Type IV Problem

Author: Edmund A. Laport

Problem. Supply two resistance loads with specified potentials in a specified phase relationship from a common generator. Load A is 250 ohms and requires a potential of 800 volts, this potential to lag the generator electromotive force by 90 degrees.

Fig. 5.34

Load B is 550 ohms, its terminal potential must be 1,200 volts, and it must lead the generator electromotive force by 45 degrees. The generator potential is 1,500 volts. What networks are required? What impedance is seen by the generator? The problem is illustrated in Fig. 5.34.

Procedure. Set up the known conditions in the form of a vector diagram of potentials and currents, as in Fig. 5.35. The problem does not specify an input impedance to either network A or network B.

Fig. 5.35

Fig. 5.36

If we choose the simplest possible circuit to make the specified potential transformations, we can use a two-element L network for each branch, provided that we are not particular about the power factor of the input impedance. Such a circuit for each branch would be that of Fig. 5.36.

In the vector form of the problem we lack only the potentials for V₁ for each network. We know I₀. We do not know I₁ but know its direction with respect to V₀. We do not know I₂ but know its direction with respect to V₁. We can draw V₁ directly since V₀ and V_g are specified.

Fig. 5.37

Thus we have enough information to complete the vector diagram of Fig. 5.37. This diagram reveals the nature and magnitude of each reactive element, and we obtain

The input impedance to the two networks in parallel is Vg/Ig, where Ig is the vector sum of I_2A and I_2B. The two series components of the input impedance are where

The entire circuit required to satisfy the problem is shown in Fig. 5.38.

In a similar manner problems having a multiplicity of loads fed in any specified amplitudes and phases of potentials can be solved readily. Some examples of what can be done in this same simple procedure are given herewith.

The following problem represents a special case of potential transformation with two equal loads for changing from a single-end generator to a balanced (push-pull) load.

Fig. 5.38

Problem. It is desired to feed two 250-ohm loads in push-pull with 500 volts across each resistance from a single-end generator having a terminal potential of 300 volts. What network is required, and what impedance is presented to the generator?

Fig. 5.39

Procedure. Set up vectors representing the desired potentials at 180 degrees for the load, and draw the vector for the generator potential at 90 degrees to each of the load potentials. Thus we have stated the problem vectorially, as shown at the left of Fig. 5.39.

The choice of the direction of Vg at 90 degrees to the load potentials is arbitrary. It could be any angle between the two. The 90-degree choice, however, has symmetry which is usually desirable.

Fig. 5.40

Fig. 5.41

On the basis of L networks, we complete the vector diagram to obtain the desired information, as in the previous problem. The usual elementary considerations provide the basis for this procedure, from which we get Fig. 5.40 and Fig. 5.41. Tabulating vector lengths,

These values yield the circuit shown in Fig. 5.42.