Nodal analysis can be used to solve virtually any complex multi-source DC electrical circuit. It is based on KCL, writing expressions involving each node in the circuit. A system of equations results, there being as many equations as there are nodes in the circuit, minus the reference node (which usually is taken as ground). The set of equations will exhibit diagonal symmetry, which can be used as a crosscheck before setting out to solve them. The solution will be a complete set of node voltages. From these, any branch current may be determined as needed.

There are two different methods of creating the system of equations. The first method is deemed the general method and will work for a mix of current sources and voltage sources. Individual currents are defined based on the node voltages and any known current sources. KCL is then applied at each node, followed by simplification and combination of terms to arrive at the end equations. The second approach uses the inspection method. If the circuit contains only current sources (or if the voltage sources are converted), this method allows direct generation of the system of equations without the need for simplification and thus is less prone to error.

Mesh analysis can be used to solve any planar complex multi-source DC electrical circuit. It is based on KVL, writing expressions involving each closed loop in the circuit. The loops are minimally sized and the set of loops must cover every component in the circuit. A system of equations results, there being as many equations as there are loops. As with nodal analysis, the set of equations will exhibit diagonal symmetry. The solution will be a complete set of mesh currents. From these, any node voltage may be determined.

Like nodal, mesh offers two different methods of creating the system of equations. The general method will work for a mix of current sources and voltage sources. Individual loops are defined based on the meshing currents passing through each component. KVL is then applied around each loop, followed by simplification and combination of terms to arrive at the end equations. In contrast, if the circuit contains only voltage sources (or if the current sources are converted), then the inspection method may be used. This method allows direct generation of the system of equations and is faster and less error prone.

Dependent sources are current or voltage sources whose value depends on the current or voltage developed in some other part of the circuit. There are four types: current controlled current source, current controlled voltage source, voltage controlled current source and voltage controlled voltage source. These sources are used commonly to model the characteristics of active devices such bipolar and field effect transistors. Techniques for solution tend to be a bit more involved than when using constant sources, however, nodal analysis in particular tends to work well.

## Review Questions

1. Describe the practical differences between nodal analysis and mesh analysis.

2. What is diagonal symmetry? Of what use is it?

3. What are the differences between the general method and the inspection method of nodal analysis?

4. What are the differences between the general method and the inspection method of mesh analysis?

5. What is a supernode?

6. What is a supermesh?

7. Describe the concept of dependent sources and how they differ from independent or constant sources.