Circuit Theorems
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This document provides information on circuit theorems including linearity property, superposition theorem, source transformation, Thevenin's theorem, and Norton's theorem. It includes examples of applying each theorem to solve for voltages and currents in circuits. The maximum power transfer theorem is also discussed and an example is provided to determine the load resistance for maximum power transfer.
![3) Find Thevenin's Equivalent at a-b,3) Find Thevenin's Equivalent at a-b,
note how R4 does not affect Voc! [Vth=28, Rth=8]note how R4 does not affect Voc! [Vth=28, Rth=8]
N.B: You can use any technique to find Vth:
• Source Transform,
• Super Position
• Nodal/Mesh Analysis,
• KCL/KVL](https://image.slidesharecdn.com/g-170228092256/85/Circuit-Theorems-21-320.jpg)
![a) Find Rth
30
b) Find Vth
c)c) Set RL = Rth, Pmax = VthSet RL = Rth, Pmax = Vth22
/(4Rth) [2.901W]/(4Rth) [2.901W]](https://image.slidesharecdn.com/g-170228092256/85/Circuit-Theorems-30-320.jpg)
![7) What is the most power we can deliver to RL? (use Vth, Rth from7) What is the most power we can deliver to RL? (use Vth, Rth from
prob 4) [256 W]prob 4) [256 W]
8) Find the Max Power that can be delivered to a load at a-b
[125 mW]](https://image.slidesharecdn.com/g-170228092256/85/Circuit-Theorems-31-320.jpg)
![3) Find Thevenin Eq at a-b3) Find Thevenin Eq at a-b note how R4 does not affect Voc!note how R4 does not affect Voc!
[Vth=28, Rth=8][Vth=28, Rth=8]
4) Find the Thèvenin equivalent circuit at the terminals A-B.4) Find the Thèvenin equivalent circuit at the terminals A-B.
Rt = 25Rt = 25Ω;Ω;Vt = 160VVt = 160V](https://image.slidesharecdn.com/g-170228092256/85/Circuit-Theorems-38-320.jpg)
![5) Find the Max Power that can be delivered5) Find the Max Power that can be delivered
to a load at a-b [125 mW]to a load at a-b [125 mW]
39
a
b](https://image.slidesharecdn.com/g-170228092256/85/Circuit-Theorems-39-320.jpg)
Circuit Theorems
- 1. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Debre Markos UniversityDebre Markos University School of EngineeringSchool of Engineering Electrical Engineering DepartmentElectrical Engineering Department ECEG 1712 Lecturer: Demsew Mitiku (MSc) October, 2016 Basics of Electricity and Electronics Device
- 2. 2 Circuit TheoremsCircuit Theorems • Motivation • Linearity Property • Superposition • Source Transformation • Thevenin’s Theorem • Norton’s Theorem • Maximum Power Transfer
- 3. 3 If you are given the following circuit, are thereIf you are given the following circuit, are there any other alternative(s) to determine theany other alternative(s) to determine the voltage across 2voltage across 2ΩΩ resistor?resistor? What are they? And how? Can you work it out by inspection? Motivation (1)
- 4. 4 Linearity Property (1)Linearity Property (1) It is the property of an element describing a linear relationship between cause and effect. A linear circuit is one whose output is linearly related (or directly proportional) to its input. Homogeneity (scaling) property v = i R → k v = k i R Additive property v1 = i1 R and v2 = i2 R → v = (i1 + i2) R = v1 + v2
- 5. 5 Linearity Property (2)Linearity Property (2) Example 1 By assume Io = 1 A, use linearity to find the actual value of Io in the circuit shown below. *Refer to in-class illustration, text book, answer Io = 3A
- 6. 6 Superposition Theorem (1)Superposition Theorem (1) It states that the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltage across (or currents through) that element due to EACH independent source acting alone. The principle of superposition helps us to analyze a linear circuit with more than one independent source by calculating the contribution of each independent source separately.
- 7. 7 We consider the effects of 8A and 20V one by one, then add the two effects together for final vo. Superposition Theorem (2)Superposition Theorem (2)
- 8. 8 Superposition Theorem (3)Superposition Theorem (3) Steps to apply superposition principle 1. Turn off all independent sources except one source. Find the output (voltage or current) due to that active source using nodal or mesh analysis. 2. Repeat step 1 for each of the other independent sources. 3. Find the total contribution by adding algebraically all the contributions due to the independent sources.
- 9. 9 Superposition Theorem (4)Superposition Theorem (4) Two things have to be keep in mind: 1.When we say turn off all other independent sources: Independent voltage sources are replaced by 0 V (short circuit) and Independent current sources are replaced by 0 A (open circuit). 2.Dependent sources are left intact because they are controlled by circuit variables.
- 10. 10 Superposition Theorem (5)Superposition Theorem (5) Example 2 Use the superposition theorem to find v in the circuit shown below. 3A is turned off by open-circuit 6V is turned off by short-circuit *Refer to in-class illustration, text book, answer v = 10V
- 11. 1) Use Superposition to find Vx (3V)1) Use Superposition to find Vx (3V)
- 12. 12 • Dependent Sources are not turned off when usingDependent Sources are not turned off when using the Superposition Theoremthe Superposition Theorem Example 3 Use superposition to find vx in the circuit below. *Refer to in-class illustration, text book, answer Vx = 12.5V 2A is discarded by open-circuit 20 Ω v1 4 Ω10 V + − (a) 0.1v1 4 Ω 2 A (b) 20 Ω 0.1v2 v2 10V is discarded by open-circuit Dependant source keep unchanged
- 13. 13 Source Transformation (1)Source Transformation (1) • An equivalent circuit is one whose v-i characteristics at some terminals (a-b) are identical with the original circuit. • Source Transformation is the process of replacing a voltage source vS in series with a resistor R by a current source iS in parallel with a resistor R, or vice versa. • The value of is or vs in the replacement circuit is chosen to have the same behavior at the terminals of interest.
- 14. 14 Source Transformation (2)Source Transformation (2) (a) Independent source transform (b) Dependent source transform •The arrow of the current source is directed toward the positive terminal of the voltage source. •The source transformation is not possible when R = 0 for voltage source and R = ∞ for current source. ++ ++ -- --
- 15. 15 Source TransformationSource Transformation RelationshipRelationship •The two circuits are equivalent IF vs = is R ++ -- •For example, if vs = 10V and R = 2 Ohm • then is = 5A in source transform •same effect at a-b !
- 16. 16 Source Transformation (3)Source Transformation (3) Example 4 Find vo in the circuit shown below using source transformation. * Vo is 1.93 V *answer vo = 100V + vo = Vo vo
- 17. •Some other tricks with Source Transform: .Resistors in Parallel with V source can be removed .Resistors in Series with I source can be removed 17 vo Vo
- 18. Multiple sources in parallel can be converted to current sources, added and reducedcan be converted to current sources, added and reduced Likewise, sources in seriesLikewise, sources in series Voltage sources, added, reducedVoltage sources, added, reduced 18 + vo -- Ans. Vo=-1.43V
- 19. 19 Thevenin’s Theorem (1) It states that a linear two-terminal circuit (Fig. a) can be replaced by an equivalent circuit (Fig. b) consisting of a voltage source VTH in series with a resistor RTH, Where: • VTH is the open-circuit voltage at the terminals. • RTH is the input or equivalent resistance at the terminals when the independent sources are turned off, sometimes called the "Lookback Resistance"
- 20. 20 Thevenin’s Theorem (2) Example 5 Using Thevenin’s theorem, find the equivalent circuit to the left of the terminals in the circuit shown below. Then find i. *Refer to in-class illustration, textbook, answer VTH = 6V, RTH = 3Ω, i = 1.5A 6 Ω 4 Ω (a) RTh 6 Ω 2A 6 Ω 4 Ω (b) 6 Ω 2A + VTh −
- 21. 3) Find Thevenin's Equivalent at a-b,3) Find Thevenin's Equivalent at a-b, note how R4 does not affect Voc! [Vth=28, Rth=8]note how R4 does not affect Voc! [Vth=28, Rth=8] N.B: You can use any technique to find Vth: • Source Transform, • Super Position • Nodal/Mesh Analysis, • KCL/KVL
- 22. 4) Find the Thèvenin equivalent circuit at the4) Find the Thèvenin equivalent circuit at the terminals A-B. Rth = 25terminals A-B. Rth = 25Ω;Ω;Vth = 160VVth = 160V
- 23. Finding Thevenin Eq. withFinding Thevenin Eq. with Dependent SourcesDependent Sources • Vth is still the open circuit voltage – across the terminals of interest • To find Rth, – we CAN turn off Independent Sources – we CANNOT turn off Dependent Sources • they depend on other circuit variables – SOLN: hook up an external source at terminals, find Vo/Io = Rth 23
- 24. 24 Thevenin’s Theorem (3)Thevenin’s Theorem (3) Example 6 Find the Thevenin equivalent circuit of the circuit shown below to the left of the terminals. *Refer to in-class illustration, textbook, answer VTH = 5.33V, RTH = 0,444Ω 6 V 5 Ω Ix 4 Ω + − (a) 1.5Ix i1 i2 i1 i2 3 Ω o + VTh − b a 1.5Ix 1 V+ − 3 Ω0.5I x 5 Ω (b) a b 4 Ω Ix i
- 25. 5) Find Thevenin at a-b (Vth=8, Rth=1)5) Find Thevenin at a-b (Vth=8, Rth=1) 24 6) Find the Thèvenin equivalent circuit at the terminals U-V. (Vth=0, Rth = 9.74)
- 26. 26 Norton’s Theorem (1)Norton’s Theorem (1) o It states that a linear two-terminal circuit can be replaced by an equivalent circuit of a current source IN in parallel with a resistor RN, Where • IN is the short circuit current through the terminals. • RN is the input or equivalent resistance at the terminals when the independent sources are turned off. The Thevenin’s and Norton equivalent circuits are related by a source transformation.
- 27. 27 Norton’s Theorem (2)Norton’s Theorem (2) Example 7 Find the Norton equivalent circuit of the circuit shown below. *Refer to in-class illustration, textbook, RN = 1Ω, IN = 10A. 2 Ω (a) 6 Ω 2vx + − + vx − + vx − 1V + −ix i 2 Ω (b) 6 Ω 10 A 2vx + − + vx − Isc
- 28. 28 Maximum Power Transfer (1)Maximum Power Transfer (1) L Th THL R V PRR 4 2 max =⇒= If the entire circuit is replaced by its Thevenin equivalent except for the load, the power delivered to the load is: The power transfer profile with different RL For maximum power dissipated in RL, Pmax, for a given RTH, and VTH, L LTh Th L R RR V RiP 2 2 + ==
- 29. 29 Example 8 Determine the value of RL that will draw the maximum power from the rest of the circuit shown below. Calculate the maximum power. 2 Ω 4 Ω 1 V+ − (a) 1 Ω 3vx + − i v0+ − vx 9 V+ − io 1 Ω + VTh − + − 3vx 2 Ω + − vx 4 Ω (b) Fig. a => To determine RTH Fig. b => To determine VTH *Refer to in-class illustration, textbook, RL = 4.22Ω, Pm = 2.901W Maximum Power Transfer (2)Maximum Power Transfer (2)
- 30. a) Find Rth 30 b) Find Vth c)c) Set RL = Rth, Pmax = VthSet RL = Rth, Pmax = Vth22 /(4Rth) [2.901W]/(4Rth) [2.901W]
- 31. 7) What is the most power we can deliver to RL? (use Vth, Rth from7) What is the most power we can deliver to RL? (use Vth, Rth from prob 4) [256 W]prob 4) [256 W] 8) Find the Max Power that can be delivered to a load at a-b [125 mW]
- 32. Summary of Thevenin/NortonSummary of Thevenin/Norton • Vth = open cct voltage at terminals • Rth = equivalent Resistance seen at terminals when all independent sources turned off • In = current through terminals when they are shorted • Rn = Rth • In = Vth/Rth, Vth = In * Rn 32
- 33. Finding RFinding Rthth • Circuit has ONLY Independent Sources: – Option A: • Turn off all sources • Find equivalent resistance between terminals – Option B: • Find Vth, In (short circuit current) • Rth = Vth/In 33
- 34. Finding RFinding Rthth • Circuit has Dependent + Independent Sources: – Option A: • Turn off all INDEP sources • Connect an External Source to terminals (Vo or Io) • Find the other quantity (Io or Vo) • Rth = Vo/Io – Option B: • Find Vth (open cct volts), In (short cct current) • Rth = Vth/In 34
- 35. Finding RFinding Rthth • Circuit has ONLY Dependent Sources: – Option A: • Connect an External Source to terminals (Vo or Io) • Find the other quantity (Io or Vo) • Rth = Vo/Io 35
- 36. 1) Use Superposition to find Vx (3V)1) Use Superposition to find Vx (3V) HandoutsHandouts
- 37. 37 2) Find v2) Find voo in the circuit shown below usingin the circuit shown below using source transformation.source transformation. *answer vo = 100V + vo --
- 38. 3) Find Thevenin Eq at a-b3) Find Thevenin Eq at a-b note how R4 does not affect Voc!note how R4 does not affect Voc! [Vth=28, Rth=8][Vth=28, Rth=8] 4) Find the Thèvenin equivalent circuit at the terminals A-B.4) Find the Thèvenin equivalent circuit at the terminals A-B. Rt = 25Rt = 25Ω;Ω;Vt = 160VVt = 160V
- 39. 5) Find the Max Power that can be delivered5) Find the Max Power that can be delivered to a load at a-b [125 mW]to a load at a-b [125 mW] 39 a b