The algebraic sum of all currents entering a node must always be zero where inis the nthcurrent. N is the number of branches.A common assignment: 1.if the current is entering the node, assign a negative “-“sign and 2.if the current is leaving the node, assign a positive “+” sign.For the following figureThe node equation can be written as To use KCL to analyze a circuit,1.Write KCL equations for the currents2.Use Ohm’s law to write currents in terms of bode voltages (one equation for each resistor)3.Solve to find values of node voltage and current

Example:
Find the current through a 20Ω resistance, and current through a 40Ω resistance

Write KCL at node
x
Write
in the circuit using Ohm’s Law
Apply last two equation into KCL at node
x
The current through a 20Ω resistance
The current through a 40Ω resistance

where
v
n
is the n
th
voltage.
N
is the number of elements in the loop
A common assignment:
1.
if the positive (+) side of the voltage is encountered first, assign a positive “+”sign to the voltage across the
element.
2.
If the negative (-) side of the voltage is encountered first, assign a negative “-”sign to the voltage across the
element.
For the following figure
To use KVL to analyze a circuit,
1.
Write KVL equations for voltages
2.
Use Ohm’s law to write voltages in terms of resistances and currents.
3.
Solve to find values of the currents and then voltages.

Examples:
Example 2
: Find the current
i
and voltage
v
over the each resistor.

KVL equations for voltages
Using Ohm’s Law
Substituting into KVL equation

Example 3: Find v1 and v2 in the following circuit(note: the arrows are signifying the positive position of the box and the negative is at the end of the box) Loop 1Loop 2Example 4: Find V1, V2, and V3.(note: the arrows are signifying the positive position of the box and the negative is at the end of the box)

Loop 1
Loop 2
Loop3
Example 5:
Find V1, V2, V3,and V4
(note: the arrows are signifying the positive position of the box and the negative is at the end of
the box)

Loop 1
Loop 2
Loop 3
Loop 4

Practice Problems:
(Click image to view solution)
Problem 1: Find V1 in the following circuit.Problem 2: Find V0 in the following circuit.Problem 3:Find V1,V2, and V3 in the following circuit.

Problem 4
: Find I
1
,I
2
, I
3
in the following circuit
Problem 5
: Find the resistor R value in the following circuit.
Exercises:

1.V1=8V, V2=-4V, V4=14V. Find V3and V5in the following circuit2.Find Vxand Vyin the following circuit3.Find Vx, Vyand Vzin the following circuit

4.Find the KCL node equations at nodes A,B,C, and D5.If I1=4A , I2=5A, and I3=3A, then using KCL find I4and, I5in the following circuit
Answers:
1.
V
3
=12V and V
5
=-2V
2.
V
x
=12V and V
y
=9V
3.
V
x
=35V, V
y
=5V, and V
z
=15V
4.
At node A:
At node B:
At node C:
At node D:
5.
I
4
=2A and I
5
=1A
Survey:

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- Kirchhoff's circuit laws, Voltage drop, Kirchhoff