# 4.8: Summary

In this chapter we have examined several new techniques and theorems to assist with the analysis of DC electrical circuits. Beginning with more practical models for voltage and current sources, we added an internal resistance which sets limits on the source's maximum output. For a voltage source, this resistance is in series, its ideal value being a short. For current sources, the resistance is in parallel, its ideal value being an open.

Source conversions allow us to create an equivalent current source for any practical voltage source and vice versa. An equivalent source is one that will create the same voltage across (and current into) whatever the new source is connected to as did the original source. In some cases, this swap allows differing sources to be combined into a single source, simplifying analysis.

The superposition theorem states that, for any multi-source linear bilateral network, the contributions of each source may be determined independent of all other sources, the final result being the summation of the contributions, cognizant of current directions and voltage polarities. Thus, the original circuit of $$N$$ sources generates $$N$$ new circuits, one for each source under consideration and with all other sources replaced by their ideal internal resistance.

Thévenin's and Norton's theorems allow the simplification of complex linear single port (i.e., two connecting points) networks. The Thévenin equivalent consists of a voltage source with series a resistance while the Norton equivalent consists of a current source with a parallel resistance. These equivalents, when replacing the original sub-circuit, will create the same voltage across the remainder of the circuit with the same current draw. That is, the remainder of the circuit will see no difference between being driven by the original sub-circuit or by either the Thévenin or Norton equivalents.

The maximum power transfer theorem states that for a simple voltage source with an internal resistance driving a single resistor load, the maximum load power will be achieved when the load resistance equals the internal resistance. At this point, efficiency will be 50%. If the load resistance is higher than the internal resistance, the load power will not be as great, however, the system efficiency will increase.

Delta-Y conversions allow the generation of equivalent “three connection point” resistor networks. Resistor networks with three elements in the shape of a triangle or delta (with one connection point at each corner) can be converted into a three element network in the shape of a Y or T, or vice versa. The two versions will behave identically to the remainder of the circuit. This allows the simplification of some circuits and eases analysis.

## Review Questions

1. Define the term linear bilateral network.

2. What are the ideal internal resistances of voltage sources and current sources?

3. Outline the process of converting a voltage source into a current source, and vice versa.

4. In general, describe the process of using superposition to analyze a multisource circuit.

5. What do Thévenin's and Norton's theorems state? How are they related?

6. What does the maximum power transfer theorem state? How might it be used with Thévenin's or Norton's theorems?

7. What are delta and Y configurations? How are they related?