminals, so the indicated Q falls short of the coil's true $\omega L/R$ by the factor $C/(C+C_d)$.

$l_s$ and $l_x$ are the balance lengths for the standard emf $V_s$ and the unknown $V_x$ at the same working current $I_w$, on a uniform wire of resistance ${\rho }_l$ per unit length; $k$ is the potential gradient (volts per unit length).

A precision resistive divider presents $V/K$ to the potentiometer ($K$ = ratio, e.g. 10 or 100) and $V_p$ is the potentiometer reading. Unlike the potentiometer itself, the divider draws current from the circuit under test.

The potentiometer reads the drop $V_p$ across a four-terminal standard resistor $R_s$ carrying the current. This is how ammeters are calibrated.

$R_x$ and a standard $R_s$ carry the same current in series, and the potentiometer reads the two drops in quick succession. Suitable for low resistances, because only the potential terminals are involved.

$V_1$ = in-phase component, $V_2$ = quadrature component, both measured against the potentiometer's reference phase; the signs of $V_1$ and $V_2$ fix the quadrant of $\theta$. The polar type gives $V$ and $\theta$ directly.

$N$ is the number of turns, $B$ the gap flux density, $A$ the coil area and $G=NBA$. With a varying current the inertia of the moving system averages the torque, so $\theta$ follows the average (DC) value. The scale is linear, and the pointer tries to deflect backwards if the current reverses.

Derived from stored energy. Torque is proportional to $i^2$, so the direction of deflection does not depend on polarity and $I$ is the rms value on AC. The scale is square-law unless the vane is shaped so that $dL/d\theta$ falls as $\theta$ increases.

$I_1$ and $I_2$ are the rms currents in the fixed and moving coils, and $\varphi$ is the angle between them (on DC, use the DC currents with $\cos \varphi =1$). Used as an ammeter or voltmeter, it reads rms and has the same calibration on DC and AC.

$T$ is one full period. A PMMC reads the signed mean $\frac{1}{T}{\int }_0^{T}vdt$, which is zero for a symmetric AC wave. Do not confuse it with $V_{\text{av}}$.

For a sine wave, $k_f=\pi /(2\sqrt{2})\approx 1.111$ and $k_c=\sqrt{2}\approx 1.414$.

$V_0$ is the DC component, and $V_1,V_2,\dots$ are the rms values of components at different frequencies, whose cross-products average to zero. For two sinusoids of the same frequency, add them as phasors first.

Applies to an average-responding full-wave meter whose scale is marked in rms of a sine. $V_{\text{av}}$ is the mean of $|v|$. The reading is correct only when the wave's $k_f=1.111$.

Applies to an average-responding half-wave meter with its scale marked in rms of a sine. Only the conducting half-cycle contributes. Reversing the leads measures the other half, which gives a different reading on an asymmetric wave.

Applies to a full-wave average-responding meter calibrated for a sine, with $k_f$ computed from the mean of $|v|$. The error is a percentage of the TRUE rms of the applied wave. Square wave ($k_f=1$): $+11.1\%$. Triangle ($k_f=1.155$): $-3.8\%$.

$m$ is the multiplying power: the new full-scale current $I$ divided by the movement's full-scale current. The shunt and the movement share the same voltage.

$V$ is the desired full-scale voltage. The total meter resistance is $V/I_{fs}$.

The voltmeter's resistance depends on the RANGE selected, not on the reading. A higher sensitivity (smaller $I_{fs}$) means less loading.

For a PMMC movement with ideal diodes and a scale marked in rms of a sine, where $S_{dc}=1/I_{fs}$. A sine of rms value $V$ has a mean of $0.9V$ after full-wave rectification and $0.45V$ after half-wave rectification, so the total resistance for full scale on an AC range $V$ is $0.9V/I_{fs}$ or $0.45V/I_{fs}$.

$V_{th}$ and $R_{th}$ form the Thevenin equivalent seen from the voltmeter's terminals, and $V_{th}$ is the true (unloaded) voltage. The error is a percentage of the true value and is always negative.

$R_{th}$ is the Thevenin resistance of the loop the ammeter is inserted into, and $R_A$ is the ammeter's resistance. The error is a percentage of the true current, and the current is always reduced.

$C_{in}$ includes the capacitance of the scope input and the cable. Example: a 9 MΩ tip resistor with a 1 MΩ ∥ 18 pF input gives 10:1 and needs $C_p=2$ pF. The circuit under test then sees 10 MΩ in parallel with 1.8 pF. When the condition holds, the division ratio is the same at all frequencies.

$K$ is the nominal ratio, primary:secondary. For example, a 100/5 A CT has $K_{CT}=20$, and an 11 kV/110 V PT has $K_{PT}=100$. Ratio and phase-angle errors are neglected here.

The ammeter carries the load current plus the voltmeter's current. This connection is preferred for low-voltage, high-current loads.

The voltmeter reads the load voltage plus the ammeter's drop. This connection is preferred for high-voltage, low-current loads.

$V$ and $I$ are rms values of sinusoids. $\varphi$ is the angle by which the current lags the voltage: for a lagging (inductive) load, $\varphi >0$ and $Q>0$.

A known non-inductive resistance $R$ is connected in series with the load. $V_1$ is the voltage across the series combination, $V_2$ the voltage across $R$ and $V_3$ the voltage across the load; $P$ and $\cos \varphi$ are the load's. It follows from the phasor sum $V_1^2=V_2^2+V_3^2+2V_2{V}_3\cos \varphi$ with $I=V_2/R$.

A known non-inductive resistance $R$ is connected in parallel with the load. $I_1$ is the total current, $I_2$ the current in $R$ and $I_3$ the load current. It follows from $I_1^2=I_2^2+I_3^2+2I_2{I}_3\cos \varphi$ with $V=I_{2}R$.

$V_0$ and $I_0$ are the DC components, $V_n$ and $I_n$ the rms values of the $n$th harmonics, and ${\varphi }_n$ the angle between them. Products of components at different frequencies average to zero, so a harmonic present in only one of $v$ and $i$ contributes no power. An electrodynamometer wattmeter with negligible PC inductance reads this $P$. $V_{\text{rms}}$ and $I_{\text{rms}}$ are the full rms values, including every harmonic.

Applies to a wattmeter calibrated on DC with a PC resistance of $R_p$, on a sinusoidal supply. The PC current lags $V$ by $\beta$. $\varphi$ is positive for a lagging load and negative for a leading load.

Multiply the reading by this factor. Equivalently, $W/P_{\text{true}}={\cos }^2\beta (1+\tan \varphi \tan \beta )$.

Valid for small $\beta$. A lagging load makes the wattmeter read high, and a leading load makes it read low. The error grows quickly as the power factor falls.

$R_c$ is the CC resistance and $R_p$ the PC-circuit resistance. With the PC on the load side, the CC carries the PC current as well as the load current. Choose the connection whose extra loss is smaller: the load-side PC for high-current, low-voltage loads, and the supply-side PC for low-current, high-voltage loads.

$V_r$ and $I_r$ are the rated (range) voltage and current, and $\cos {\varphi }_r$ is the rated power factor: 1 for a standard wattmeter, and the stated low value (for example 0.2) for a low power-factor wattmeter. The multiplying factor is in watts per division. When the wattmeter is fed through a CT and a PT, multiply further by $K_{CT}{K}_{PT}$.

For a balanced sinusoidal load, with $\varphi$ the load phase angle (positive for lagging). With phase sequence RYB, current coils in lines R and Y and both pressure coils returned to B, $W_1$ is the wattmeter in line R.

$P=W_1+W_2$ holds for ANY 3-wire load. $Q$ and $\tan \varphi$ hold only for a balanced sinusoidal load. A negative reading enters every formula with its sign.

For a balanced sinusoidal load, with $W_1$ the larger reading and $W_2$ entered with its sign, so $-1\le r\le 1$. It follows from the $\tan \varphi$ expression. Checks: $r=1$ gives 1, $r=0$ gives 0.5, and $r=-1$ gives 0.

For a balanced sinusoidal 3-wire load, with the current coil in one line and the pressure coil across the other two lines (for sequence RYB: CC in R, PC from Y to B). Reversing the pressure-coil connection or the phase sequence reverses the sign of the reading.

The one-wattmeter method needs the neutral, or an artificial star point, for its pressure coil. The last expression is for a balanced sinusoidal load. $V_L$ and $I_L$ are line values, and $\varphi$ is the phase angle of each phase's load impedance.

$K$ is in rev/kWh (imp/kWh for an electronic meter), $E$ in kWh and $P$ in kW. Check the units given: some meters state the constant in kWh per revolution or W·s per revolution instead.

The error is a percentage of the true energy. A positive error means the meter registers more than was consumed (it runs fast).

$n$ is the number of pulses counted in $t$ seconds, and $K$ is in imp/kWh.

The indirect forms use wattmeter, voltmeter and ammeter readings. For a balanced three-phase load, the two-wattmeter method gives $\tan \varphi$ directly.

Read off the rating plate and fixed for the transformer. For example, 1000/5 A gives $K_n=200$ and 11000/110 V gives $K_n=100$. Readings are normally multiplied by $K_n$.

The ratio of the actual primary magnitude to the actual secondary magnitude at the operating point. It is not constant: it changes with the primary current or voltage, the burden and the burden power factor.

Both are defined so that $n\approx K_n$. A bar (single-turn) primary CT has $N_p=1$, so $n=N_s$. With turns compensation, $n$ is made slightly smaller than $K_n$.

An RCF above 1 means the nominal-ratio figure is too low. For an ideal transformer RCF = 1. In terms of RCF, ratio error $=\left(\frac{1}{\text{RCF}}-1\right)\times 100\%$.