Kirchoffs Circuit Law
Kirchoffs Circuit Law
We saw in the Resistors
tutorial that a single equivalent resistance, ( RT ) can be found when
two or more resistors are connected together in either series, parallel or combinations of both, and that these circuits
obey Ohm's Law. 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 First Law - The Current Law, (KCL)
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.
Kirchoffs Current Law
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. We can use Kirchoff's current law when analysing parallel circuits.
Kirchoffs Second Law - The Voltage Law, (KVL)
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.
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 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 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.
Find the current flowing in the 40Ω Resistor, R3
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 : I1 + I2 = I3
At node B : I3 = I1 + I2
Using Kirchoffs 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 = 20I2 + 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.
Application of Kirchoffs 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 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 Theory we will look at
Mesh Current Analysis to do just that.