Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Engineering 43 Chp 8 [3-4] Magnetic Coupling Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. Outline – Magnetic Coupling • Mutual Inductance • Behavior of inductors sharing a common magnetic field • Energy Analysis • Used to establish relationship between mutual reluctance and self-inductance Last Time

3. Outline – Magnetic Coupling cont. • Ideal Transformer • Device modeling of components used to change voltage and/or current levels • Safety Considerations • Important issues for the safe operation of circuits with transformers

4. Consider Now two Coils Wrapped Around a Closed Magnetic (usually iron) Core. A  Area The Ideal Transformer • The Iron Core Strongly confines the Magnetic Flux, , to the Interior of the Closed Ring • All Turns, N1 & N2, of Both Coils are Linked by the Core Flux • Again, this is a NONconductive (no wires) connection

5. The Coils, N1 & N2, are Flux-Linked:  = N Ideal X-Former Physics/Math • Then the Ratio of v1:v2 • Next Apply Ampere’s Law (One of Maxwell’s Eqns) • By Faraday’s Induction Law for Both Coils • Where • H  Magnetic Field Strength (Amp/m)

6. Ampere’s Law Ideal X-Former Physics cont.1 • H=0 in Ampere’s Law • Now Manipulate Ampere’s Law Eqn • The Path for the Closed Line Integral is a path Within the Iron Core • If the Magnetic Core is IDEAL, then

7. The Ideal Xformer Ideal X-Former Physics cont.2 • But v•i = POWER, and by the previous Eqn the total power used by the Xformer is ZERO • Thus in Ideal Form, a Transformer is LOSSLESS • Thus the INput Power = OUTput Power • But By Flux Linkage • So in the ideal Case Ampere’s Law

8. From The Device Physics; The Ideal Xformer Eqns Ideal X-Former Circuit Symbol • The Circuit Symbol Iron Core • The Main practical Application for This Device: • TRANSFORM one AC Voltage-Level to Another • Since an Xformer is Two Coupled Inductors, Need the DOT Convention to Track Polarities

9. When a Voltage is Transformed, Give the INput & OUTput sides Special Names INput, v1, Side  PRIMARY Circuit OUTput, v2, Side  SECONDARY Circuit Transformer Application • The Voltage Xformer • Pictorial Representation Src • Then the Circuit Symbol Usage as Applied to a Real Circuit Load

10. The Actual Ckt Symbol As used On an Engineering Dwg Transformer Practical App • The Practical Symbol does NOT Use DOTS • NEMA has Established Numbering Schemes That are Functionally Equivalent to the Dots • The Multiple “Taps” on the Primary Side Allow The Transformation of More Than One Voltage Level • See next Slide for a REAL Xformer Design • The Parallel (||) lines Between the Coils Signify the Magnetic Core

11. 208:115 Vac StepDown Xformer 208:24 Vac StepDown Xformer 208Vac, 80 kVA Input

12. Have TWO Choices for Polarity Definitions Symmetrical Sign (Dot) Conventions • Thus The Form of the Governing Equations Will Depend on the Assigned: • DOT POSITION • VOLTAGE POLARITY • CURRENT DIRECTION • INput/OUTput (I/O)

13. In Practice, The Vast Majority of Xformers are Used in AC Circuits Recall The Symmetrical Ideal-Xformer Eqns Phasor Analysis • Illustration: InPut IMPEDANCE = V1/I1 • Notice That This is an I/O Model; So the Eqns • These are LINEAR in i & v, so PHASOR Analysis Applies

14. An I/O Xformer in Phasor Domain Phasor Analysis: Input Z • Solve for V1 • Now The Input Impedance Z1 • Now Apply Ohm’s Law to the Load, ZL • Now Sub for V2 and I2 From I/O Xformer Eqns • For 10X stepDOWN (N1:N2 = 10:1) Z1 is 100X ZL

15. An I/O Xformer Phasor Domain Input Impedance Phasor Analysis: Input Z cont.1 • Thus ZL is Said to be REFLECTED to the Input Side (by [N1/N2]2) • For Future Reference • For a LOSSLESS Primary/Secondary Xformer the INput impedance is a Fcn ONLY of the LOAD Impedance, ZL • Ideal Xformer Phasor Eqns

16. Given the Ckt Below Find all I’s and V’s Numerical Example • Using stepDOWNXformer • Find I1 by Ohm • Note: n = N2/N1 = 1/4 • Game Plan • reflect impedance into the primary side and make the transformer “transparent to Source”

17. The Intermediate Ckt Numerical Example cont.1 Z2 • About the Same Hassle-Factor; use ZI stepDOWNXformer • Now Find V1 by Ohm or V-Divider • Next Determine SIGNS • Which is Easier?

18. The Original Ckt Numerical Example cont.2 stepDOWNXformer • Compare Current Case to I/O Model • Voltage-2 is OPPOSITE (NEGATIVE at Dot) • Current-2 is OPPOSITE (INTO Dot) • Then In This Case • Recall Now the I/O Model Eqns OUT IN

19. The Original Ckt Numerical Example cont.3 • The Output Voltage stepDOWNXformer • Then Output Current • Using the Signs as Determined by Dots and Polarities • Note: On Calculator aTan(–4.72/–20.33) = 13.07° • Recall RANGE of aTan = –90° to + 90°

20. Given the Ckt Below Find I1 & Vo Illustration stepUPXformer • Note: n = N2/N1 = 2/1 • Game Plan • reflect impedance into the primary side and make the transformer “transparent to Src” • Again using

21. Given the Ckt Below Find I1 & Vo Illustration cont.1 Z2 stepUPXformer • Thus I1 by Ohm • Next Find Vo by I2 and Ohm’s Law • Define I2 Direction per I/O Model (V2 is ok)

22. Given I/O XFormer Ckt Xformer Thevenin Equivalent • Find the The Thevenin Equivalent at 2-2’ • First Find the OPEN Ckt Voltage at 2-2’ • Note: The Dots & Polarities Follow the I/O Model

23. Now find ZTH at Terminals 2-2’ Xformer Thevenin Equiv. cont.1 • Thus the Thevenin Equivalent at 2-2’ • “Back Reflect” Impedance into SECONDARY • The Xformer has been “made Transparent” to the Secondary Side • Next: Find Thevenin Equiv at Terminals 1-1’

24. Given I/O XFormer Ckt Thevenin Equiv. from Primary • As in Open Ckt • Thevenin impedance will be the Secondary impedance reflected into the PRIMARY Ckt • Find the The Thevenin Equivalent at 1-1’ • Then The Open Ckt Voltage Depends on VS2

25. Primary v. Secondary Thevenin • The Base Ckt • Thevenin From Primary • Thevenin From Secondary • Equivalent circuit reflecting into primary • Equivalent circuit reflecting into secondary

26. Exmpl: Draw Thevenin Equiv’s Equivalent circuit reflecting into SECONDARY Equivalent circuit reflecting into PRIMARY

27. Given the Ckt Below Find I1 Example 1 • Note The Dot Locations 1’ • Note: n = N2/N1 = 2/1 • Game Plan • Find Thevenin Looking from PRIMARY Side • Draw the Ckt

28. Example: Safety Considerations • Two Houses Powered By DIFFERENT XFormers • Utility Circuit Breaker X-Y OPENS: Powering DOWN House-B • The Well-Meaning Neighbor Runs Extension Cord House-A → House-B • This POWERS the 2nd-ary Side of the House-B Pole Xformer

29. Example: Safety Consid cont.1 • Transformers are BIdirectional Devices • They can step-UP or step-DOWN Voltages • Thus the 120Vac/15A Extension Cord Produces 7200 Vac Across Terminals X-Z • The Service Engineer (SE) Now Goes to the Breaker to ReMake the Connection to House-B • The SE expects ZERO Volts at X-Z; If She/He Does NOT Check by DMM, then He/She Could Sustain a Potentially FATAL 7.2 kV Electric-Shock! 

30. At the Generating Facility (e.g. Diablo Canyon) Electricity is Generated at 15-25 kVac But Xformers are used to Set-UP the Voltage-Level to 400-765 kVac Why? → Line SIZE (and others) Exmple  Power Transmission

31. Case Study: Transmit 225 MW over 100 Miles of Wire 2 Conductors 95% Efficiency Cu Wire w/ Resistivity ρ = 80 nΩ-m Find the Wire Diameter, d, for: V = 15 kVac V = 500 kVac Exmpl – Power Xmission cont.1

32. By Solid-State Physics (c.f. ENGR-45) Exmpl – Power Xmission cont.2 • By Power Rln for Resistive Ckts • Where for the Wire • ρ Resistivity (Ω-m) • l  Length (m) • A  X-Section Area (m2) • In This Case • Solve for d • Then the Power Loss

33. Finally Solve for Transmission Cable Diameter Exmpl – Power Xmission cont.3 • d15 = 5.03” • Pretty BIG & HEAVY • d500 = 0.15” • MUCH Better

34. Consider Now two Coils Wrapped Around a Closed Magnetic (usually iron) Core. A  Area Summary: Ideal Transformer • The Iron Core Strongly confines the Magnetic Flux, , to the Interior of the Closed Ring • All Turns, N1 & N2, of Both Coils are Linked by the Core Flux

35. The Ideal Xformer Eqns Faraday’s Law Ideal X-Former Circuit Symbol • The Circuit Symbol Iron Core • Ampere’s Law • The Main practical Application for This Device: • TRANSFORM one AC Voltage-Level to Another • As Two Coupled Inductors, Xformers Use the DOT Convention to Track Polarities

36. When a Voltage is Transfrormed, Give the INput & OUTput sides Special Names INput, v1, Side  PRIMARY Circuit OUTput, v2, Side  SECONDARY Circuit Transformer Application • The Voltage Xformer • Pictorial Representation • Then the Circuit Symbol Usage as Applied to a Real Circuit Load

37. Have TWO Choices for Polarity Definitions Symmetrical Sign (Dot) Conventions • The Form of the Governing Equations • INput/OUTput (I/O)

38. In Practice, The Vast Majority of Xformers are Used in AC Circuits Ideal-Xformer Eqns Are LINEAR in i&v so PHASOR Analysis Applies Phasor Analysis • Illustration: InPut IMPEDANCE • Notice That This is an I/O Model; So the Eqns Yield

39. WhiteBoard Work • Let’s Work This Nice Problem

40. 208:115 Vac StepDown Xformer 208:24 Vac StepDown Xformer 208Vac, 80 kVA Input