Kirchhoff's Laws

 


  • Kirchhoff’s Current Law (KCL)

  • Kirchhoff’s Voltage Law (KVL)

Kirchhoff’s Current Law (KCL):

The algebraic sum of all currents entering a node must always be zero

where in is the nth current. N is the number of branches.

A common assignment:

  1. if the current is entering the node, assign a negative “-“sign and
  2. if the current is leaving the node, assign a positive “+” sign.

For the following figure

The node equation can be written as

To use KCL to analyze a circuit,

  1. Write KCL equations for the currents
  2. Use Ohm’s law to write currents in terms of bode voltages (one equation for each resistor)
  3. Solve to find values of node voltage and current

  Example: Find the current through a 20Ω resistance, and current through a 40Ω resistance


Write KCL at node x

Kirchhoff

Write Kirchhoff in the circuit using Ohm’s Law

    

Apply last two equation into KCL at node x

Kirchhoff

Kirchhoff

Kirchhoff

The current through a 20Ω resistance

The current through a 40Ω resistance

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Kirchhoff’s Voltage Law (KVL):

The algebraic sum of all voltage around the closed loop must be always zero.

where vn is the nth voltage. N is the number of elements in the loop

A common assignment:

  1. if the positive (+) side of the voltage is encountered first, assign a positive “+”sign to the voltage across the element.
  2. If the negative (-) side of the voltage is encountered first, assign a negative “-”sign to the voltage across the element.

For the following figure

To use KVL to analyze a circuit,

  1. Write KVL equations for voltages
  2. Use Ohm’s law to write voltages in terms of resistances and currents.
  3. Solve to find values of the currents and then voltages.

Examples:

Example 2: Find the current i and voltage v over the each resistor.

KVL equations for voltages

Using Ohm’s Law

Substituting into KVL equation

    

    

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Example 3: Find v1 and v2 in the following circuit
(note: the arrows are signifying the positive position of the box and the negative is at the end of the box)

Fig1.6
Loop 1

KVL
KVL

Loop 2

KVL
KVL

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Example 4: Find V1, V2, and V3.
(note: the arrows are signifying the positive position of the box and the negative is at the end of the box)

Fig1.7
Loop 1

Ohm's law
KVL

Loop 2

KVL
KVL

Loop3

KVL
KVL
KVL

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Example 5: Find V1, V2, V3,and V4
(note: the arrows are signifying the positive position of the box and the negative is at the end of the box)

circuit-5
Loop 1

Ohm's law
KVL

Loop 2

KVL
KVL
KVL

Loop 3

KVL
KVL
KVL

Loop 4

KVL
KVL
KVL

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Practice Problems:

(Click image to view solution)

Problem 1: Find V1 in the following circuit.

View Solution

practice 1 circuit

Solution:

By KVL

KVL Equation
KVL Equation

By KVL for inner loop
KVL Equation
KVL Equation
KVL Equation

         

Problem 2: Find V0 in the following circuit.

View Solution

Solution:

KVL Outher Loop

KVL
KVL
KVL

KVL right inner loop

KVL
KVL

         

Problem 3: Find V1,V2, and V3 in the following circuit.

View Solution

practice-3.gif

Solution:

Outher lopp

KVL
KVL

V2 and 10V source loop

KVL
KVL

V3 and 10V source loop

KVL
KVL

  

Problem 4: Find  I1,I2, I3in the following circuit

View Solution

practice-4.gif

Solution:

Loop 1
Ohm's law
Loop 2
Ohm's law
Loop 3
Ohm's law

From Loop 3
Ohm's law

Substitute into loop  2 , the results is
Ohm's law
Ohm's law (4)

Substitute into loop  1, the results is
Ohm's law
Ohm's law

Substitute into l(4), the results is
Ohm's law
Ohm's law

Substitute into loop  3, the results is
Ohm's law
Ohm's law
Ohm's law

      

Problem 5: Find the resistor R value in the following circuit.

View Solution

practice-5.gif

Solution:

200 ohms and 50 Ohms resistors have same voltage and they are parallel

Ohm's law
Ohm's law

The voltage on R resistor is

Ohm's law

Then, we can find the resistor value as
Ohm's law

         

Exercises:

  1. V1=8V, V2=-4V, V4=14V. Find V3 and V5 in the following circuit

    exercise 1
  2. Find Vx and Vy in the following circuit
    exercise 2
  3. Find Vx , Vy and Vz in the following circuit
    exercise 3
  4. Find the KCL node equations at nodes A,B,C, and D

    exerice 4
  5. If I1=4A , I2 =5A, and I3 =3A, then using KCL find I4 and, I5 in the following circuit
    exerice 5
    Answers:
    1. V3 =12V and V5=-2V
    2. Vx =12V and Vy=9V
    3. Vx =35V, Vy=5V, and Vz=15V
    4. At node A:
      Ohm's law
      At node B:
      Ohm's law
      At node C:
      Ohm's law
      At node D:
      Ohm's law
    5. I4=2A and I5=1A


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