This appendix discusses three additional theorems that are especially useful in the simplification of circuits containing dependent sources. In our previous studies ^[1] we discussed the superposition principle and Thevenin's and Norton's theorems.

C.1 The Substitution Theorem

The substitution theorem states that if the voltage across a branch with nodes x and y of a network is v _xy and the current through this branch is i _xy, a different branch may be substituted in its place in the network provided that the voltage across the substitute branch is also v _xy and the current through it is also i _xy. The most common use of this theorem is to replace an impedance by a voltage or current source, or vice versa. The substitution theorem can best illustrated with the simple circuit and the substitute branches shown in Figure C.1.

Figure C.1: Illustration of the substitution theorem

For the simple resistor circuit of Figure C.1(a) we find by series-parallel resistance combinations that v _xy = 6 V and i _xy = 3 A. According to the substitution theorem, the 2 ? resistor across terminals x and y can be replaced with a source with a 6 V source as shown in Figure C.1(b) and the rest of the network will be unaffected. The current in the branch will be 3 A as before.

Other substitutions are possible also. For instance, the substitute branch may consist of a...