Kirchoff's Voltage Law

Kirchoff's Second Law - The Voltage Law, (KVL)

Kirchoff's Voltage Law or KVL, states that "in any closed loop network, the total voltage around the loop is equal to the sum of all the voltage drops within the same loop" which is also equal to zero. In other words the algebraic sum of all voltages within the loop must be equal to zero. This idea by Kirchoff is known as the Conservation of Energy.

Kirchoff's Voltage Law

Kirchoffs Voltage Law
Starting at any point in the loop continue in the same direction noting the direction of all the voltage drops, either positive or negative, and returning back to the same starting point. It is important to maintain the same direction either clockwise or anti-clockwise or the final voltage sum will not be equal to zero. We can use Kirchoff's voltage law when analysing series circuits.
When analysing either DC circuits or AC circuits using Kirchoff's Circuit Laws a number of definitions and terminologies are used to describe the parts of the circuit being analysed such as: node, paths, branches, loops and meshes. These terms are used frequently in circuit analysis so it is important to understand them.
  • Circuit - a circuit is a closed loop conducting path in which an electrical current flows.
  • Path - a line of connecting elements or sources with no elements or sources included more than once.
  • Node - a node is a junction, connection or terminal within a circuit were two or more circuit elements are connected or joined together giving a connection point between two or more branches. A node is indicated by a dot.
  • Branch - a branch is a single or group of components such as resistors or a source which are connected between two nodes.
  • Loop - a loop is a simple closed path in a circuit in which no circuit element or node is encountered more than once.
  • Mesh - a mesh is a single open loop that does not have a closed path. No components are inside a mesh.
  • Components are connected in series if they carry the same current.
  • Components are connected in parallel if the same voltage is across them.
Kirchoffs Circuit Law

Example No1

Find the current flowing in the 40Ω Resistor, R3
The circuit has 3 branches, 2 nodes (A and B) and 2 independent loops.
Using Kirchoff's Current Law, KCL the equations are given as;
At node A :    I1 + I2 = I3
At node B :    I3 = I1 + I2
Using Kirchoff's Voltage Law, KVL the equations are given as;
Loop 1 is given as :    10 = R1 x I1 + R3 x I3 = 10I1 + 40I3
Loop 2 is given as :    20 = R2 x I2 + R3 x I3 = 20I2 + 40I3
Loop 3 is given as :    10 - 20 = 10I1 - 20I2
As I3 is the sum of I1 + I2 we can rewrite the equations as;
Eq. No 1 :    10 = 10I1 + 40(I1 + I2)  =  50I1 + 40I2
Eq. No 2 :    20 = 20I1 + 40(I1 + I2)  =  40I1 + 60I2
We now have two "Simultaneous Equations" that can be reduced to give us the value of both I1 and I2 
Substitution of I1 in terms of I2 gives us the value of I1 as -0.143 Amps
Substitution of I2 in terms of I1 gives us the value of I2 as +0.429 Amps
As :    I3 = I1 + I2
The current flowing in resistor R3 is given as :     -0.143 + 0.429 = 0.286 Amps
and the voltage across the resistor R3 is given as :     0.286 x 40 = 11.44 volts
The negative sign for I1 means that the direction of current flow initially chosen was wrong, but never the less still valid. In fact, the 20v battery is charging the 10v battery.