Lecture 3

•Review:•Ohm’s Law, Power, Power Conservation

•Kirchoff’s Current Law•Kirchoff’s Voltage Law•Related educational modules:

–Section 1.4

Review: Ohm’s Law

• Ohm’s Law• Voltage-current characteristic of ideal resistor:

)t(iR)t(v

Review: Power• Power:

• Power is positive if i, v agree with passive sign convention (power absorbed)

• Power is negative if i, v contrary to passive sign convention (power generated)

)t(i)t(v)t(p

Review: Conservation of energy

• Power conservation:• In an electrical circuit, the power generated is the same

as the power absorbed.

• Power absorbed is positive and power generated is negative

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• Two new laws today:

• Kirchoff’s Current Law

• Kirchoff’s Voltage Law

• These will be defined in terms of nodes and loops

Basic Definition – Node• A Node is a point of connection between two or more

circuit elements• Nodes can be “spread out” by perfect conductors

Basic Definition – Loop• A Loop is any closed path through the circuit which

encounters no node more than once

Kirchoff’s Current Law (KCL)• The algebraic sum of all currents entering (or

leaving) a node is zero• Equivalently: The sum of the currents entering a node

equals the sum of the currents leaving a node• Mathematically:

• We can’t accumulate charge at a node

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Kirchoff’s Current Law – continued

• When applying KCL, the current directions (entering or leaving a node) are based on the assumed directions of the currents• Also need to decide whether currents entering the node

are positive or negative; this dictates the sign of the currents leaving the node

• As long all assumptions are consistent, the final result will reflect the actual current directions in the circuit

KCL – Example 1

• Write KCL at the node below:

KCL – Example 2

• Use KCL to determine the current i

Kirchoff’s Voltage Law (KVL)• The algebraic sum of all voltage differences around

any closed loop is zero• Equivalently: The sum of the voltage rises around a closed

loop is equal to the sum of the voltage drops around the loop

• Mathematically:

• If we traverse a loop, we end up at the same voltage we started with

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Kirchoff’s Voltage Law – continued• Voltage polarities are based on assumed polarities

• If assumptions are consistent, the final results will reflect the actual polarities

• To ensure consistency, I recommend:• Indicate assumed polarities on circuit diagram• Indicate loop and direction we are traversing loop• Follow the loop and sum the voltage differences:

• If encounter a “+” first, treat the difference as positive• If encounter a “-” first, treat the difference as negative

KVL – Example• Apply KVL to the three loops in the circuit below. Use the

provided assumed voltage polarities

Circuit analysis – applying KVL and KCL

• In circuit analysis, we generally need to determine voltages and/or currents in one or more elements

• We can determine voltages, currents in all elements by:• Writing a voltage-current relation for each element (Ohm’s

law, for resistors)• Applying KVL around all but one loop in the circuit• Applying KCL at all but one node in the circuit

Circuit Analysis – Example 1• For the circuit below, determine the power absorbed by each

resistor and the power generated by the source. Use conservation of energy to check your results.

Example 1 – continued

Circuit Analysis – Example 2• For the circuit below, write equations to determine the

current through the 2 resistor

Example 2 – Alternate approach

Circuit Analysis• The above circuit analysis approach (defining all “N”

unknown circuit parameters and writing N equations in N unknowns) is called the exhaustive method

• We are often interested in some subset of the possible circuit parameters• We can often write and solve fewer equations in order to

determine the desired parameters

Circuit analysis – Example 3• For the circuit below, determine:

(a) The current through the 2 resistor(b) The current through the 1 resistor(c) The power (absorbed or generated) by the source

Circuit Analysis Example 3 – continued