ECE 220 Network Analysis I

Lesson 3. Circuit Solving with Kirchhoff's Laws


Using Kirchhoff's Laws Systematically

Before going on to the node-voltage and mesh-current methods of solving circuits, you must thoroughly understand how to solve circuits by applying Kirchhoff's Laws.

To completely "solve" a circuit, we must know the voltage across and the current through each element. The technique is as follows:

  1. Label the circuit:
    1. Assign a current (showing its direction) in every element. Elements in the same branch should be assigned the same current. Don't forget to assign currents to voltage sources.
    2. Assign a voltage across every element. If it's a resistor, inductor, or capacitor the + sign should be placed where the current enters the element. Don't forget to assign voltages to current source.
    3. Identify each essential node (where 3 or more wires join). The node need not be a single point; it may stretch across a circuit and go around corners. Label each node with a letter of the alphabet.
    4. Identify the simple loops (meshes). These are the "window panes." They are loops that don't contain other loops. Label each one with a number.
  2. Write an Ohm's Law equation for each resistor. V = IR.
  3. Count your nodes. Decide which node to ignore (usually the most complex one). Write a Kirchhoff's Current Law equation for each of the remaining nodes. Take the currents leaving the node as positive.
  4. Write a Kirchhoff's Voltage Law equation for each mesh. Go clockwise around each mesh and take voltage drops as positive. Don't forget voltages across current sources.
  5. Count your equations and unknowns. They should be equal.
  6. Solve the set of simultaneous equations.

Systematic Solution Example

We will employ the process above to completely solve the circuit shown in Figure 1 below.

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Figure 1.  Circuit to be solved.

Step 1a is to assign currents to every element. This is shown in Figure 2, below.

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Figure 2.  Circuit with currents defined.

The current in the 6 V battery and the 9 Ω resistor is already known to be 3 A, so it is not necessary to define a current in that branch.  Note the source in the upper branch.  This is a dependent voltage source.  Its voltage is dependent on the current in the center branch.  Although it is dependent on a current, it is still a voltage source, not a current source.  It is therefore necessary to define a current in the branch with the dependent source.  The direction of current I8 is arbitrary.  The current may be defined in either direction.

Step 1b is to assign voltages, as shown below in Figure 3.

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Figure 3.  Circuit with currents and voltages defined.

For each of the resistors, a voltage is assigned.  The polarity of each voltage is chosen with the + sign where the current comes in.  A voltage must also be assigned to the 3 A current source.  The polarity of this voltage is arbitrary.

Step1c requires that essential nodes be identified circled and labled.  This is shown below in Figure 4.  Step 1d is to identify and label meshes.  This is also done in Figure 4.

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Figure 4.  Circuit with all necessary labels.

Now that the circuit is completely labeled, it's time to write the equations.  Step 2 calls for writing the Ohm's Law equations.  We write one for each resistor.  These are shown below.

V4 = 4I4
V8 = 8I8
V9 = 3×9

In step 3, we write the Kirchhoff's Current Law Equations.  There are two essential nodes.  We write one fewer equation than there are nodes, so we need only one equation.  We can write it at either node A or node B.  Let's use node B:

3 + I4 - I8 = 0

In step 4, we write  Kirchhoff's Voltage Law Equations.  There are two meshes.  We write one equation for each mesh:

Mesh 1:  V8 - 5I4 + V4 = 0
Mesh 2:  V9 - V4 + 6 - V3 = 0

In step 5, we count the equations and unknowns.  There are 6 equations and 6 unknowns (V3, V4, V8 , V9, I4, I8), therefore we can be confident that we can solve the circuit.

Solving the circuit (step 6) yields the following results:

V3 = 46.714 V
V4 = -13.714 V
V8 =  -3.429 V
V9 = 27 V
I4 = -3.429 A
I8 = -0.429 V

Most of the results were negative.  Choosing the other direction for I8 would have resulted in additional positive values, but that doesn't matter.

Solving Simultaneous Equations

Step 6, above, "Solve the set of simultaneous equations," can be greatly simplified by use of modern tools.  High-end calculators can solve sets of simultaneous equations.  If you have such a calculator, you should learn to use this feature.  Computer algebra systems such as Maple can also be used to solve simultaneous equations.

Simulations

Below are five links to simulations of Kirchhoff's Laws.  They were created by Sergey Kiselev and Tanya Yanovsky-Kiselev.
* Single loop
* Double loop with three sources
* Double loop with three resistors
* Double loop with three resistors and two sources
* Double loop with three resistors and three sources

Here's another link to a site that allows you to design and build a circuit with resistors, light bulbs, ammeters, voltmeters, etc.  It was created by the Article 19 Group.  It requires the Shockwave plugin.


Before going on to the homework, you should complete Tutorial 3 on circuit solving with Kirchhoff's Laws.


Homework Problems

Note:  If you use a simultaneous-equation-solving calculator or a computer algebra system to solve a homework problem, note this fact in your solution.  This applies for all homework problems, not just the ones in this lesson.

  1. Completely label the diagram. Use Ohm's Law and Kirchhoff's Laws to write enough equations to solve for the current through and the voltage across each element. Do not solve.  Note:  The purpose of this exercise is to give you practice in carefully labeling circuits and writing the circuit equations.  Caution:  There is a dependent source in this problem; is it a voltage source or a current source?  Another caution:  Make sure your current directions match your voltage polarities for  resistors.)
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  2. Label the diagram as appropriate. Write enough equations to solve for the unknown V4.  Do not solve.  Do not include more equations than you need. The purpose of this exercise is to help you recognize the features of a circuit that are not necessary for a solution.  Hint: Write the equations to completely solve the circuit, then cross out the ones that aren't needed.  Further hint:  If an unknown appears in but one equation, and that unknown is not needed, the equation (and unknown) can be eliminated.  (Caution:  There is a current source in this problem.  Does a current source have a voltage across it?)
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  3. E = 3I, R1 = 8 Ω, R2 = 7 Ω, Is = 6 A.  Find I. The answer is an integer.  (Caution:  There is a dependent source in this problem.  Is it a voltage source or a current source?)
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Bonus (no partial credit).  Find the current I in the 10 Ω resistor.  The answer is not an integer.
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