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1.1 KIRCHOFF’S LAWS

• 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).

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1.1.1 Simple Applications of Kirchoff’s Laws

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1.1.1.1 - 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.

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1.1.1.2 - 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).

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1.1.2 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,

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1.1.3 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,

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1.1.4 More Advanced Applications

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1.1.4.1 - 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,

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1.1.4.2 - 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.

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1.1.4.3 - 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,