Chapter 2 Resistive Circuits Quiz: Exercise 2.9 (Node-voltage Analysis)

Chapter 2 Resistive Circuits 2.4.2 Circuit with Voltage Sources * Can’t write KCL for node containing voltage source. * Super-node combines several node. * KCL: the net current flowing through any closed boundary is zero.

Chapter 2 Resistive Circuits 2.4.2 Circuit with Voltage Sources * To solve for v, we will need another equation; for the other super-node: * We obtain two identical equations! * We will obtain dependent equations, if we use all the nodes in writing current equations. * We can use KVL to obtain another independent equation:

Chapter 2 Resistive Circuits Exercise 2.11

Chapter 2 Resistive Circuits 2.4.3 Circuits with Controlled Sources – Example 2.9 KCL equations at each node: (Controlling variable) In terms of node voltage, Substitute back:

Chapter 2 Resistive Circuits 2.4.3 Circuits with Controlled Sources – Example 2.10 Super-node containing voltage source: KCL: For node 3: For the reference node: KVL (back to the super-node):

Chapter 2 Resistive Circuits Summary of Sec. 2.4 – Node-Voltage Analysis 1. Select a reference node and assign variables for the unknown node voltages. If the reference node is chosen at one end of an independent voltage source, one node voltage is known at the start, and fewer need to be computed. 2. Write network equations. First, use KCL to write current equations for nodes and supernodes. Write as many current equations as you can without using all of the nodes. Then if you do not have enough equations because of voltage sources connected between nodes, use KVL (super-node) to write additional equations. 3. If the circuit contains dependent sources, find expressions for the controlling variables in terms of the node voltages. Substitute into the network equations, and obtain equations having only the node voltages as unknowns. 4. Put the equations into standard form and solve for the node voltages. 5. Use the values found for the node voltages to calculate any other currents or voltages of interest.

Chapter 2 Resistive Circuits Quiz – Exercises 2.13b Find

Chapter 2 Resistive Circuits Exercise 2.13b

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits – Additional Example Find the unknown shown

Chapter 2 Resistive Circuits – Additional Example Find the unknown shown

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits 2.5 Mesh-Current Analysis 2.5.1 Basic Procedures * Branch –Current and Mesh-Current Analyses * Mesh currents flow around closed paths, it automatically satisfy KCL.

Chapter 2 Resistive Circuits 2.5 Mesh-Current Analysis * Procedures of Mesh-Current Analysis: (1) Choosing mesh current; (2) Writing KVL equations for each mesh; (3) Solving for mesh currents. This is a loop but not a mesh (current)

Chapter 2 Resistive Circuits Example 2.12 – Mesh Current Analysis

Chapter 2 Resistive Circuits 2.5.2 Mesh Current in Circuits Containing Current Sources * We usually avoid writing KVL equations containing current sources, since we don’t know the voltage drop across a current source. Another equation is needed, we may use: Write current source in terms of mesh currents

Chapter 2 Resistive Circuits 2.5.2 Mesh Current in Circuits Containing Current Sources We can’t write KVL equations for meshes 1 and 2. We have for the current source: Still, one more equation is needed. We create a super-mesh containing meshes 1 and 2 and apply KVL:

Chapter 2 Resistive Circuits 2.5.3 Circuits Containing Controlled Sources – Example 2.13 Super mesh containing meshes 1 and 2 eq. (1) For the source current: eq. (a) We also know that: eq. (b) From eqs. (a) and (b), we have: eq. (2) Solving eqs. (1) and (2), we have:

Chapter 2 Resistive Circuits 2.5.3 Standard Mesh-Current Analysis Procedures 1. If necessary, redraw the network without crossing conductors or elements. Then define the mesh currents flowing around each of the open areas defined by the network. For consistency, we usually select a clockwise direction for each of the mesh currents, but this is not a requirement. 2. Write network equations, stopping after the number of equations is equal to the number of mesh currents. First, use KVL to write voltage equations for meshes that do not contain current sources. Next, if any current sources are present, write expressions for their currents in terms of the mesh currents. Finally, if a current source is common to two meshes, write a KVL equation for the supermesh. 3. If the circuit contains dependent sources, find expressions for the controlling variables in terms of the mesh currents. Substitute into the network equations, and obtain equations having only the mesh currents as unknowns.

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits Quiz – Exercises 2.20 (b) and 2.21 (b) Exercise 2.20 (b) Exercise 2.21 (b)

Chapter 2 Resistive Circuits Exercise 2.20 (b)

Chapter 2 Resistive Circuits Exercise 2.21 (b)

Chapter 2 Resistive Circuits 2.6 Thevenin and Norton Equivalent Circuits * To replace two-terminal circuits by simple equivalent circuits. 2.6.1 Thevenin Equivalent Circuits * Thevenin theorem: Any two-terminal circuit consisting resistances and sources can be expressed as an independent voltage source in series with a resistance.

Chapter 2 Resistive Circuits 2.6.1 Thevenin Equivalent Circuits * The Thevenin voltage equals the open circuit voltage of the original circuit. * The short circuit is the same for the original circuit and for the Thevenin equivalent.

Chapter 2 Resistive Circuits 2.6.1 Thevenin Equivalent Circuits – Example 2.14

Chapter 2 Resistive Circuits 2.6.2 Zeroing Sources to Find Thevenin Resistance * In zeroing a voltage source, we reduce its voltage to zero, that is, to short the source. * In zeroing a current source, we reduce its current to zero, that is, to open the source. * We can find the Thevenin resistance by zeroing the sources in the original network and then computing the resistance between the terminals, if there is no dependent source.

Chapter 2 Resistive Circuits Example 2.15 – Zero Sources to Find Thevenin Resistance (1) Zero the sources: (2) Find the short-circuit current of the original circuit,

Chapter 2 Resistive Circuits Example 2.16 - Thevenin Equivalent of a Circuit with a Dependent Source (1) We can’t fine the Thevenin resistance by zeroing the sources. (2) Find the open-circuit voltage first (by node-voltage analysis). (3) Consider the short-circuit condition ?

Chapter 2 Resistive Circuits Quiz – Exercise 2.22 : Find the Thevenin Equivalent

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits – Quiz

Chapter 2 Resistive Circuits – Additional Example ?

Chapter 2 Resistive Circuits 2.6.3 Norton Equivalent Circuits * Norton’s Theorem: Any two terminal circuit consisting resistances and sources can be expressed as an independent current source in parallel with a resistance. * The resistance in Norton equivalent is the same as Thevenin resistance. (1) If we zero the source, the two circuits are the same. (2) The Norton current is equal to the short-circuit current.

Chapter 2 Resistive Circuits 2.6.3 Norton Equivalent Circuits

Chapter 2 Resistive Circuits Example 2.17 - Norton Equivalent Circuits (1) Find open-circuit voltage (2) Find short-circuit current

Chapter 2 Resistive Circuits Quiz – Exercise 2.25 - Find Norton Equivalent Circuits

Chapter 2 Resistive Circuits Quiz – Exercise 2.25 - Find Norton Equivalent Circuits

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits – Additional Example

Chapter 2 Resistive Circuits 2.6.4 Source Transformations In this situation, both circuits have the same open-circuit voltage and short-circuit current: “Source Transformation” can be used to simplify problems.

Chapter 2 Resistive Circuits 2.6.4 Example 2.18 – Using Source Transformations (1) Approach I - Transform current source to voltage source (2) Approach II - Transform voltage source to current source

Chapter 2 Resistive Circuits – Additional Example Use source transformation to find the voltage

Chapter 2 Resistive Circuits 2.6.5 Maximum Power Transfer What is the maximum power which can be delivered to the load? We replace the original circuit by its Thevenin Equivalent When the load resistance equals the Thevenin resistance, it can absorb the maximum power from a two-terminal circuit.

Chapter 2 Resistive Circuits Example: Find the maximum power can be transferred.

Chapter 2 Resistive Circuits 2.7 Superposition Principle Suppose that we have a circuit composed of resistances, linear dependent sources and n independent sources, the superposition principle states that the total response is the sum of the responses to each of the independent sources acting individually: Linear dependent source

Chapter 2 Resistive Circuits 2.7 Superposition Principle

Chapter 2 Resistive Circuits 2.8 Wheatstone Bridge * Wheatstone bridge is used to precisely measure unknown resistance. * When the bridge is in balance:

Chapter 2 Resistive Circuits Example 2.21 Wheatstone Bridge

Chapter 2 Resistive Circuits 2.8 Wheatstone Bridges

Chapter 2 Resistive Circuits Additional Example: Wheatstone Bridge

Chapter 2 Resistive Circuits Additional Example: Wheatstone Bridge

Chapter 2 Resistive Circuits - SUMMARY

Chapter 2 Resistive Circuits - SUMMARY

Chapter 2 Resistive Circuits - SUMMARY

Chapter 2 Resistive Circuits - SUMMARY