Electrical Measurements and Bridges

Bridge balance condition

A bridge is a null-type network in which an unknown impedance is compared with known arms. It is balanced when the detector (galvanometer or voltmeter) reads zero, so the two detector terminals are at the same potential. For a four-arm bridge with impedances Z_1, Z_2, Z_3, Z_4 arranged so that Z_1 and Z_2 form one divider and Z_3 and Z_4 form the other, the balance condition is Z_1 Z_4 = Z_2 Z_3, meaning the products of impedances in opposite arms are equal. For a resistive Wheatstone bridge this gives R_1/R_2 = R_3/R_4, so the unknown is R_4 = R_2R_3/R_1. When the arms are complex, the balance equation has real and imaginary parts, and each part gives one condition, so two unknown elements can be found together, for example a resistance and a capacitance. For an AC bridge, the frequency at balance follows from requiring the reactive terms to match. If balance requires \omega L = R, then f = R/(2\pi L). Some bridges have balance conditions free of \omega, namely the Wheatstone, Maxwell and Schering bridges, whereas the Wien bridge requires \omega^2 = 1/(R_1R_2C_1C_2) and therefore balances at only one frequency. When the bridge is unbalanced, the detector voltage is the difference of the two divider voltages. The sensitivity is the ratio of the change in detector voltage to a small change in one arm resistance, evaluated at the operating point. In a balanced bridge no current flows through the detector, so each arm current can be found by treating the remaining network as simple series-parallel paths.

Two-wattmeter method

The two-wattmeter method measures the total active power of a three-phase, three-wire load using two single-phase wattmeters. Their current coils are placed in two lines, and their pressure coils are connected between the lines with the third line as the common reference. Each wattmeter reads the average of the product of its pressure-coil voltage and current-coil current, which equals VI\cos\theta, where \theta is the angle between those two phasors. A wattmeter therefore reads zero when its voltage and current phasors are in quadrature, and this can happen even in a balanced circuit. Each wattmeter reads the active power of the part of the circuit it measures, so the two readings together give the total active power. The sum of the two readings is the total power P = W_1 + W_2. The difference is related to the power factor by \tan\phi = \sqrt{3}\,(W_1 - W_2)/(W_1 + W_2), which gives W_1 - W_2 = (P/\sqrt{3})\tan\phi. For example, at a total of 60 kW and a power factor of 0.5 lagging, \phi = 60^\circ, so W_1 - W_2 = 60 kW and the readings are 60 kW and 0 kW. A lagging load therefore gives a larger reading on one wattmeter, and at unity power factor the two readings are equal.

Error propagation in power measurement

When a derived quantity is a product or quotient of measured quantities, the worst-case relative error is the sum of the relative errors of the factors. For sums and differences, the absolute errors add instead. For a quotient such as the power factor \cos\phi = P/(VI), the relative errors of the wattmeter reading and of both the voltage and current readings add, so the worst-case error is their sum. For a product P = VI, if the voltage has \pm 2\% error and the current has \pm 3\%, the power has \pm 5\% worst-case error. The absolute uncertainty is that percentage times the nominal value, so an indicated 400 W with \pm 5\% error is (400 \pm 20) W, which is also written as 400 W \pm 5\%. A bound quoted as an absolute value must be converted before comparison with a percentage bound.

Frequency and period counter-timer

An electronic counter-timer works in two basic modes. In frequency mode, a gate of known duration T_g opens and the counter registers the number of input cycles N, so f = N/T_g. A gate of 1 s makes the displayed count equal the frequency in hertz. In period mode, the input signal controls the gate, which stays open for one input period T = 1/f, while the counter counts pulses from a time base of frequency f_s. The displayed count is N = f_s T = f_s/f. For example, an input of 500 Hz with a 1 MHz time base gives a count of 10^6/500 = 2000. The count is always uncertain by \pm 1 pulse because the gate and the clock are not synchronized. When frequency is measured by counting clock pulses over one input period, this uncertainty gives a frequency error \Delta f \approx f^2 t_c, where t_c is the clock period. The error therefore grows rapidly with input frequency. For example, a 10 kHz input with a 1 \mus clock has \Delta f \approx 100 Hz.

Current transformer ratio and phase error

A current transformer (CT) reduces a large primary current to a standard secondary current, ideally with I_s = I_p N_p/N_s. In practice, a small excitation current is needed to magnetize the core, so the secondary ampere-turns differ from the primary ampere-turns. The difference is the excitation current I_e referred to the primary. Its component in phase with the induced EMF is the watt-loss component. For a unity power factor burden, this component mainly produces the ratio error, and the magnetizing component, which lags the EMF by 90^\circ, mainly produces the phase error. The phase error is the angle between the reversed secondary current and the primary current, and it is often given in minutes of arc, where 30 minutes equals 0.5^\circ. The burden is the impedance connected to the secondary, and the rated VA equals the square of the secondary current times the burden resistance. A 5 VA rating at 1 A secondary therefore fixes the secondary EMF at 5 V. For a measurement chain, the worst-case error is the sum of the errors from each element. For example, a class-1 CT with a \pm 1\% ratio error and a burden tolerance of \pm 0.5\% adds to the meter's reading error and digit uncertainty. A 3\frac{1}{2}-digit true-RMS meter with \pm(0.1\% of reading \pm 2 digits) contributes both a reading term and a fixed digit term to the total.

Low resistance measurement (Kelvin) methods

For resistances below about 1 \Omega, the resistance of the connecting leads and the contact resistances at the terminals are of the same order as the quantity being measured. A two-terminal or three-terminal connection then gives a large error. The four-terminal (Kelvin) method uses separate current terminals, which carry the test current, and potential terminals, which sense the voltage drop directly across the specimen. Almost no current flows through the potential leads, so lead and contact resistances are excluded. A short copper wire of a few centimetres length and sub-millimetre diameter has a resistance of only a few milliohms, which is a typical case for four-wire measurement. Ellipsometry is an optical technique for measuring film thickness and refractive index, so it is not a resistance-measurement method. The Kelvin double bridge is the standard instrument for low resistances. It has four terminals and a second set of ratio arms that cancels the effect of the connecting resistance, so it measures low resistances accurately. High-valued resistances are measured by other bridges, not by the Kelvin double bridge.

Wattmeter rating and power factor

A low-power-factor (LPF) wattmeter is designed with its full-scale power rated at the product of rated voltage, rated current and a low rated power factor. The full-scale power is P_{FS} = V_{rated} I_{rated} \cos\phi_{rated}. A low-power-factor wattmeter can therefore be used on loads with a power factor no higher than a maximum value, because the reading must not exceed full scale. At rated voltage and rated current, the largest allowed power factor is \cos\phi_{max} = P_{FS}/(V_{rated} I_{rated}). For example, a 250 V, 10 A LPF wattmeter with a full scale of 1000 W has a rated power factor of 0.4, and it cannot be used on a 250 V, 10 A load at a higher power factor without exceeding full scale.

Rectifier-type meter reading of non-sinusoidal wave

A rectifier-type (average-responding) moving-coil meter rectifies its input and deflects according to the mean of the rectified waveform. Its scale, however, is calibrated to show the RMS value of a sine wave. For a sine wave of peak V_m, the mean of the rectified wave is 2V_m/\pi and the RMS is V_m/\sqrt{2}, so the calibration factor is \pi/(2\sqrt{2}) \approx 1.11. The meter indication is therefore 1.11 times the mean of the absolute value of the input. This matches the true RMS only for sine waves. For a symmetrical square wave, the mean of the absolute value equals the peak value, and the true RMS also equals the peak. The meter therefore reads 1.11 times the peak, which is about 11% above the true RMS. A true-RMS meter does not have this error, because it measures the heating effect of the waveform.