4.1 KIRCHOFF'S LAWS
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Kirchoff's Current Law: "The sum of currents at any node in a circuit must equal zero" - keep in mind that current is a flow rate for moving electrons. And, electrons do not appear and disappear from the circuit. Therefore, all of the electrons flowing into a point in the circuit must be flowing back out.

Kirchoff's Voltage Law: "The sum of all voltages about a closed loop in a circuit is equal to zero" - Each element will have a voltage (potential) between nodes. If any two points on the closed loop are chosen, and different paths chosen between them, the potentials must be equal or current will flow in a loop indefinately (Note: this would be perpetual motion).

4.1.1 Simple Applications of Kirchoff's Laws

4.1.1.1 - Parallel Resistors

Let's consider one on the most common electrical calculations - that for resistors in parallel. We want to find the equivalent resistance for the network of resistors shown.

4.1.1.2 - Series Resistors

Now consider another problem with series resistors. We can use Kirchoff's voltage law to sum the voltages in the circuit loop. In this case the input voltage is a voltage rise, and the resistors are voltage drops (the signs will be opposite).

4.1.2 Node Voltage Methods

If we consider that each conductor in a circuit has a voltage level, and that the components act as bridges between these, then we can try some calculations.

This method basically involves setting variables, and then doing a lot of algebra.

This is a very direct implementation of Kirchoff's current law.

First, let's consider an application of The Node Voltage method for the circuit given below,

4.1.3 Current Mesh Methods

If we consider Kirchoff's Voltage law, we could look at any circuit as a collection of current loops. In some cases these current loops pass through the same components.

We can define a loop (mesh) current for each clear loop in a circuit diagram. Each of these can be given a variable name, and equations can be written for each loop current.

These methods are quite well suited to matrix solutions

Lets consider a simple problem,

4.1.4 More Advanced Applications

4.1.4.1 - Voltage Dividers

The voltage divider is a very common and useful circuit configuration. Consider the circuit below, we add a current loop, and assume there is no current out at Vo,

4.1.4.2 - The Wheatstone Bridge

The wheatstone bridge is a very common engineering tool for magnifying and measuring signals. In this circuit a supply voltage Vs is used to power the circuit. Resistors R1 and R2 are generally equal, Rx is a resistance to be measured, and R3 is a tuning resistor. An ammeter is shown in the center, and resistor R3 is varied until the current in the center Ig is zero.

4.1.4.3 - Tee-To-Pi (Y to Delta) Conversion

It is fairly common to use a model of a circuit. This model can then be transformed or modified as required.

A very common model and conversion is the Tee to Pi conversion in electronics. A similar conversion is done for power circuits called delta to y.

We can find equivalent resistors considering that,

To find the equivalents the other way,