Basic Electronics on the Go - Kirchhoffs Circuit Law

From http://www.electronics-tutorials.ws/dccircuits/dcp_4.html

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, I₁, I₂, I₃ are all positive in value and the 2 currents leaving the node, I₄ and I₅ are negative in value. Then this means we can also rewrite the equation as;

I₁ + I₂ + I₃ – I₄ – I₅ = 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.

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, R₃

 

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 :    I₁ + I₂ = I₃

At node B :    I₃ = I₁ + I₂

Using Kirchhoffs Voltage Law, KVL the equations are given as;

Loop 1 is given as :    10 = R₁ I₁ + R₃ I₃ = 10I₁ + 40I₃

Loop 2 is given as :    20 = R₂ I₂ + R₃ 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 = 20I₂ + 40(I₁ + I₂)  =  40I₁ + 60I₂

We now have two “Simultaneous Equations” that can be reduced to give us the values of 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.

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)