# Circuit Theory/Circuit Definition

Circuit Analysis is reverse engineering. Given a circuit, figure out the currents, voltages, and powers associated with each component. The project or problem that produced the circuit or the purpose of the circuit is not of concern. Most circuits are designed to illustrate a concept or practice the math rather than do something useful. Towards the end of this book, the "useful" circuits are studied that related to wireless communication.

### Untangle

First untangle a circuit. Flatten into two dimensions. Put all the components in the same orientation: up and down.

### Naming

This is an exercise in naming unknowns. There are only two things to remember:

• Devices in series share the same current.
• Devices in parallel share the same voltage.

The consequence of not doing this is merely another simple equation to write down and solve.

Now name the voltages and currents following these rules:

• Name series voltage with the same subscript as the component such as $v_{r1}v_{c1}v_{L1}$ . Choose a different naming convention for the shared voltage of parallel components such as $v_{1}v_{2}$ . Do not guess the polarity of the voltage yet.
• Name the parallel currents with the same subscript as the component such as $i_{r1}i_{c1}i_{L2}$ . Choose a different subscript for the shared current of series circuits such as $i_{1}i_{2}$ . Don't put arrow heads on wires yet indicating current direction.
• Don't forget to put voltages across current sources and name currents coming out of voltage sources.

### Loops

Loops are the smallest circles of components. Count them. Don't count "loops containing loops" or "loops of loops." Say there are 3 loops. Now any three loops of components can be considered (including loops containing loops) that follow this rule: Collectively each component must be mentioned in a loop at least once.

Some loops are trivial. Components in parallel will form a loop, but they share the same voltage. If you call them a loop, then you will end up with an equation that says $V_{r1}-V_{r2}=0$ . Don't count trivial loops unless you can not see devices sharing the same EMF difference.

Draw a loop with a different color pen and name the loop something like $L_{1}L_{2}L_{3}$ . Put an arrowhead describing the direction of the loop. Now go around the loop in the chosen direction, put a + on the wire entering each component and - on the wire leaving the component.

Current supplies in parallel with a component will share the same voltage as the component so nothing needs to be done.

Current supplies without a parallel component will need a voltage assigned to them. The polarity doesn't matter. But a polarity has to be chosen.

Voltage supplies can be symbolized by alternating plates (long and short) or a circle with + and - in it. Assign + to the long plate, and - to the short plate. Otherwise use the + and - in the voltage source symbol.

If two loops overlap, leave the first + and -. Don't put conflicting + and - signs on the same component.

Don't use the word "mesh" here. "Mesh analysis" is covered later. It is a different type of analysis that is not as general, but often is much easier than Kirchhoff's Law. Kirchhoff's Law always works and is described next.

### Junctions

A "junction" is a collection of wires at the same EMF (not voltage .. electro motive force is an absolute, voltage is relative). Wires touching each other have the same energy per charge (EMF) in them. Junctions stop at any component.

Some "Junctions" are trivial. These are the junctions where a single wire connects only two components. Don't count these unless you can not identify series components "sharing" a current.

Count the "non-trivial" "Junctions." Say there are 3. Subtract 1. This makes 2. You can write two junction equations. There typically more than 2 possible junction equations. Make sure the two written mention all the currents at least once.

At this point equations can be written around a node or a "cut set" which is similar to at big loop with smaller loops within it. Explore the electrical version of cut sets here. They can make analyzing just part of a circuit easier (discussed later). Again the equations have to include each unknown current at least once.

Name the "Junctions" with symbols like $J_{1}J_{2}$ .

Now put arrow heads for current direction for passive (resistors) and reactive (capacitors and inductors) components based upon the voltage polarity. The current should move from + to -. The arrow head should be into the + and out of the -.

Current sources will already have a current direction.

Voltage sources with a series component share the same current. The direction of this current is determined by the voltage polarity on the component.

Voltage sources that don't have a series component can be given current in any direction. Remember that the polarity of these voltages and currents is capturing the circuit topology, not predicting the polarity of the ultimate numeric answer.

Don't use the word "node" here. "Node analysis" is covered later. It is a different type of analysis that is not as general, but often is much easier than Kirchoff's Law. Kirchoff's Law always works and is described later.