Introduction to Electrical and
Computer Engineering
Kirchhoff’s Laws
Electrical & Computer Engineering
Kirchhoff’s Laws (1 of 15)
Kirchhoff’s Laws
• Kirchhoff’s Voltage Law (KVL)
• Kirchhoff’s Current Law (KCL)
– Named after German scientist Gustav Kirchhoff, these
laws are used to help analyze more complicated circuits
Electrical & Computer Engineering
Kirchhoff’s Laws (2 of 15)
1
Kirchhoff’s Voltage Law (KVL)
• The algebraic sum of all voltages around any closed
path is equal to zero.
– A closed path is one that ends at the node it began at,
and in which no node is encountered more than once
-VS + V1 + V2 + V3 = 0
V1 + V2 + V3 = VS
Electrical & Computer Engineering
Kirchhoff’s Laws (3 of 15)
KVL Example
-30 + V1 + V2 = 0
V1 + V2 = 30
Using Ohm’s Law, we find the
current coming from the
voltage source
V1 = 5*I and V2 = 10*I
So, 15*I = 30, or I = 2 A.
That makes V1 = 10 V and
V2 = 20 V
Electrical & Computer Engineering
Kirchhoff’s Laws (4 of 15)
2
Series Connections
Elements are said to be “in series” when only 2 elements are
connected to a single node
Write KVL:
-VS + V1 + V2 + V3 = 0
VS = V1 + V2 + V3
Apply Ohm’s Law: VS = ISR1 + ISR2 + ISR3
= IS (R1 + R2 + R3)
VS = ISReq
Where Req = R1 + R2 + R3
Each element in series, carries a common
current, I (satisfies KCL at each node)
Electrical & Computer Engineering
Kirchhoff’s Laws (5 of 15)
Series Connections (continued)
General expression for k resistors connected in
series is:
⋯
Note: Req > largest R
Electrical & Computer Engineering
Kirchhoff’s Laws (6 of 15)
3
Kirchhoff’s Current Law (KCL)
• KCL: the algebraic sum of currents entering any node is
zero (conservation of charge)
– A node is a point of connection of two or more elements. Can also
be stated: the sum of currents entering a node is equal to the sum of
currents leaving a node
Write KCL at the top node:
IS = I1 + I2 + I3
Now Apply Ohms’ Law.
IS
= VS/R1 + VS/R2 + VS/R3
= Vs [1/R1 + 1/R2 + 1/R3]
The elements have the same voltage across
them (satisfies KVL in each loop)
IS = VS/Req
Where 1/Req = 1/R1 + 1/R2 + 1/R3
Electrical & Computer Engineering
Kirchhoff’s Laws (7 of 15)
Parallel Connections
• Elements are said to be in parallel if they form a
loop containing no other elements
– Think of this as elements that have the same top and
bottom nodes
1
1
∑
When you have 3 Resistors (6, 10, and 25 
each) in parallel:
1/Req = 1/6 + 1/10 + 1/25 = 0.306
Req = 3.3 
Equivalent resistance 3.3  is less than the
smallest R (6 )
Note: Req < smallest R
Electrical & Computer Engineering
Kirchhoff’s Laws (8 of 15)
4
Parallel Connections (continued)
Special case with only 2 resistors
1
1
1
2 Resistors (3 and 6  each) in parallel
1/Req = 1/3 + 1/6 = (2/6 + 1/6) = 3/6 = ½
so
Req = 2 
OR product over sum yields answer more
quickly (3*6)/(3+6) = 18/9 = 2 
Electrical & Computer Engineering
Kirchhoff’s Laws (9 of 15)
Parallel Connections (continued)
What happens when R1 = R2?
Req = (RR)/2R = R/2 (half the original value)
2 Resistors (6  and 6  each) in parallel:
6*6/(6+6) = 36/12 = 3 
or more quickly…6/2 = 3 
Electrical & Computer Engineering
Kirchhoff’s Laws (10 of 15)
5
Series/Parallel Combinations
Find Rab
Electrical & Computer Engineering
Kirchhoff’s Laws (11 of 15)
Series/Parallel Combinations
• First step, combine the outer two resistors in series
into a single equivalent resistance
– 1+2=3
• Second step, combine the calculated equivalent
resistance in parallel with the 6 resistor
– (3*6)/(3+6)=2
•
•
•
•
Continue the process with the remaining resistors
Third step, 2+4=6
Fourth step, (6*12)/(6+12)=4
Fifth step, 4+6=10=Rab
Electrical & Computer Engineering
Kirchhoff’s Laws (12 of 15)
6
Series/Parallel Combination Example
How much power is the 5 A source supplying?
First step, find the equivalent resistance.
Electrical & Computer Engineering
Kirchhoff’s Laws (13 of 15)
Series/Parallel Combination Example
6
10
*64 /
16
64
7.2
12.8
*30 /
30
Electrical & Computer Engineering
16*64 / 16 64
7.2
12.8
20
20*30 / 20 30
12
Kirchhoff’s Laws (14 of 15)
7
Series/Parallel Combination Example
5 12=300W
Electrical & Computer Engineering
Kirchhoff’s Laws (15 of 15)
8