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:
v( t )  R  i( t )
Review: Power
• Power:
p( t )  v( t )  i( t )
• 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)
Review: Conservation of energy
• Power conservation:
• In an electrical circuit, the power generated is the same
as the power absorbed.
p 0
All elements
• Power absorbed is positive and power generated is
negative
• 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:
N
i
k 1
k
(t )  0
• We can’t accumulate
charge at a node
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:
N
v
k 1
k
(t )  0
• If we traverse a loop, we end up
at the same voltage we started with
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