# Unit 3A Resistors in Series and in Parallel; Voltage Division and Current Division

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Chapter 4 – Techniques of Circuit Analysis

## Study Guide

Objectives:

1. Understand and be able to use the node-voltage method to solve a circuit.
2. Understand and be able to use the mesh-current method to solve a circuit.
3. Be able to decide whether the node-voltage method or the mesh-current method is the preferred approach to solving a particular circuit.
4. Understand source transformation and be able to use it to solve a circuit.
5. Understand the concept of the Thevenin- and Norton-equivalent circuits and be able to construct a Thevenin or Norton equivalent for a circuit.
6. Know the condition for maximum power transfer to a resistive load and be able to calculate the value of the load resistor that satisfies this condition.

Mastering the Objectives:

1. Read the Introduction and Section 4.1.

a)Redraw the circuit of Fig. 4.1(b). Label all essential nodes with capital letters and label all remaining nodes with lower-case letters.

1. How many nodes does this circuit have?
2. How many essential nodes does this circuit have?
3. How many independent KCL equations can you write?
4. How many meshes does this circuit have?
5. List the meshes in the circuit using the technique of Example 4.1(e).
6. How many independent KVL equations can you write?

b)Solve Chapter Problem 4.1.

1. Read Sections 4.2 – 4.4.

a)The node-voltage method usually results in two or more simultaneous equations that you must solve. It is strongly recommended that you use a calculator to solve the simultaneous equations. This will require that you place the equations in standard form before entering them into the calculator. The standard form for an equation collects the terms that involve unknowns on the left-hand side and the constant terms on the right-hand side. The standard form for Eqs. 4.5 – 4.6 is

It is important to keep the unknowns, in this case the voltages v1 and v2, in the same order in each equation.

1. Write the node-voltage equations from Example 4.2 in standard form.
2. Write the first three equations in Example 4.3 in standard form.
3. Write Eqs. 4.7 – 4.8 in standard form

b)Solve Assessment Problem 4.1 and Chapter Problem 4.9. Place your equations in standard form and use a calculator to solve them.

c)Each dependent source in a circuit requires a constraint equation. If the controlling quantity is a current, you will need to use Ohm’s law to write the constraint equation. This is illustrated in Example 4.3. If the controlling quantity is voltage, you will not use Ohm’s law; instead, the constraint equation will define the controlling voltage in terms one or more of the node voltages. Look at the dependent source in Fig. P4.25. The controlling quantity is the voltage vx. Assuming the bottom node is the reference node and the remaining nodes have voltages v1, v2, and v3 (labeled from the left), then

Practice writing just the dependent source constraint equation for the following circuits:

1. P4.17
2. P4.19
3. P4.27
4. P4.28

d)Solve Assessment Problem 4.3 and Chapter Problem 4.20. Remember to place the simultaneous equations in standard form and use a calculator to solve them.

e)The special cases that arise in the node-voltage method involve voltage sources, as discussed in Section 4.4. If a voltage source is the only element between an essential node and the reference node, the voltage at the essential node must be the value of the voltage source. Therefore, there is no need to write a KCL equation at that essential node. This case is illustrated in Fig. 4.12. When a voltage source is the only element between two non-reference essential nodes you must write one supernode equation and one supernode constraint equation. This is illustrated in Fig. 4.14. Look at Assessment Problems 4.4 – 4.6. Assume the reference node is the lower node in each circuit.

1. Which circuits have a voltage source between an essential node and the reference node?
2. Which circuits have a voltage source between two non-reference essential nodes?

f)Before you write the simultaneous equations needed to solve a circuit using thenode-voltage method, you should determine how many equations you will need to write. To do so, follow these steps:

1. Find the essential nodes and choose a reference node. Plan to write one KCL equation at each non-reference essential node unless a subsequent step eliminates a KCL equation.
2. Find the dependent sources. You will write a dependent source constraint equation for each dependent source.
3. Find the voltage sources. Are any voltage sources the only element between the reference node and another essential node? If so, eliminate the planned KCL equation at that essential node.
4. Are any of the voltage sources between two non-reference essential nodes? For each such voltage source eliminate the two planned KCL equations at the two essential nodes, add a KCL equation for the supernode, and add a supernode constraint equation.

We illustrate these steps using Fig. 4.13:

1. There are four essential nodes. We choose the lower node as the reference node and plan to write three KCL equations at the remaining essential nodes.
2. There is one dependent source so we plan to write one dependent source constraint equation.
3. There is one voltage source between an essential node and the reference node, so there is no need to write the planned KCL equation at node 1.
4. There is one voltage source between two non-reference essential nodes, so instead of writing the planned KCL equations at nodes 2 and 3 we write one supernode KCL equation and one supernode constraint equation.

In summary,

• The voltage at node 1 is known (50 V).
• Write one supernode KCL equation (Eq. 4.12).
• Write one supernode constraint equation (Eq. 4.13).
• Write one dependent source constraint equation (Eq. 4.14).

Practice identifying the number and type of equations to write. For the following circuits, determine the number of KCL equations, the number of supernode KCL equations, the number of supernode constraint equations, and the number of dependent source constraint equations. Do not write any equations!

1. P4.20
2. P4.25
3. P4.27
4. P4.29

g)Solve Assessment Problems 4.4 – 4.6. Determine the number and type of equations you will need to write before you write them. Put the equations in standard form and solve them using a calculator.

1. Read Sections 4.5 – 4.7.

a)It is important to place mesh-current equations in standard form and use a calculator to solve them. This is illustrated in Example 4.4. Solve Assessment Problem 4.7 and Chapter Problem 4.31. Remember to put your equations in standard form and use a calculator to solve them.

b)Each dependent source in a circuit requires a constraint equation. If the controlling quantity is a voltage, you will write the constraint equation using Ohm’s law to find the controlling voltage from one or more mesh currents. As an example, look at Fig. P4.42. If the left mesh current is i1, clockwise, then the constraint equation for the dependent current source is

If the controlling quantity is a current then the constraint equation does not use Ohm’s law. Instead, the controlling current is expressed in terms of one or more of the mesh currents. Example 4.5 illustrates such a constraint equation. To practice, write the dependent source constraint equations for the following circuits:

1. Assessment Problem 4.8
2. Assessment Problem 4.9
3. Fig. P4.34
4. Fig. P4.39

c)Solve Assessment Problem 4.8 and Chapter Problem 4.33. Remember to place the equations in standard form and solve them using a calculator.

d)The special cases that arise in the mesh-current method involve current sources, as discussed in Section 4.7. If a current source is positioned on the perimeter of a mesh, so is not shared by two meshes, the current in that mesh must equal the value of the current source. Therefore, there is no need to write a KVL equation for that mesh. This case is illustrated in Fig. P4.37. In this circuit there are three mesh currents, but the mesh current in the lower mesh must have the same values as the current source, 30 A. Therefore you need to write only two KVL equations to solve this circuit using the mesh-current method. When two meshes share a current source you must write one supermesh KVL equation and one supermesh constraint equation. This is illustrated in Fig. 4.25. Note from Fig. 4.26 that the supermesh consists of the two meshes that share the current source, with the branch containing the current source eliminated. Look at Assessment Problems 4.10 – 4.12.

1. Which circuits have a current source on the perimeter of a single mesh?
2. Which circuits have a current source that is shared between two meshes?

e)Before you write the simultaneous equations needed to solve a circuit using the mesh-current method, you should determine how many equations you will need to write. To do so, follow these steps:

1. Count the number of meshes. Plan to write one KVL equation for each mesh unless a subsequent step eliminates a KVL equation.
2. Find the dependent sources. You will write a dependent source constraint equation for each dependent source.
3. Find the current sources. Are any current sources on the perimeter of a single mesh? If so, eliminated the planned KVL equation for that mesh.
4. Do two meshes share any of the current sources? For each such current source eliminate the two planned KVL equations for the two meshes, add a KVL equation for the supermesh, and add a supermesh constraint equation.

We illustrate these steps using Fig. P4.47:

1. There are four meshes so we plan to write four KVL equations, one for each mesh.
2. There are two dependent sources so we plan to write two dependent source constraint equations.
3. There is one current source on the perimeter of a single mesh, so we can eliminate the planned KVL equation for that mesh.
4. There is one current source shared between two meshes, so we eliminate the planned KVL equations for those meshes, add a supermesh KVL equation and a supermesh constraint equation.

In summary,

• The current in the left mesh is known (19 A).
• Write a KVL equation for the top mesh.
• Write one supermesh KVL equation for the remaining two meshes.
• Write one supermesh constraint equation.
• Write two dependent source constraint equations.

Practice identifying the number and type of equations to write. For the following circuits, determine the number of KVL equations, the number of supermesh KVL equations, the number of supermesh constraint equations, and the number of dependent source constraint equations. Do not write any equations!

1. P4.38
2. P4.40
3. P4.43
4. P4.46

f)Solve Assessment Problems 4.10 – 4.12. Determine the number and type of equations you will need to write before you write them. Put the equations in standard form and solve them using a calculator.

a)In deciding whether to use the node-voltage method or the mesh-current method to solve a particular circuit, you can use the techniques in Study Guide sections 2(f) and 3(e) to determine the number and type of equations needed for each method. You should also take into account what quantity you are asked to find – if a current is requested, the mesh-current method may be preferred because it produces currents, and if a voltage is requested the node-voltage method may be preferred because it produces voltages. Practice comparing the node-voltage method and the mesh-current method for the following circuits, but do not write or solve any equations:

1. Chapter Problem 4.6
2. Chapter Problem 4.12
3. Chapter Problem 4.25
4. Chapter Problem 4.39
5. Chapter Problem 4.47
6. Chapter Problem 4.56

b)Solve Assessment Problems 4.13 and 4.14. Compare the node-voltage method with the mesh current method by determining the number and type of equations required for each method before choosing the method you will use.

a)The node-voltage method and the mesh current method can be used in any circuit containing essential nodes and meshes, respectively. The source-transformation method discussed in this section can only be used in circuits with specific configurations amenable to this method. Usually, circuits that can be simplified using source transformation look like ladders. Look at the circuits in Figs. 4.9 – 4.15 and determine which circuits could be simplified using source transformation.

b)Use source transformation on the circuit in Fig. 4.37 in Example 4.8 to find the power delivered by the 40 V source. To do this you will need to work from the left side of the circuit toward the right side of the circuit.

c)Solve Assessment Problem 4.15 and Chapter Problem 4.55.

1. Read Sections 4.10 and 4.11.

a)The Thevenin- and Norton-equivalent methods can be used to simplify circuits, just as the source-transformation method and series and parallel combinations of resistors. When faced with the problem of determining a Thevenin or Norton equivalent for a particular circuit, you need to determine which of the following three categories the circuit belongs to:

1. A circuit with one or more independent sources, no dependent sources, and one or more resistors. In these circuits you can calculate any two of the following three quantities: The open circuit voltage, voc = vTh; the short-circuit current, isc = iN; or the Thevenin-equivalent resistance, RTh = RN. It is usually quite easy to calculate RTh by replacing all voltage sources with short circuits, all current sources with open circuits, and making series and parallel combinations of resistors to find the equivalent resistance.
2. A circuit with one or more independent sources, at least one dependent source, and one or more resistors. In these circuits you cannot calculate RTh by removing the independent sources and combining resistors. Instead, you must solve two different circuits, one to find the open circuit voltage and the other to find the short circuit current, and then use RTh = voc/isc to find the equivalent resistance.
3. A circuit with no independent sources. In these circuits, vTh = 0 V, so the Thevenin equivalent and Norton equivalent circuits consist of a single resistor. If the circuit has no dependent sources, the problem is reduced to making series and parallel combinations of resistors to find a single equivalent resistance. If the circuit has one or more dependent sources, you must use the test-source method.

Look at the circuits in Figs. P4.59 – P4.65 and P4.73 – P4.74 and determine the appropriate category for each circuit with respect to determining the Thevenin equivalent.

b)Find the Thevenin resistance Rab in the circuit of Fig. 4.45 by eliminating the independent source and making series and parallel combinations of resistors.

c)Solve Assessment Problems 4.16 and 4.17, and Chapter Problem 4.62.

d)Example 4.11 illustrates the use of a test source to find RTh in a circuit with no independent sources. It is often easier to apply a 1 V source and use familiar circuit analysis techniques to find the current through the 1 V source. For the circuit in Fig. 4.54 of Example 4.11, apply a 1 V source in place of vT. Then,

Use this technique to solve Chapter Problem 4.73.

e)Solve Assessment Problem 4.19.