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### Basic Circuit Analysis

Once you have a general understanding of how electricity works, the concepts discussed in the
basic circuit theory guide can be put to use analyzing a few simple circuits. Once again, reading this guide will not suddenly render you capable of solving the most complex circuit problems, nor will it allow you to design the most useful of circuits for yourself. There are many analysis methods available (tools in your toolbox, as one of my professors liked to say), but I will only go over a few of the simpler concepts here.

## Primitive Circuit Elements

All circuits have to have some sort of source for current to flow. These are represented by one of the following symbols, depending on the type of source in use.

Voltage Source:                                                   Current Source: With an independent source, the supplied voltage or current does not change, but with a dependent source, the supply is dependent upon some other part of the circuit. For example, a voltage controlled current source supplies a current with a value equal to some constant times the voltage at some other place in the circuit.

There are also a few principles that one must understand. A node is where two or more circuit elements are connected. Series elements all have the same current - equal current flows through each element, but the voltage may vary. Parallel elements have the same voltage - the voltage across all elements is identical, but the current may vary.

The most basic circuit component is the resistor. Although a resistor is an actual component consisting of a set resistance, any thing connected in a circuit has some value of resistance, although other types are generally referred to as impedance instead, but more on that later.

Resistors True resistors are an ohmic material and are measured in ohms (Ω) or volts per amp (V/A). The power dissipated by a resistor is calculated by  multiplying the resistor current and voltage, dividing the square of the resistor voltage by the resistance, or multiplying the square of the resistor current by the resistance.

Resistors can also be combined to form other resistances. Series resistances are added together while the sum of the inverse parallel resistors will yield the equivalent resistance. ## Basic Methods

There are two laws that form the basics of circuit analysis: Kirchhoff's Current Law and Kirchhoff's Voltage Law.

KCL: The sum of all currents entering a node must equal the sum of all currents leaving that same node.
KVL: The sum of all voltage rises must equal the sum of all voltage drops in a closed circuit loop.

Using these rules and Ohm's Law, a few useful realizations about resistive circuits can be made. Basically, the voltage at a node or current through a loop can easily be distinguished by using voltage or current divider equations.

Voltage Divider:                                                    Current Divider: ### Mesh Analysis This method solves for the currents of a closed loop planar circuit. A circuit is planar if it can be drawn without any wires crossing each other. Once the circuit is drawn in a planar fashion, a current path can be drawn inside each loop. Because each loop will have its own current, the number of unidentified currents will be equal to the number of loops; hence, there will be as many equations as there are unknowns.

With the current paths drawn, simple equations can be created using Kirchhoff's Voltage Law. In the example shown to the left, there are two loops forming currents I1 and I2. It is assumed that the values for the resistors and voltage source are known. The voltage source must be equal to the sum of the voltage drops across R1 and R2. Knowing that resistance multiplied by current equals voltage, this is an easy problem to solve. Because R2 is in the path of two currents, each must be taken into consideration.

### Supermesh Whenever a current source is situated in between two meshes, a supermesh is said to have occurred. In this case, the circuit is treated as if the current source were not there to find the first equation. The second necessary equation comes from the relationship between the two mesh currents and the current source.

Vs
Ri1 + Ri2    ;    is = i2 - i1

### Nodal Analysis This method can solves for the voltages at any node in the circuit by using Kirchhoff's Current Law. This method is preferred over mesh because it does not require the circuit to be planar. Just as before, equations are created until there are enough to solve for each unknown. Knowing that the sum of the currents entering a node must equal the sum of the currents leaving a node, and that current is equal to voltage divided by resistance, many simple equations can be made.

In the example shown to the left, there are three distinct current paths involving the central node Vx. The current through R1 is the difference between Vs and Vx divided by R1. Because there are no other nodes beyond Vx, the current through R2 and R3 is simply the voltage Vx divided by those resistances. The directions of the currents are not really important, as long as the polarity of any voltage sources are correctly used.

### Supernode In the example shown, the placing of of voltage source VA creates a supernode situation. This source is said to be floating because both terminals are of unknown voltages. To solve for this circuit, an extra equation relating V1, V2, and VA must used. Although, in this particular circuit, using mesh analysis would probably be easier.

### Superposition

This method is only useful if there are multiple independent sources in a linear circuit - can only contain independent sources, linear elements (resistors, capacitors, & inductors), and linear dependent sources.
1. Only consider one source at a time: replace voltage sources with a short and current sources with an open circuit.
2. Solve for any desired current or voltage characteristics using the one source still "on."
3. Repeat for any other current or voltage sources present.
4. Add up each of the individually solved voltage or current characteristics for the final results.

This method cannot be used directly to find power because it is non-linear (P = I2 * R = V2 / R).

## Equivalent Transformations

Another popular way to analyze a difficult circuit is to transform it into something else which is easier to deal with. Doing so will create an equivalent circuit which can be used to find the same results as the original. There really is no one size fits all rule to determine when to use these methods. It really comes down to what you think will work and what you are comfortable working with.

### Source Transformation

When it is more ideal to work with one type of source (current or voltage) over the other, then one can be transformed into the other using a couple of theorems. Thévenin's Theorem: In a linear, two-terminal, electrical network, any combination of voltage or current sources and resistors is equivalent to one single voltage source and one series resistor. Norton's Theorem: In a linear, two-terminal, electrical network, any combination of voltage or current sources and resistors is equivalent to one single current source and one parallel resistor.

These circuits models will have identical current/voltage characteristics. To solve for the unknown parameters, a few simple equations can be used on the original "black box" circuit.

1. Vth = Open Circuit Voltage -> Open VL (IL = 0) and solve for the open circuit voltage.
2. IN  = Short Circuit Current -> Short VL (VL = 0) and solve for the short circuit current.
3. Rth = Network Resistance -> [Independent Sources Only] Short all voltage sources and open all current sources. Then use resistor reduction methods to solve for the network resistance.
4. Rth = Network Resistance -> [Any Source Present] Attach a 1A "test current source" across the output terminals with ITEST flowing against IL and calculate the terminal voltage. The equivalent resistance is this voltage divided by ITEST.

### Source Transformation Examples

Thévenin: Norton: ### Delta-Wye Transform

Although this is really just a type of resistor reduction and not a complete cuicuit analysis tool, it is a more advanced transformation of one circuit into an equivalent, and so I am placing it here. 