Gustav Kirchhoff’s Current Law is one of the fundamental laws used for circuit analysis. His current law states that for a parallel path the total current entering a circuits junction is exactly equal to the total current leaving the same junction.

In other words the algebraic sum of ALL the currents entering and leaving a junction must be equal to zero as: Σ IIN = Σ IOUT.

This idea by Kirchhoff is commonly known as the Conservation of Charge, as the current is conserved around the junction with no loss of current.

Here, the 3 currents entering the node, I

_{1}, I

_{2}, I

_{3}are all positive in value and the 2 currents leaving the node, I

_{4}and I

_{5}are negative in value. Then this means we can also rewrite the equation as;

I

_{1}+ I_{2}+ I_{3}– I_{4}– I_{5}= 0The term

**Node**in an electrical circuit generally refers to a connection or junction of two or more current carrying paths or elements such as cables and components. Also for current to flow either in or out of a node a closed circuit path must exist.

## Kirchhoffs Second Law – The Voltage Law, (KVL)

**Kirchhoffs 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.

### Kirchhoffs 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.

### Common DC Circuit Theory Terms:

- • Circuit – a circuit is a closed loop conducting path in which an electrical current flows.
- • Path – a single line of connecting elements or sources.
- • 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. There are no components inside a mesh.

### A Typical DC Circuit

## Kirchhoffs Circuit Law Example No1

Find the current flowing in the 40Ω Resistor, R

_{3}
The circuit has 3 branches, 2 nodes (A and B) and 2 independent loops.

Using

**Kirchhoffs Current Law**,**KCL**the equations are given as;
At node A : I

_{1}+ I_{2}= I_{3}
At node B : I

_{3}= I_{1}+ I_{2}
Using

**Kirchhoffs Voltage Law**,**KVL**the equations are given as;
Loop 1 is given as : 10 = R

_{1}I_{1}+ R_{3}I_{3}= 10I_{1}+ 40I_{3}
Loop 2 is given as : 20 = R

_{2}I_{2}+ R_{3}I_{3}= 20I_{2}+ 40I_{3}
Loop 3 is given as : 10 – 20 = 10I

_{1}– 20I_{2}
As I

_{3}is the sum of I_{1}+ I_{2}we can rewrite the equations as;
Eq. No 1 : 10 = 10I

_{1}+ 40(I_{1}+ I_{2}) = 50I_{1}+ 40I_{2}
Eq. No 2 : 20 = 20I

_{2}+ 40(I_{1}+ I_{2}) = 40I_{1}+ 60I_{2}
We now have two “

**Simultaneous Equations**” that can be reduced to give us the values of I_{1}and I_{2}
Substitution of I

_{1}in terms of I_{2}gives us the value of I_{1}as -0.143 Amps
Substitution of I

_{2}in terms of I_{1}gives us the value of I_{2}as +0.429 Amps
As : I

_{3}= I_{1}+ I_{2}
The current flowing in resistor R

_{3}is given as : -0.143 + 0.429 = 0.286 Amps
and the voltage across the resistor R

_{3}is given as : 0.286 x 40 = 11.44 volts
The negative sign for I

_{1}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.### Application of Kirchhoffs Circuit Laws

These two laws enable the Currents and Voltages in a circuit to be found, ie, the circuit is said to be “Analysed”, and the basic procedure for using

**Kirchhoff’s Circuit Laws**is as follows:**1.**Assume all voltages and resistances are given. ( If not label them V1, V2,… R1, R2, etc. )**2.**Label each branch with a branch current. ( I1, I2, I3 etc. )**3.**Find Kirchhoff’s first law equations for each node.**4.**Find Kirchhoff’s second law equations for each of the independent loops of the circuit.**5.**Use Linear simultaneous equations as required to find the unknown currents.

### We can also use loop analysis to calculate the currents in each independent loop which helps to reduce the amount of mathematics required by using just Kirchhoff's laws. In the next tutorial about DC circuits, we will look at Mesh Current Analysis to do just that.

(to be updated)