Kirchoffs Voltage Law
Kirchoffs Voltage Law (KVL) states that the algebraic sum of the voltages across any set of branches
in a closed loop is zero. i.e.
B
X
Vb = 0
b=1
Where B is the number of branches in the circuit and Vb is the voltage across each branch.
Figure 1 shows a single loop circuit. The KVL computation is expressed graphically in that voltages around a loop are summed up by traversing (figuratively walking around) the loop.
path of traversal to sum voltages
10V
path of traversal to sum
Vr1
Vr1
R1
R1
assumed current
direction
R2
Vr2
10V
assumed curr
direction
R3
R3
Vr3
Vr3
Resulting KVL Equation: Vr1 + Vr2 + Vr3 − 10 = 0
KVL equation with clockw
Vr1 + Vr2 + Vr3 +
Figure 1: A single loop circuit
The KVL equation is obtained by traversing a circuit loop in either direction and writing down
path of traversal to sum
unchanged the voltage of each element whose + terminal is entered first and writing down the
Vr1
negative of every elements voltage where the minus sign is first met. The loop must start and end
at the same point. It does not matter where you start on the loop.
R1
Note that a current direction must have been assumed. The assumed current creates10V
a voltage
across each resistor and fixes the position of the + and - signs so that the passive sign convention
is obeyed. The assumed current direction and polarity of the voltage across each resistor must be
in agreement with the passive sign convention for KVL analysis to work.
assumed curr
direction
R3
Vr3
The voltages in the loop may be summed in either direction. It makes no difference except to
change all the signs in the resulting equation. Mathematically speaking, its as if the KVL equation
KVL equation assuming cloc
is multiplied by -1. See figure 2.
Vr1 + Vr2 + Vr3 +
1
path of traversal to sum voltages
Vr2
3 Vr2
− 10 = 0
10V
path of traversal to sum voltages
Vr1
Vr1
R1
R1
assumed current
path of traversal
to sum voltages
direction
R2
10V
Vr2
assumed current
path of traversal
to sum voltages
direction
Vr1
R3
Vr1
R3
R1
Vr3
R1
Vr3
current
10VKVL equationassumed
with clockwise
summation:
R2
Vr2
direction
Vr1 + Vr2 + Vr3 + 10 = 0
R3
R2
Vr2
current
10V
KVL equation assumed
with counterwise
summation:
R2
Vr2
direction
−Vr3 − Vr2 − Vr1 + 10 = 0
R3
Figure
Vr3 2: Voltage summation can be done in either direction
Vr3
path of traversal to sum voltages
− 10 = 0
path of traversal to sum voltages
KVL equation with counterwise summation:
KVL equation with clockwise summation:
The case on the right ofVr1
figure 3 will obviously result in negative result
Vr1 for the current. This is
−Vr3 − Vr2 − Vr1 + 10 = 0
Vr1 + Vr2 + Vr3 + 10 = 0
correct considering the current arrow is pointing in the opposite direction.
R1
10V
10V
assumed current
path of traversal
to sum voltages
direction
R1
R2
10V
Vr2
assumed current
path of traversal
to sum voltages
direction
Vr1
R3
Vr1
R3
R1
Vr3
R1
Vr3
assumed current
direction
10V
R2
Vr2
assumed current
direction
R2
Vr2
KVL equation assuming clockwise current flow:
R3 + − 10 = 0
Vr1 + Vr2 + Vr3
R2
Vr2
KVL equation assuming counter clockwise current flow:
−Vr1 − Vr2 R3
− Vr3 − 10 = 0
Vr3
Vr3
KVL equation assuming clockwise current flow:
Vr1 + Vr2 + Vr3 + − 10 = 0
KVL equation assuming counter clockwise current flow:
−Vr1 − Vr2 − Vr3 − 10 = 0
Figure 3: The assumed current direction fixes the voltage references
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