## Kirchhoff’s Laws

Topics Covered in this ppt

Kirchhoff’s Current Law (KCL)

Kirchhoff’s Voltage Law (KVL)

Method of Branch Currents

Node-Voltage Analysis

Method of Mesh Currents

## Kirchhoff’s Current Law (KCL)

- The sum of currents entering any point in a circuit is equal to the sum of currents leaving that point.
- Otherwise, a charge would accumulate at the point, reducing or obstructing the conducting path.
- Kirchhoff’s Current Law may also be stated as
*I**IN** = I**OUT*

Fig. 9-2: Series-parallel circuit illustrating Kirchhoff’s laws.

The 6-A IT into point C divides into the 2-A I3 and 4-A I4-5

I4-5 is the current through R4 and R5

IT − I3 − I4-5 = 0

6A − 2A − 4A = 0

At either point C or point D, the sum of the 2-A and the 4-A branch currents must equal the 6A line current.

Therefore, Iin = Iout

**Loop Equations**

- A
**loop** is a closed path.
- This approach uses the algebraic equations for the voltage around the loops of a circuit to determine the branch currents.
- Use the
*IR* drops and KVL to write the loop equations.
- A loop equation specifies the voltages around the loop.

- Loop Equations
*Σ**V = V**T* means the sum of the IR voltage drops must equal the applied voltage. This is another way of stating Kirchhoff’s Voltage Law.

In Figure 9-2, for the inside loop with the source VT, going counterclockwise from point B,

90V + 120V + 30V = 240V

If 240V were on the left side of the equation, this term would have a negative sign.

The loop equations show that KVL is a practical statement that the sum of the voltage drops must equal the applied voltage.

- The algebraic sum of the voltage rises and
*IR* voltage drops in any closed path must total zero.

For the loop CEFDC without source the equation is

- −V4 − V5 + V3 = 0
- −40V − 80V + 120V = 0
- 0 = 0

## Method of Branch Currents

Application of Kirchhoff’s laws to a circuit with two sources in different branches.

**V****R****1**** = I****1****R****1**

**V****R****2**** = I****2****R****2**

**V****R****3**** = I****3****R****3**

**V****R****3**** = (I****1****+I****2****)R****3**

**Loop equations:**

V1 – I1R1 – (I1+I2) R3 = 0

V2 – I2R2 – (I1+I2) R3 = 0

**Solving for currents**

Using the method of elimination, multiply the top equation by 3 to make the I2 terms the same in both equations

9I1 + 3I2 = 42

1I1 + 3I2 = 7

**Subtracting**

7I1 = 35

I1 = 5A

To determine I2, substitute 5 for I1

2(5) + 3I2 = 7

3I2 = 7 − 10

3I2 = −3

I2 = −1A

This solution of −1A for I2 shows that the current through R2 produced by V1 is more than the current produced by V2.

The net result is 1A through R2 from C to E

**Calculating the Voltages**

VR1 = I1R1 = 5 x 12 = 60V

VR2 = I2R2 = 1 x 3 = 3V

VR3 = I3R3 = 4 x 6 = 24V

**Note**: VR3 and VR2 have opposing polarities in loop 2.

This results in the −21V of V2

**Checking the Solution**

At point C: 5A = 4A + 1A

At point D: 4A + 1A = 5A

Around the loop with V1 clockwise from B,

84V − 60V − 24V = 0

Around the loop with V2 counterclockwise from F,

21V + 3V − 24V = 0

## Node-Voltage Analysis

- A
**principal node** is a point where three or more currents divide or combine, other than ground.
- The method of
**node voltage analysis** uses algebraic equations for the node currents to determine each node voltage.
- Use KCL to determine node currents
- Use Ohm’s Law to calculate the voltages.
- The number of current equations required to solve a circuit is one less than the number of principal nodes.
- One node must be the reference point for specifying the voltage at any other node.
- Finding the voltage at a node presents an advantage: A node voltage must be common to two loops so that voltage can be used for calculating all voltages in the loops.

Fig. 9-7: Method of node-voltage analysis for the same circuit as in Fig. 9-5.

### Calculating All Voltages and Currents

**Node Equations**

- Applies KCL to currents in and out of a node point.
- Currents are specified as
*V/R* so the equation of currents can be solved to find a node voltage.

**Loop Equations**

- Applies KVL to the voltages in a closed path.
- Voltages are specified as
*IR* so the equation of voltages can be solved to find a loop current.

## Method of Mesh Currents

- A
**mesh** is a simplest possible loop.
**Mesh currents** flow around each mesh without branching.
- The difference between a mesh current and a branch current is that a mesh current does not divide at a branch point.
- A mesh current is an assumed current; a branch current is an actual current.
*IR* drops and KVL are used for determining mesh currents.
- The number of meshes is the number of mesh currents. This is also the number of equations required to solve the circuit.

- A clockwise assumption is standard. Any drop in a mesh produced by its own mesh current is considered positive because it is added in the direction of the current.
- Mesh A: 18IA − 6IB = 84V
- Mesh B: 6IA + 9IB = −21V

Use either the rules for meshes with mesh currents or the rules for loops with branch currents, but do __not__ mix the two methods.

To eliminate IB and solve for IA, divide the first equation by 2 and the second by 3. then

9IA − 3IB = 42

−2IA + 3IB = −7

Add the equations, term by term, to eliminate IB. Then

7IA = 35

IA = 5A

To calculate IB, substitute 5 for IA in the second equation:

−2(5) + 3IB = −7

3IB = −7 + 10 =3

IB = 1A

The positive solutions mean that the electron flow for both IA and IB is actually clockwise, as assumed.