Table of Contents
Basic Laws
Ohm's Law
Ohm's Law states that the voltage, v, across a resistor is directly proportional to the current, i, flowing through the resistor.
- Determined experimentally in 1826.
- In math speak, we write Ohm's law as:
- The resistance, R of an element expresses its ability to resist the flow of electric current.
- Resistance is expressed in units of ohms (
). - One ohm is one volt per ampere, i.e.,
V/A. - A circuit element with a resistance approaching zero is called a short circuit.
- A circuit element with a resistance approaching infinity is called an open circuit.
- A resistor that obeys Ohm's law is known as a linear resistor. (We'll deal with these this quarter.)
Nodes, Branches and Loops
- The configuration of an electric circuit is known as a network topology.
- A network consists of a collection of interconnected elements.
- A circuit is a network containing one or more closed paths.
- Our study of network topology deals with the geometric configuration of the network.
- We define the terms branch, node, and loop to help describe specific network topologies.
A node is the point of connection between two or more elements.
- Often we will indicate a node in a circuit with a dot.
- Two nodes, connected by a short circuit (a wire with negligible resistance) constitute a single node.
- Usually we'll only be interested in nodes that connect three or more elements.
A branch represents any two-terminal element in a circuit.
- Fundamentally, a branch is very simple. It consists of an element with two terminals:
- A voltage source.
- A resistor.
- A current source.
A loop is a closed path in a circuit.
- A closed path is formed by starting a given node and passing through a set of unique nodes ending at the starting node.
- A loop is said to be an independent loop if it contains at least one branch that is not part of any other independent loop.
- We will write an equation for each independent loop in a circuit.
- Each equation is an independent equation.
- The set of independent equations may be solved simultaneously in order to determine the behavior of the circuit.
Fundamental theorom of network topology
For a given network with b branches, l loops and n nodes in the given network:
Example:
Find the number of branches, independent loops, and nodes in the above network.
Parallel and Series
Using the network topology terminology, we can define two specific topologies:
- Two elements are in parallel if they are connected to the same two nodes.
- The voltage across both elements must be the same.
- Two elements are said to be in series if they exclusively share a single node.
- That is, the two elements are connected and no other elements share that connection point.
- The current through both elements must be the same.
Krichhoff's Laws
Krichhoff's Current Law
Kirchoff's Current Law (KCL) states that the algebraic sum of currents entering a node is zero.
where N is the number of branches connected to the node and 
is the nth current entering (or leaving) the node.
- Must be when defining the direction of current.
- We will use signed values to describe each current.
- Simply stated, KCL says: current into the node = current out of the node.
Example:
Verify that the current flowing into the node in the center of the circuit is equal to the current flowing out of the circuit.
Krichhoff's Voltage Law
Kirchoff's Voltage Law (KVL) states that the algebraic sum of voltages around a loop is zero.
where M is the number of voltages in the loop and 
is the mth voltage.
- This implies that we can increase the voltage by placing voltage sources in series.
- Simply stated, KVL says: in a loop, the voltage rise = voltage drop.
Example:
Verify that the voltages for each loop sum to zero.