However, sometimes in complex circuits such as bridge or T networks, we can not simply use Ohm’s Law alone to find the voltages or currents circulating within the circuit. For these types of calculations we need certain rules which allow us to obtain the circuit equations and for this we can use **Kirchoffs Circuit Law**.

In 1845, a German physicist, **Gustav Kirchoff** developed a pair or set of rules or laws which deal with the conservation of current and energy within electrical circuits. These two rules are commonly known as: *Kirchoffs Circuit Laws* with one of Kirchoffs laws dealing with the current flowing around a closed circuit, **Kirchoffs Current Law, (KCL)** while the other law deals with the voltage sources present in a closed circuit, **Kirchoffs Voltage Law, (KVL)**.

**Kirchoffs Current Law** or KCL, states that the “*total current or charge entering a junction or node is exactly equal to the charge leaving the node as it has no other place to go except to leave, as no charge is lost within the node*“. In other words the algebraic sum of ALL the currents entering and leaving a node must be equal to zero, I_{(exiting)} + I_{(entering)} = 0. This idea by Kirchoff is commonly known as the **Conservation of Charge**.

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} = 0

The 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. We can use Kirchoff’s current law when analysing parallel circuits.

**Kirchoffs 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**.

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 **Kirchoffs 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 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.

__Note that:__

Components are said to be connected in Series if the same current flows through component.

Components are said to be connected in Parallel if the same voltage is applied across them.

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 **Kirchoffs 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 **Kirchoffs Voltage Law**, **KVL** the equations are given as;

Loop 1 is given as : 10 = R_{1} x I_{1} + R_{3} x I_{3} = 10I_{1} + 40I_{3}

Loop 2 is given as : 20 = R_{2} x I_{2} + R_{3} x 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.

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 **Kirchoff’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 Kirchoff’s first law equations for each node.**4.**Find Kirchoff’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.

As well as using **Kirchoffs Circuit Law** to calculate the various voltages and currents circulating around a linear circuit, 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 Kirchoff's laws. In the next tutorial about DC circuits, we will look at Mesh Current Analysis to do just that.

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Is kirchoffs law has it’s limit to solve combination circuit?

I thank this website for giving me the information about kirchoffs voltage and current law stay bless. God bless u all

there is no limitation in using Kirchoffs law. the only problem here is when there are many circuit wicdow involved, you needed more variables to used in computing the electrical parameter inside the circuit. unlike other method such as mesh, nothon and thievenen they can manipulate the circuit into more simpler way.

You write up is very helpful

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lost it at the simultaneous equations. Mine did not agree with your calculations. It is possible to show this section. Thanks

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good stuff,will check it out regularly

Ok yes, this was weath it knowing how and so i would also like to thanks for the idea

me and my thoughts loop master oudy.

nice

why is loop 3 clock wise? can someone explain why it is orientated that way?

English only.

We impose any direction either clockwise or not

loop 3 is clock wise just because the more of the current is passing by that side if we look at the diagram there is the resistance of the left side resistance lower then the resistance of right side resistor which is 20 thats why the left side resistor 10 allow more current then that of the 20 one so thats why the loop 3 runs in that side.

Electrical Man can be like its