Electronic Devices and Amplifier Circuits

Appendix FSubstitution, Reduction, and Miller’s TheoremsThis appendix discusses three additional theorems that are especially useful in the simplificationof circuits containing dependent sources. In our previous studies * we discussed thesuperposition principle and Thevenin’s and Norton’s theorems.F.1 Substitution TheoremThe substitution theorem states that if the voltage across a branch with nodes x and y of a network isv xyand the current through this branch is i xy, a different branch may be substituted in its place inthe network provided that the voltage across the substitute branch is also v xyand the currentthrough it is also i xy. The most common use of this theorem is to replace an impedance by a voltageor current source, or vice versa. The substitution theorem can best illustrated with the simplecircuit and the substitute branches shown in Figure F.1.2 Ω24 V y( a)4 Ω2 Ωv xy= 6 Vi xy= 3 A2 Ωxx x x xy( b)3 A6 V3 V 3 VyFigure F.1. Illustration of the substitution theoremFor the simple resistor circuit of Figure F.1(a) we find by series−parallel resistance combinationsthat v xy = 6 V and i xy = 3 A. According to the substitution theorem, the 2 Ω resistor across terminalsx and y can be replaced with a source with a 6 V source as shown in Figure F.1(b) and therest 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 resistanceof 1 Ω and a voltage source of 3 V as shown in Figure F.1(c), or it may consist of a resistanceof 3 Ω and a voltage source of 3 V of opposite polarity as shown in Figure F.1(d).The voltage source in Figure F.1(a) could be DC or AC. If it is an AC source, we can substitute the2 Ω resistor across terminals x and y with a branch that has capacitive reactance in series with ay( c)1 Ω( d)3 Ω X C = 8 ⁄ 3 Ωy( e)V' = 10∠53.1°V* For a detailed discussion of Thevenin’s and Norton’s theorems and the superposition principle please refer to CircuitAnalysis I with MATLAB Applications, ISBN 978−1−934404−17−9.Electronic Devices and Amplifier Circuits with MATLAB® / Simulink® / SimElectronics® Examples, Third EditionCopyright © Orchard PublicationsF−1