Kirchhoff’s Laws - Current Law (KCL), Voltage Law (KVL) | web4study

Kirchhoff’s Laws – Current Law (KCL), Voltage Law (KVL)



Kirchhoff’s Laws

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
  • IIN = IOUT

Kirchhoff’s Current Law (KCL)

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
  • ΣV = VT 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

Method of Branch Currents

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

VR1 = I1R1

VR2 = I2R2

VR3 = I3R3

VR3 = (I1+I2)R3

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.

Node-Voltage Analysis

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

Node-Voltage Analysis

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.

Method of Mesh Currents

  • 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.


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