Finite Load Resistance

The real power output of the a-c source with zero output impedance, when the resistance of the gate windings and the core losses are neglected, as under the assumed ideal conditions, must be zero. This is also evident from the current and voltage waves shown in Fig. 7-13(a), from which it is apparent that the average of the instantaneous power taken over a complete cycle is zero. When a noninductive load R_L is connected in series with the gate windings, the real power output of the source must equal (k_fI_L)²(R_L + 2R_G), where R_G is the resistance of each gate winding. The current wave is then advanced in phase from the wave shown in Fig. 7-13(a). If the value of R_L + 2R_G is in the range for linear operation, i.e., low enough so that the law of equal ampere turns is practically satisfied, the current wave form will again be rectangular and of an amplitude determined by Eq. 7-37. The wave forms of applied voltage and output current including the fundamental component of output current for a noninductive load are shown in Fig. 7-14. Since sinusoidal voltage is assumed, the real power output of the a-c source is

[7-40]

where I_L1 is the rms value of the fundamental component in the gate current. It was shown in Section 5-4 that only the harmonics that are present in both the voltage and the current waves contribute to the real power. The applied voltage has no higher harmonics, and for that reason, only the fundamental in the current need be considered for the real power.

The values of V and I_L1 in Eq. 7-40 are rms. The fundamental component of a periodic rectangular wave has an amplitude that is 4/π times that of the rectangular wave. Therefore

and

[7-41]

Figure 7-13. Wave forms of (a) a-c supply voltage and output current; (b) and (c) gate voltages, and (d) voltage across impedance in series with control circuit of saturable reactor supplying a noninductive load when free even-harmonics are suppressed.

The angle α, found by substituting Eq. 7-41 in Eq. 7-40, is

[7-42]

where R_G is the resistance of each gate winding.

Figure 7-14. Source voltage and output current for resistance load when free even-harmonics are suppressed in control current.

For a given value of a-c supply voltage and premagnetizing current, if the load resistance is increased beyond a certain value, the gate current is no longer entirely a function of the control current, but of the load resistance as well. This can be seen from the extreme case in which the load resistance R_L approaches infinity, causing the gate current to approach zero, regardless of the value of premagnetizing current i_c. With a given load resistance R_L, increasing the premagnetization causes the angle α to decrease and to shift the current wave to the left, which follows from Eq. 7-40. However, when the resistance reaches a value such that (2V/R_L + 2R_G) sin α is less than N_Ci_c/NG, the wave form of the current is no longer rectangular, and the critical value of a is determined as follows

[7-43]

from Eq. 7-49

[7-44]

Hence, from Eqs. 7-43 and 7-44, we get

[7-45]

and

Therefore, the range of load resistance over which the wave form of the output is rectangular is

[7-46]

where V is the rms value of the a-c supply voltage.

When the average value V_av instead of the rms value V of the sinusoidal supply voltage is used, Eq. 7-46 becomes

[7-47]

since V = 1.11 V_av.