Question

What is KVL? Give an example of the application of KVL. In your example, please include details of your circuit and all the KVL equations

What is KCL? Give an example of the application of KCL. In this example, make sure to include all the KCL equations.

Answer 1

Kirchhoff’s Voltage Law

Kirchhoff’s Voltage Law (KVL) is Kirchhoff’s second law that deals with the conservation of energy around a closed circuit path.

EXAMPLE:

Suppose a circuit with two parallel paths (loops) and a single voltage source (DC), as shown in the diagram below. Find the current and voltage of each element of the circuit for the following given circuit parameter using Kirchhoff’s voltage law.

R1 = 5Ω

R2 = 10Ω

R3 = 5Ω

R4 = 10Ω

V = 20Volts

SOLUTION:

ASSIGN LOOPS NAME:

There are two loops (closed paths) in the circuit, loop 1 with two resistors and a single voltage source, where in loop 2 there is no voltage source, three resistors only.

ASSIGN SIGN TO CIRCUIT ELEMENTS:

Now, assign a positive sign to the resistor terminal which is closer to voltage source positive terminal and negative sign to whom which are closer to negative terminals. OR you may assign (+ -) sign to each component terminal by going through the loop. But make sure that the sign of voltage rise (- +) is exactly opposite to sign of voltage drop (+ -). Both will give you same results, even same equations.

ASSIGN CURRENT NAMES:

Assign a name to the current of each loop and as discussed in KCL, write equations for current at each node. For the above circuit KCL equations will be:

at node a: I1=I1

at node b: I1=I2+I3

at node c: I2+I3=I1

WRITE LOOP EQUATIONS:

The next step is to write equations for each loop. Based on the sign and current name assigned, as shown below.

For loop1: V=I1R1+I2R4

For loop2: 0=I3R2+I3R3−I2R4

Notice the negative sign in the second equation, it is because of being opposite in direction of loop arrow i.e. for R2 and R3 it is (+ -) but for R4 it is (- +).

Now, put the value of I1, which will give you two simultaneous equations with unknown variable I2 and I3. The value of these currents can easily find out using cramer’s rule for the matrix.

V=(R1+R4)I2+R1I3

0=–R4I2+(R3+R2)I3

Let’s put the values of resistors and the voltage source, and see what happen

20=(5+10)I2+5I3—−(1)

0=−10I2+(5+10)I3—–(2)

Now multiply the equation (1) by -3 and add with the equation (2)

0=−10I2+(15)I3—−(2)

−60=−45I2−15I3—−(1)

SLOVE THE EQUATIONS:

Adding the above two equation will give us the equation (3)

−60=−55I2+0−(3) OR I2=−60−55=1.09Amp

Now putting the current I2 in the equation (2) to get current I3

0=−10(1.09)+15I3

OR

I3=10(1.09)15=0.72Amp

By applying KCL to the node b, the following equation can be obtained

I1=I2+I3

Now putting the values of the current will give us the value of I1

I1=1.09+0.72=1.81Amp

Now, it is easy to find any parameter of the circuit, just use the ohm’s law, as shown below

V3=I3R3=0.72(5)=3.6Volts

And

V2=I3R2=0.72(10)=7.2Volts

V4=I2R4=1.09(10)=10.9Volts

V1=I1R1=1.81(5)=9.05Volts

Krichoffs Current law

Kirchhoff’s Current Law, often shortened to KCL, states that “The algebraic sum of all currents entering and exiting a node must equal zero.”

This law is used to describe how a charge enters and leaves a wire junction point or node on a wire.

Armed with this information, let’s now take a look at an example of the law in practice, why it’s important, and how it was derived.

Parallel Circuit Review

Let’s take a closer look at that parallel example circuit:

parallel circuit example

Solving for all values of voltage and current in this circuit:

At this point, we know the value of each branch current and of the total current in the circuit. We know that the total current in a parallel circuit must equal the sum of the branch currents, but there’s more going on in this circuit than just that. Taking a look at the currents at each wire junction point (node) in the circuit, we should be able to see something else:

At each node on the positive “rail” (wire 1-2-3-4) we have current splitting off the main flow to each successive branch resistor. At each node on the negative “rail” (wire 8-7-6-5) we have current merging together to form the main flow from each successive branch resistor. This fact should be fairly obvious if you think of the water pipe circuit analogy with every branch node acting as a “tee” fitting, the water flow splitting or merging with the main piping as it travels from the output of the water pump toward the return reservoir or sump.

If we were to take a closer look at one particular “tee” node, such as node 6, we see that the current entering the node is equal in magnitude to the current exiting the node:

From the top and from the right, we have two currents entering the wire connection labeled as node 6. To the left, we have a single current exiting the node equal in magnitude to the sum of the two currents entering. To refer to the plumbing analogy: so long as there are no leaks in the piping, what flow enters the fitting must also exit the fitting. This holds true for any node (“fitting”), no matter how many flows are entering or exiting. Mathematically, we can express this general relationship as such:

I exiting = I entering

I exiting + (- I entering) = 0

Convention: current entering a node is positive; while leaving a node is negative

KCL equation:

i1 – i5 + i4 + i3 – i2 = 0

i1 + i3 + i4 = i2 – i5   

Alternate KCL: The sum of currents entering a node is equal to the sum of currents leaving the node.

Example: Write KCL on node ‘a’ and find out Ι T

Solution

So, an application of KCL is to combine current source in parallel into one equivalent current source.

A circuit cannot contain two different currents Ι1 and Ι2 in series unless Ι1=i2; otherwise KCL will be violated

Summarized in a phrase, Kirchhoff’s Current Law reads as such:

“The algebraic sum of all currents entering and exiting a node must equal zero”

That is, if we assign a mathematical sign (polarity) to each current, denoting whether they enter (+) or exit (-) a node, we can add them together to arrive at a total of zero, guaranteed.