KIRCHHOFF’S VOLTAGE
LAW
ENGR. VIKRAM KUMAR
B.E (ELECTRONICS)
M.E (ELECTRONICS SYSTEM ENGG:)
MUET JAMSHORO
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BRANCHES AND NODES
 Branch:
Elements connected end-to-end,
nothing coming off in between (in series)
OR
A circuit element between two nodes
2
Node: Place where elements are joined—entire wire
OR
Any point where 2 or more circuit elements are
connected together
Wires usually have negligible resistance
Each node has one voltage (w.r.t. ground)
3
 Loop – a collection of branches that form a closed path
returning to the same node without going through any
other nodes or branches twice
4
Example
 How many nodes, branches & loops?
R1
+
+
-
Vs
Is
R2
R3
Vo
-
5
Example
 Three nodes
R1
+
+
-
Vs
Is
R2
R3
Vo
-
6
Example
 5 Branches
R1
+
+
-
Vs
Is
R2
R3
Vo
-
7
Example
 Three Loops, if starting at node A
A
B
R1
+
+
-
Vs
Is
R2
R3
Vo
-
C
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Basic Laws of Circuits
Kirchhoff’s Voltage Law:
 Kirchhoff’s voltage law tells us how to handle voltages in an
electric circuit.
 Kirchhoff’s voltage law basically states that the
algebraic sum of the voltages around any closed path
(electric circuit) equal zero. The secret here, as in
Kirchhoff’s current law, is the word algebraic.
 There are three ways we can interrupt that the algebraic sum of
the voltages around a closed path equal zero. This is similar to
what we encountered with Kirchhoff’s current law.
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Kirchoff’s Voltage Law (KVL)
 The algebraic sum of voltages around each loop is
zero
 Beginning with one node, add voltages across
each branch in the loop (if you encounter a +
sign first) and subtract voltages (if you
encounter a – sign first)
 Σ voltage drops - Σ voltage rises = 0
 Or Σ voltage drops = Σ voltage rises
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KIRCHOFF’S VOLTAGE LAW (KVL)
•
The sum of the voltage drops around any closed loop is zero.
We must return to the same potential (conservation of energy).
Path
Path
+
V1
-
“drop”
-
“rise” or “step up”
V2
(negative drop)
+
Closed loop: Path beginning and ending on the same node
Our trick: to sum voltage drops on elements, look at the first sign you encounter on
element when tracing path
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KVL EXAMPLE
+ v2 
b

a
v3
+
Examples of
three closed
paths:
2
1
+
vb
+
va

c
-
+
vc

3
Path 1:
 v a  v 2  vb  0
Path 2:
 vb  v3  vc  0
Path 3:
 va  v2  v3  vc  0
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Basic Laws of Circuits
Kirchhoff’s Voltage Law:
Consideration 1: Sum of the voltage drops around a circuit
equal zero. We first define a drop.
We assume a circuit of the following configuration. Notice that
no current has been assumed for this case, at this point.
_ v2
+
+
v1
+
_
_
_
v3
v4
+
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Basic Laws of Circuits
Kirchhoff’s Voltage Law:
Consideration 1.
We define a voltage drop as positive if we enter the positive terminal
and leave the negative terminal.
+ v1
_
The drop moving from left to right above is + v1.
_
v1
+
The drop moving from left to right above is – v1.
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Basic Laws of Circuits
Kirchhoff’s Voltage Law:
Consider the circuit of following Figure once
again. If we sum the voltage drops in the clockwise direction around the
circuit starting at point “a” we write:
- v1 – v2 + v4 + v3 = 0
_ v2
 drops in CW direction starting at “a”
+
+
v1
+
_
“a” •
_
_
v3
- v3 – v4 + v2 + v 1 = 0
v4
+
 drops in CCW direction starting at “a”
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Basic Laws of Circuits
Kirchhoff’s Voltage Law:
Consideration 2: Sum of the voltage rises around a circuit
equal zero. We first define a drop.
We define a voltage rise in the following diagrams:
_
v1
+
The voltage rise in moving from left to right above is + v1.
+ v1
_
The voltage rise in moving from left to right above is - v1.
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Basic Laws of Circuits
Kirchhoff’s Voltage Law:
Consider the circuit of Figure 3.7 once
again. If we sum the voltage rises in the clockwise direction around the
circuit starting at point “a” we write:
+ v1 + v2 - v4 – v 3 = 0
 rises in the CW direction starting at “a”
_
v2 +
+
v1
+
_
“a” •
_
_
+ v 3 + v4 – v2 – v1 = 0
v3
v4
+
 rises in the CCW direction starting at “a”
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Basic Laws of Circuits
Kirchhoff’s Voltage Law:
Consideration 3: Sum of the voltage rises around a circuit
equal the sum of the voltage drops.
Again consider the circuit of following Figure in which we start at
p
point
“a” and move in the CW direction. As we cross elements
1 & 2 we use voltage rise: as we cross elements 4 & 3 we use
voltage drops. This gives the equation,
_ v2
+
2
+
v1
_
v1 + v2 = v4 + v3
+
3
1
_
_
v4
4
v3
+
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Basic Laws of Circuits
Kirchhoff’s Voltage Law: Comments.
• We note that a positive voltage drop = a negative voltage rise.
• We note that a positive voltage rise = a negative voltage drop.
• We do not need to dwell on the above tongue twisting statements.
• There are similarities in the way we state Kirchhoff’s voltage
and Kirchhoff’s current laws: algebraic sums …
However, one would never say that the sum of the voltages
entering a junction point in a circuit equal to zero.
Likewise, one would never say that the sum of the currents
around a closed path in an electric circuit equal zero.
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Example
 Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Assign current variables and directions
Use Ohm’s law to assign voltages and polarities consistent with
passive devices (current enters at the + side)
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Example
 Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Starting at node A, add the 1st voltage drop: + I1R1
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Example
 Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Add the voltage drop from B to C through R2: + I1R1 + I2R2
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Example
 Kirchoff’s Voltage Law around 1st Loop
A
I1
+
I1R1
-
B
R1
I2
+
-
Vs
+
+
Is
R2 I2R2
R3
Vo
-
C
Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0
Notice that the sign of each term matches the polarity encountered 1st
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Kirchhoff’s Voltage Law: Further details.
For the circuit of Figure there are a number of closed paths.
Three have been selected for discussion.
-
+ v 2
-
v1
+
Figure
-
v3
- v5 +
v4
-
Path 1
+
Path 2
v6
+
+ v7 -
+
Multi-path
Circuit.
Path 3
v8
+
+
+
v12
v10
-
+ v11 -
- v9 +
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Kirchhoff’s Voltage Law: Further details.
For any given circuit, there are a fixed number of closed paths
that can be taken in writing Kirchhoff’s voltage law and still
have linearly independent equations. We discuss this more, later.
Both the starting point and the direction in which we go around a closed
path in a circuit to write Kirchhoff’s voltage law are arbitrary. However,
one must end the path at the same point from which one started.
Conventionally, in most text, the sum of the voltage drops equal to zero is
normally used in applying Kirchhoff’s voltage law.
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Kirchhoff’s Voltage Law: Illustration from Figure
“b”
-
•
Using sum of the drops = 0
+ v 2
-
v1
+
-
v3
- v5 +
v4
+
+ v7 -
v12
v10
-
+ v11 -
- v7 + v10 – v9 + v8 = 0
•
“a”
v8
+
+
+
Blue path, starting at “a”
v6
+
+
-
- v9 +
Red path, starting at “b”
+v2 – v5 – v6 – v8 + v9 – v11
– v12 + v1 = 0
Yellow path, starting at “b”
+ v2 – v5 – v6 – v7 + v10 – v11
- v12 + v1 = 0
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THE END
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