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Network Reduction Techniques and Network Theorems

Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I. 1. Reading Assignment: Sections 4.9 - 4.16, in Electric Circuits, 9 th Ed. by Nilsson . Network Reduction Techniques and Network Theorems Chapter 4, Sections 9-16, in the text by Nilsson covers several useful network

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Network Reduction Techniques and Network Theorems

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  1. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 1 Reading Assignment: Sections 4.9 - 4.16, in Electric Circuits, 9th Ed. by Nilsson • Network Reduction Techniques and Network Theorems • Chapter 4, Sections 9-16, in the text by Nilsson covers several useful network • reduction techniques and network theorems. • The purposes of these techniques and theorems are: • To provide alternate analysis methods • To provide methods for simplifying circuits • To provide methods for representing circuits in the simplest possible form • To gain insight into circuit behavior • Topics to be covered • Source Transformations • Superposition • Thevenin’s and Norton’s Theorems • Maximum Power Transfer Theorem

  2. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 2 Source Transformations Before covering the actual technical of source transformations, some background on types of sources is necessary. There are two broad categories of voltage and current sources: 1) Ideal sources 2) Real sources (or practical sources) So far in the course we have only considered ideal voltage and current sources. We will now consider real sources. • Example: • The voltage supplied by an ideal 12V sourcewill maintain 12V for any current required by the circuit. • The voltage supplied by a real 12V source(such as a car battery) will drop as the current required increases.

  3. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 3 I R + S + V _ V S _ I Real Voltage Source V Real Voltage Source A real voltage source is modeled using an ideal voltage source, VS and a series resistance, RS. Develop an expression for I as a function of V and put it in the form y = mx + b. Plot the characteristics of a real voltage source. Also show the characteristics of an ideal voltage source.

  4. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 4 • Example: A car battery has a voltage of 13V and a current of 0A when nothing • is being powered by the battery, but the voltage drops to 9V and the current is • 300A while starting the car. • A) Draw the characteristics for the battery. • B) Determine the resistance of the battery, RS. • Draw a model of the battery.

  5. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 5 • Example: (continued) • If two headlight were left on (Sylvania 9006ST Halogen Headlamps, rated for 55W at 12.8V), use the model to determine the current. Show the point on the graph. • If someone was “jump-starting” another car with this battery and accidentally touched the jumper cables together, determine the current through the cables (before they melted)! Show this point on the graph.

  6. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 6 I + R P I V P _ I Real Current Source V Real Current Source A real current source is modeled using an ideal current source, IP and a parallel resistance, RP. Develop an expression for I as a function of V and put it in the form y = mx + b. Plot the characteristics of a real current source. Also show the characteristics of an ideal current source.

  7. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 7 I I + + R S + Load Load R V V V _ P I S P _ _ Real Voltage Source Real Current Source • Source Transformations (or source conversion) • There are two types of source transformations: • Transform a real voltage source into a real current source • Transform a real current source into a real voltage source In order for the two types of sources to be equivalent, they should provide the same voltage and current to any load. This can be accomplished by equating their characteristics. In particular, the x-intercepts, y-intercepts, and slopes should be equal (actually equating any 2 of these 3 items fixes the remaining item).

  8. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 8 I IP Slope = 1/RP I V IP RP Real voltage source characteristics Real current source characteristics VS /RS Slope = 1/RS V VS Equating the y-intercepts yields: VS/RS = IP Equating the x-intercepts yields: VS = IPRP Equating the slopes yields: RS = RP

  9. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 9 I I Convert V - source to I - source 4 + + + 20 V V _ 5 A 4 V _ _ Convert I - source to V - source Source Transformations - Conversion Formulas: Simple Example:

  10. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 10 3 2 + + 27 V 6 _ 5 V X _ Example: Determine VX in the circuit below using source transformations.

  11. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 11 Example: Determine VX in the circuit below using source transformations.

  12. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 12 Example: Determine VX in the circuit below.

  13. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 13 + V1 - I + R S + Load Load V2 RP V _ I S P _ Important notes on source transformations: 1) Transformed sources are equivalent in that they provide the same terminal voltage and terminal current (V and I) to any connected load. 2) Transformed sources are not equivalent internally. For example, the current through or the voltage across RS and RP is not the same. To assume that they are the same is a common error. Example: In the circuit below, V1 V2. I 3) All sources are not transformable. Note that a voltage source MUST have a SERIES resistor to be transformable. Note that a current MUST have a PARALLEL to be transformable. 4) Dependent sources can be transformed also.

  14. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 14 Source Transformations - not always possible The last page stated that all sources are not transformable so source transformations cannot be used on all circuits. Example: Draw a circuit that cannot be analyzed using source transformations.

  15. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 15 Example: Determine VX in the circuit below. Show a correct and an incorrect approach.

  16. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 16 Example: Determine VX in the circuit below.

  17. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 17 • Superposition • The superposition theorem essentially states that independent sources act separately. In particular, the current or voltage in any part of the circuit can be calculated by determining the current or voltage due to each independent source (with all other independent sources killed) and then by adding the results algebraically. • Independent sources are killed by: • Shorting voltage sources (which is equivalent to setting the voltage to zero) • Opening current sources (which is equivalent to setting the current to zero) • Notes: • Never kill a dependent source • Superposition applies to voltage and current, but not to power

  18. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 18 Example: Determine the current IX using superposition.

  19. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 19 12 5 6 + _ + 2 A 10 V _ 30 V V 6 + X _ Example: Determine the voltage VX using superposition.

  20. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 20 Example: Determine the voltage VX using superposition.

  21. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 21 Example: A) Determine the voltage VX using superposition.

  22. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 22 Example: (continued) B) Show that superposition does not apply to power (i.e., show that PT P1 + P2 for the top right 18 ohm resistor.)

  23. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 23 Thevenin’s & Norton’s Theorems Thevenin’s and Norton’s theorems are two related theorems that allow us to represent any circuit by a simple equivalent circuit. The equivalent circuit is easier to understand than the original circuit and gives us insight into circuit behavior. Engineer’s often use equivalent systems to help them understand the original system. In Statics, for example, multiple forces are often replaced by an equivalent resultant force, as illustrated in the example below. F1 FR F3 F2 F4 Original System Equivalent System

  24. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 24 Thevenin’s & Norton’s Theorems Any one-port network N may be represented by either of the following types of equivalent circuits: Thevenin Equivalent Circuit (TEC) – consisting of a voltage source and a series impedance Norton Equivalent Circuit (NEC) – consisting of a current source and a parallel impedance where

  25. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 25 + Network N Network N VOC Load _ ISC Network N Network N Load Illustration of VOC and ISC: Remove the load and the voltage across the open terminals is VOC Replace the load by a short circuit (wire) and the current through the short is ISC

  26. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 26 There are 3 ways to find the TEC or NEC for a given circuit: Examples using each of the three methods will be provided on the following pages.

  27. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 27 Example: Find the TEC and the NEC seen by R in the circuit shown below. Discuss which method to use.

  28. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 28 Example: Find the TEC seen by R in the circuit shown below. Discuss which method to use.

  29. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 29 Example: Find VL for R = 10, 20, 40, and 80 ohms in the circuit shown below. Hint: Use the TEC instead of the original circuit.

  30. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 30 Example: Find the TEC seen by R in the circuit shown below. Discuss which method to use.

  31. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 31 Finding Thevenin resistance by adding external sources As indicated in Method 2 for finding a TEC or NEC, RTH can also be found as follows: Discussion: If the dead circuit really acts like a resistor (RTH), then we can add any voltage source across the circuit (resistor), solve for the current, and use Ohm’s Law to find RTH. Similarly, we can add any value current source and solve for the voltage. This technique is illustrated below. Case 1: Add an external voltage source Case 2: Add an external current source IT Dead Circuit (independent sources killed) Dead Circuit (independent sources killed) + + _ VT 10V = VT 2A = IT _ Add any value of voltage source and solve for IT Add any value of current source and solve for VT

  32. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 32 Example: Find the RTH seen by R in the circuit shown below using Method 2 (i.e., add an external source). Compare the results to the value found previously using Method 3.

  33. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 33 Maximum Power Transfer Theorem Suppose that a general network N has a resistive load as shown below. • Now we might consider two questions: • For what value of RL is maximum power delivered to RL? • What is the maximum power that can be delivered to RL? To answer these questions (see next page), 1) Replace N by a Thevenin Equivalent Circuit 2) Determine a general expression for power to RL 3)

  34. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 34 Maximum Power Transfer Theorem Show that:

  35. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 35 The relationship between PL and RL can be illustrated by the graph shown below.

  36. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 36 Example: A) For what value of R in the circuit below is maximum power delivered to R? B) Determine the maximum power that could be delivered to R.

  37. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 37 Example: C) Determine the power delivered to R when R = 20, 50, 100, 200, 400, 800, and 2000 ohms. D) Plot PL versus R.

  38. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 38 30 60 20 + 100 V _ 60 R Example: Find R such that maximum power is delivered to R. Also find Pmax. (Recall that this circuit was used in an earlier example.)

  39. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 39 Applications of the Maximum Power Transfer Theorem The maximum power transfer theorem is commonly used by engineers. Sometimes the requirement that RL = RTH is referred to as “impedance matching.” Max Power Transfer Theorem application - HF transmitter and antenna

  40. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 40 Max Power Transfer Theorem application - stereo amplifier and speaker

  41. Chapter 4, Sections 9-16 EGR 271 – Circuit Theory I 41 • PSPICE Demonstration: • Reference: • PSPICE Assignment #2 • Read Chapters 1 - 3 in Schematic Capture Using Cadence PSPICE by Herniter • Handout: PSPICE Example - Maximum Power Transfer (Varying a Component Value) • Handout: PSPICE Example - Op Amp Circuit using a Library Model ( uA741) • Handout: PSPICE Example - Op Amp Example using a General Op Amp Model • DC Sweep: Illustrate how to vary the following quantities: • A voltage source • A current source • A resistor • Insert the part PARAM when varying a resistor value. • Use a potentiometer symbol (part R_VAR) for the resistor • Be sure to change the property SET from 0.5 to 1 • PROBE: Illustrate with PSPICE’s graphing window how to: • Add traces • Add text, arrows, boxes • Add one or two cursors (control with left or right mouse and with + or Shift +) • Mark points on a graph • Find maxima and minima

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