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CIRCUIT ANALYSIS ENGR. VIKRAM KUMAR B.E (ELECTRONICS) M.E (ELECTRONICS SYSTEM ENGG:) MUET JAMSHORO 1 KIRCHHOFF’S VOLTAGE LAW
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CIRCUIT ANALYSIS
ENGR. VIKRAM KUMAR
B.E (ELECTRONICS)M.E (ELECTRONICS SYSTEM ENGG:)MUET JAMSHORO
KIRCHHOFF’S VOLTAGE LAW
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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
3
Node: Place where elements are joined—entire wireOR
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)
4
Loop – a collection of branches that form a closed path returning to the same node without going through any other nodes or branches twice
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ExampleHow many nodes, branches & loops?
+ -
Vs Is
R1
R2 R3
+
Vo
-
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ExampleThree nodes
+ -
Vs Is
R1
R2 R3
+
Vo
-
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Example5 Branches
+ -
Vs Is
R1
R2 R3
+
Vo
-
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ExampleThree Loops, if starting at node A
+ -
Vs Is
R1
R2 R3
+
Vo
-
A B
C
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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 ofthe voltages around a closed path equal zero. This is similar to what we encountered with Kirchhoff’s current law.
Basic Laws of Circuits
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Kirchoff’s Voltage Law (KVL)The algebraic sum of voltages around
each loop is zeroBeginning 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 = 0Or 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).
+
-
V2
Path“rise” or “step up”(negative drop)
+
-
V1
Path
“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
Path 1: 0vvv b2a Path 2: 0vvv c3b Path 3: 0vvvv c32a
vcva
+
+
3
21
+
vb
v3v2
+
+
-
Examples of three closed paths:
a b c
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Kirchhoff’s Voltage Law:
Basic Laws of Circuits
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 thatno current has been assumed for this case, at this point.
+
+
+
+
_
_
_
_
v1
v2
v4
v3
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Basic Laws of CircuitsKirchhoff’s Voltage Law:Consideration 1.
We define a voltage drop as positive if we enter the positive terminaland 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 CircuitsKirchhoff’s Voltage Law:
+
+
+
+
_
_
_
_
v1
v2
v4
v3
Consider the circuit of following Figure onceagain. If we sum the voltage drops in the clockwise direction around thecircuit starting at point “a” we write:
- v1 – v2 + v4 + v3 = 0
- v3 – v4 + v2 + v1 = 0
•“a”
drops in CW direction starting at “a”
drops in CCW direction starting at “a”
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Basic Laws of CircuitsKirchhoff’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
+ _v1
The voltage rise in moving from left to right above is + v1.
The voltage rise in moving from left to right above is - v1.
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Basic Laws of CircuitsKirchhoff’s Voltage Law:Consider the circuit of Figure 3.7 once
again. If we sum the voltage rises in the clockwise direction around thecircuit starting at point “a” we write:
+
+
+
+
_ _
_
v1
v2
v4
v3
•“a”
+ v1 + v2 - v4 – v3 = 0
+ v3 + v4 – v2 – v1 = 0
rises in the CW direction starting at “a”
rises in the CCW direction starting at “a”
_
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Basic Laws of CircuitsKirchhoff’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 point “a” and move in the CW direction. As we cross elements1 & 2 we use voltage rise: as we cross elements 4 & 3 we use voltage drops. This gives the equation,
+
+
+
+
_
_
_
_
v1
v2
v4
v3
v1 + v2 = v4 + v3
1
2
3
4
p
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Basic Laws of CircuitsKirchhoff’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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ExampleKirchoff’s Voltage Law around 1st Loop
+ -
Vs Is
R1
R2 R3
+
Vo
-
A B
C
I2
I1
+
I2R2
-
+ I1R1 -
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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ExampleKirchoff’s Voltage Law around 1st Loop
+ -
Vs Is
R1
R2 R3
+
Vo
-
A B
C
I2
I1
+
I2R2
-
+ I1R1 -
Starting at node A, add the 1st voltage drop: + I1R1
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ExampleKirchoff’s Voltage Law around 1st Loop
+ -
Vs Is
R1
R2 R3
+
Vo
-
A B
C
I2
I1
+
I2R2
-
+ I1R1 -
Add the voltage drop from B to C through R2: + I1R1 + I2R2
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ExampleKirchoff’s Voltage Law around 1st Loop
+ -
Vs Is
R1
R2 R3
+
Vo
-
A B
C
I2
I1
+
I2R2
-
+ I1R1 -
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.
+
+
+
+ +
+
+
+
+
+
+
-- -
-
-
-
--
-
-
-v1
v2
v4
v3
v12
v11 v9
v8
v6
v5
v7
v10
+
-
Figure Multi-pathCircuit.
Path 1
Path 2
Path 3
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Kirchhoff’s Voltage Law: Further details.
For any given circuit, there are a fixed number of closed pathsthat can be taken in writing Kirchhoff’s voltage law and stillhave 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
+
+
+
+ +
+
+
+
+
+
+
-- -
-
-
-
--
-
-
-v1
v2
v4
v3
v12
v11 v9
v8
v6
v5
v7
v10
+
-
“a”•
Blue path, starting at “a”
- v7 + v10 – v9 + v8 = 0
•“b”
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
Using sum of the drops = 0
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THE END
