V, I, power & energy formulas

$N$ 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.