Kirchhoff's Voltage Law (KVL)
Voltage conservation law
Analogy: Energy Balance Water Distribution Network
Typical network includes the energy sources and loads
Energy
source
Load
2
1
1.0
4
5
0
2nd floor
1st floor
6
Potential Energy
Tap water
3
3rd floor
0.8
0.6
0.4
0.2
0.0
0
Pond/Lake/River/Sea
1
2 3 4 5 6
Position in the network
Water potential energy increases from zero to high level by the pump and then
gradually goes back to zero as the water passes through the load
7
Energy Balance in Electric Networks
ϕ3
ϕ2
ϕ4
ϕ5
9V
ϕ1
ϕ6
Electric Potential
10
5
0
0
1
2 3 4 5 6 7 8 (same
9 as 1)
Position in the network
Voltage Change along the loop
V21 = ϕ2 − ϕ1
ϕ3
ϕ2
V32 = ϕ3 − ϕ2
ϕ4
ϕ5
9V
ϕ1
ϕ6
V43 = ϕ 4 − ϕ3
V54 = ϕ5 − ϕ 4
V54 = ϕ5 − ϕ 4
V65 = ϕ6 − ϕ5
V16 = ϕ1 − ϕ6
∑V = 0
Kirchhoff's Voltage Law (KVL)
"The algebraic sum
of all voltages in a loop
must equal zero“
discovered in 1847 by Gustav R. Kirchhoff, a German scientist
KVL for multi-mesh multi-node circuits
Example: potential distribution along the mesh “b”
-
+
“3”
b
“0”
“1”
“2”
1. Assume the current directions, ib (through Rb), i4 (R4) and i3 (R3)
2. Enumerate all the different nodes along the mesh path
3. Move along the path and account for the potential changes (i.e. the voltages).
KVL for multi-mesh circuits
Let us analyze the potential distribution along the mesh “b”
Node
Voltage
Potential
0
-
“0”
0
1
+Eb
+Eb
2
-ib*Rb
+Eb -ib*Rb
“3”
3
-i4*R4
+Eb -ib*Rb -i4*R4
b
0
-i3*R3
+Eb -ib*Rb -i4*R4 ib*R3
+
“1”
“2”
(!) When passing the resistor along the
current direction, the potential decreases
(the voltage is negative);
When passing the resistor against the
current direction, the potential increases
(the voltage is positive);
The total potential change along the
close path is equal to zero (the starting
point and the end point are the same!).
Therefore, the KVL for the mesh “b” is
+Eb -ib*Rb –i4*R4 –i3*R3 = 0
Series electric circuits
Current flow
Electrons
flow
Three resistors (labeled R1, R2, and R3), connected in a chain from one terminal
of the battery to the other.
In a series circuit (or a sub-circuit), there is only one path for current to flow.
Series electric circuits
ϕ1
ϕ2
Current flow
VB
Electrons
flow
ϕ4 = 0
ϕ3
Let us assign the potentials to all the nodes.
Let us find the voltages across the battery and all the resistors:
VB = ϕ1 – ϕ4
V21 = ϕ2 – ϕ1
V32 = ϕ3 – ϕ2
V43 = ϕ4 – ϕ3
VB+ V21+V32+V43 = (ϕ1 – ϕ4)+ (ϕ2 – ϕ1)+ (ϕ3 – ϕ2)+ (ϕ4 – ϕ3)=0
Series electric circuits
ϕ1
ϕ2
Current flow
VB
Electrons
flow
ϕ3
ϕ4 = 0
VB+ V21+V32+V43 = 0 – this is the KVL
From the Ohm’s law:
V21 = - IR1
V32 = - IR2
V43 = - IR3
VB = IR1 + IR2 + IR3
I=
VB
R1 + R2 + R3
Simple series circuits
+
V1=V12
+
V2=V23
E=
-
+
V3=V34
I = E / (R1+ R2+ R3);
I = 9V/18kΩ = 0.5 mA;
V1 = I×R1;
V1 = 0.5 mA × 3 kΩ = 1.5 V
V2 = I×R2 = 5 V;
V3 = I×R3 = 2.5 V;
Σ(Vij) = 1.5+5+2.5 = 9V = E
Series circuit rule:
V1
+
+
V2
E=
-
+
V3
Input voltage
Circuit Current =
Sumof all the resistances
E
I=
R1 + R2 + R3 + ...
The voltage across any resistor can be found from the Ohm’s law:
For instance,
V1 = V12 = I ×R1
Equivalent resistance
Req
Can we replace these three resistors with just one to simplify the circuit?
What replacement do we call “equivalent”?
Single resistor is equivalent to several resistors connected in series if the
current in the circuit remains the same.
In the actual circuit:
I = E / (R1+ R2+ R3);
In equivalent circuit:
I = E / Req;
From these:
Req = R1 + R2 + R3 ;
For the series connection,
the equivalent resistance = sum (all the series resistances)
Equivalent resistance of the series circuit
Req
If some resistors in the network or a part of it, are
connected in series, then the equivalent resistance is:
Reqs = R1 + R2 + R3 + …;
Equivalent resistance of the series circuit
Any series connection of two or more resistors can be replaced by the
equivalent resistor, not necessarily the whole circuit
Req67 = R6 + R7 = 4 kOhm
Req34 = R3 + R4 = 4 kOhm
Series circuit as a voltage divider
+
+
VOUT = V2
E=
-
Using the series circuit rule,
+
E
I=
R1 + R2
R2
V2 = I×R2 = E
R1 + R2
The output voltage V2 is a fraction of the input voltage E, defined by the ratio
of the output resistance over the “total” circuit resistance.
Voltage distribution in a series circuit
V1
+
+
E=
V2
-
+
R1
V1 = I × R1 = E
R1 + R2
R2
V2 = I × R2 = E
R1 + R2
V1
R1
=
V 2 R2
Series and parallel circuit summary
Series
Parallel
Same current flows through all the series
components.
I1 = I2 = I3 = ….
Same voltage is applied across all the parallel
components.
V1 = V2 = V3 = ….
Total voltage drop is equal to the sum of
voltage drops across the series components:
VT = V1 + V2 + V3+…
Total current is equal to the sum of the
currents through all the parallel components:
IT = I1 + I2 + I3+…
The equivalent resistance is equal to the sum of
all the series resistances:
REQ = R1 + R2 + R3+…
The equivalent conductance is equal to the sum
of all the parallel conductances:
or
GEQ = G1 + G2 + G3+…
1/REQ = 1/R1 + 1/R2 +1/R3+…
Voltage division (for 2-element circuit):
V1 R1
=
V2 R2
Current division (for 2-element circuit):
I1 R2 G1
=
=
I 2 R1 G2