GATE Electrical Measurements formula sheet

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).

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.

$N$ 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.

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.

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.

$N$ = input cycles counted in the gate time $T_g$. A 1 s gate reads to 1 Hz and a 10 s gate to 0.1 Hz.

$f_c$ = clock (time-base) frequency and $M$ = number of input periods the gate stays open for ($M=1$ for a single period). Averaging makes the resolution $1/(M{f}_c)$.