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## Kirchhoff's Laws

Terminologies for electric circuits:

• node: a point where three or more current-carrying elements (branches) are connected;
• branch: a path containing one or more elements in series and connecting two nodes;
• loop: a closed path progressing from node to node and returning to the starting node.
Conventionally, constant variables are represented in upper-case letters (e.g., DC voltage and current , etc.), and time-varying variables are treated as time functions and represented in lower-case letters (e.g., AC voltage and current ).

Also note that in a circuit diagram, the direction of a current and the polarity of a voltage source can be assumed arbitrarily. To determined the actual direction and polarity, the sign of the values also should be considered. For example, a current labeled in left-to-right direction with a negative value is actually flowing right-to-left.

Kirchhoff's Laws

Based on the principle of conservation of electric charge, the Kirchoff current Law (KCL) states that:

The algebraic sum of the currents into a node at any instant is zero.

Based on the principle of conservation of energy, the Kirchoff voltage Law (KVL) states that:

The algebraic sum of the voltages around a loop at any instant is zero.

Assume currents going into the node are positive and those leaving the node negative, KCL states: .

Assume the current goes around a clockwise loop, KVL states: ; alternatively, if assume the current goes around a counter clockwise loop, we have ;

Example: Given circuit below, find , , , and .

According to Ohm's law, we have .

Apply KVL to the loop on the right to get:

According to Ohm's law, we have .

Apply KCL to the middle node on top to get:

Again by Ohm's law, we get .

Apply KVL to the loop on the left to get:

Voltage Divider:

According to Ohm's law, the voltage across the kth resistor can be found to be:

In particular if , we have

Resistors in series: According to KVL, the sum of voltages across the resistors is equal to the input voltage:

i.e.,

Resistors in parallel: According to KCL, the sum of currents through the resistors is equal to the input current:

i.e.,

or

In particular, when ,

Current Divider:

According to Ohm's law, the current through the kth resistor can be found to be:

In particular if , we have

Inductors in series: According to KVL, the sum of voltages across the inductors is equal to the input voltage:

i.e.,

Inductors in parallel: According to KCL, the sum of currents through the inductors is equal to the input current:

we get

Capacitors in parallel: According to KCL, the sum of currents through the resistors is equal to the input current:

i.e.,

Capacitors in series: According to KVL, the sum of voltages across the capacitors is equal to the input voltage:

i.e.,

 Resistor Inductor Capacitor Governing Equation Series connection Parallel connection

Next: Energy Sources Up: Chapter 1: Basic Quantities Previous: Sinusoidal Variables
Ruye Wang 2012-07-02