ENGR-43_Lec-10-2_Transformers.ppt 1 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engineering 43 Chp 8 [3-4] Magnetic Coupling

ENGR-43_Lec-10-2_Transformers.ppt 2 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis 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

ENGR-43_Lec-10-2_Transformers.ppt 3 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis 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

ENGR-43_Lec-10-2_Transformers.ppt 4 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis The Ideal Transformer  Consider Now two Coils Wrapped Around a Closed Magnetic (usually iron) Core.  The Iron Core Strongly confines the Magnetic Flux, , to the Interior of the Closed Ring All Turns, N 1 & N 2, of Both Coils are Linked by the Core Flux –Again, this is a NONconductive (no wires) connection A  Area

ENGR-43_Lec-10-2_Transformers.ppt 5 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Ideal X-Former Physics/Math  The Coils, N 1 & N 2, are Flux-Linked: = N   Then the Ratio of v 1 :v 2  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)

ENGR-43_Lec-10-2_Transformers.ppt 6 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Ideal X-Former Physics cont.1  Ampere’s Law  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

ENGR-43_Lec-10-2_Transformers.ppt 7 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Ideal X-Former Physics cont.2  The Ideal Xformer  But vi = 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

ENGR-43_Lec-10-2_Transformers.ppt 8 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Ideal X-Former Circuit Symbol  From The Device Physics; The Ideal Xformer Eqns  The Circuit Symbol  Since an Xformer is Two Coupled Inductors, Need the DOT Convention to Track Polarities  The Main practical Application for This Device: TRANSFORM one AC Voltage-Level to Another Iron Core

ENGR-43_Lec-10-2_Transformers.ppt 9 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Transformer Application  When a Voltage is Transformed, Give the INput & OUTput sides Special Names INput, v 1, Side  PRIMARY Circuit OUTput, v 2, Side  SECONDARY Circuit  The Voltage Xformer  Then the Circuit Symbol Usage as Applied to a Real Circuit  Pictorial Representation L o a d S r c

ENGR-43_Lec-10-2_Transformers.ppt 10 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Transformer Practical App  The Actual Ckt Symbol As used On an Engineering Dwg  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

ENGR-43_Lec-10-2_Transformers.ppt 11 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis 208Vac, 80 kVA Input 208:115 Vac StepDown Xformer 208:24 Vac StepDown Xformer

ENGR-43_Lec-10-2_Transformers.ppt 12 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Sign (Dot) Conventions  Have TWO Choices for Polarity Definitions Symmetrical  Thus The Form of the Governing Equations Will Depend on the Assigned: DOT POSITION VOLTAGE POLARITY CURRENT DIRECTION INput/OUTput (I/O)

ENGR-43_Lec-10-2_Transformers.ppt 13 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Phasor Analysis  In Practice, The Vast Majority of Xformers are Used in AC Circuits  Recall The Symmetrical Ideal-Xformer Eqns  Illustration: InPut IMPEDANCE = V 1 /I 1  These are LINEAR in i & v, so PHASOR Analysis Applies  Notice That This is an I/O Model; So the Eqns

ENGR-43_Lec-10-2_Transformers.ppt 14 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Phasor Analysis: Input Z  An I/O Xformer in Phasor Domain  Solve for V 1  Now Apply Ohm’s Law to the Load, Z L  Now The Input Impedance Z 1  Now Sub for V 2 and I 2 From I/O Xformer Eqns For 10X stepDOWN (N 1 :N 2 = 10:1) Z 1 is 100X Z L

ENGR-43_Lec-10-2_Transformers.ppt 15 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Phasor Analysis: Input Z cont.1  An I/O Xformer Phasor Domain Input Impedance  Thus Z L is Said to be REFLECTED to the Input Side (by [N 1 /N 2 ] 2 )  For Future Reference  For a LOSSLESS Primary/Secondary Xformer the INput impedance is a Fcn ONLY of the LOAD Impedance, Z L  Ideal Xformer Phasor Eqns

ENGR-43_Lec-10-2_Transformers.ppt 16 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Numerical Example  Given the Ckt Below Find all I’s and V’s  Using  Note: n = N 2 /N 1 = 1/4  Game Plan reflect impedance into the primary side and make the transformer “transparent to Source”  Find I 1 by Ohm stepDOWN Xformer

ENGR-43_Lec-10-2_Transformers.ppt 17 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Numerical Example cont.1  The Intermediate Ckt  About the Same Hassle-Factor; use ZI  Now Find V 1 by Ohm or V-Divider  Next Determine SIGNS  Which is Easier? Z2Z2 stepDOWN Xformer

ENGR-43_Lec-10-2_Transformers.ppt 18 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Numerical Example cont.2  The Original Ckt  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 IN OUT stepDOWN Xformer

ENGR-43_Lec-10-2_Transformers.ppt 19 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Numerical Example cont.3  The Original Ckt  The Output Voltage  Using the Signs as Determined by Dots and Polarities  Then Output Current stepDOWN Xformer  Note: On Calculator aTan(–4.72/–20.33) = 13.07° Recall RANGE of aTan = –90° to + 90°

ENGR-43_Lec-10-2_Transformers.ppt 20 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Illustration  Given the Ckt Below Find I 1 & V o  Note: n = N 2 /N 1 = 2/1  Game Plan reflect impedance into the primary side and make the transformer “transparent to Src”  Again using stepUP Xformer

ENGR-43_Lec-10-2_Transformers.ppt 21 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Illustration cont.1  Given the Ckt Below Find I 1 & V o  Thus I 1 by Ohm  Next Find V o by I 2 and Ohm’s Law Define I 2 Direction per I/O Model (V 2 is ok) Z2Z2 stepUP Xformer

ENGR-43_Lec-10-2_Transformers.ppt 22 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Xformer Thevenin Equivalent  Given I/O XFormer Ckt  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

ENGR-43_Lec-10-2_Transformers.ppt 23 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Xformer Thevenin Equiv. cont.1  Now find Z TH at Terminals 2-2’  “Back Reflect” Impedance into SECONDARY  Thus the Thevenin Equivalent at 2-2’  The Xformer has been “made Transparent” to the Secondary Side Next: Find Thevenin Equiv at Terminals 1-1’

ENGR-43_Lec-10-2_Transformers.ppt 24 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Thevenin Equiv. from Primary  Given I/O XFormer Ckt  Find the The Thevenin Equivalent at 1-1’  Then The Open Ckt Voltage Depends on V S2  As in Open Ckt  Thevenin impedance will be the Secondary impedance reflected into the PRIMARY Ckt

ENGR-43_Lec-10-2_Transformers.ppt 25 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Primary v. Secondary Thevenin  Thevenin From Primary  Equivalent circuit reflecting into primary  Thevenin From Secondary  Equivalent circuit reflecting into secondary  The Base Ckt

ENGR-43_Lec-10-2_Transformers.ppt 26 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Exmpl: Draw Thevenin Equiv’s Equivalent circuit reflecting into SECONDARY Equivalent circuit reflecting into PRIMARY

ENGR-43_Lec-10-2_Transformers.ppt 27 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Example  Given the Ckt Below Find I 1  Note The Dot Locations  Note: n = N 2 /N 1 = 2/1  Game Plan  Find Thevenin Looking from PRIMARY Side Draw the Ckt 1 1’

ENGR-43_Lec-10-2_Transformers.ppt 28 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis 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

ENGR-43_Lec-10-2_Transformers.ppt 29 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis 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! 

ENGR-43_Lec-10-2_Transformers.ppt 30 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Exmple  Power Transmission  At the Generating Facility (e.g. Diablo Canyon) Electricity is Generated at kVac  But Xformers are used to Set-UP the Voltage- Level to kVac  Why? → Line SIZE (and others)

ENGR-43_Lec-10-2_Transformers.ppt 31 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Exmpl – Power Xmission cont.1  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: a)V = 15 kVac b)V = 500 kVac

ENGR-43_Lec-10-2_Transformers.ppt 32 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Exmpl – Power Xmission cont.2  By Solid-State Physics (c.f. ENGR-45) Where for the Wire –ρ  Resistivity (Ω-m) –l  Length (m) –A  X-Section Area (m 2 )  In This Case  Then the Power Loss  By Power Rln for Resistive Ckts  Solve for d

ENGR-43_Lec-10-2_Transformers.ppt 33 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Exmpl – Power Xmission cont.3  Finally Solve for Transmission Cable Diameter  d 15 = 5.03” Pretty BIG & HEAVY  d 500 = 0.15” MUCH Better

ENGR-43_Lec-10-2_Transformers.ppt 34 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Summary: Ideal Transformer  Consider Now two Coils Wrapped Around a Closed Magnetic (usually iron) Core.  The Iron Core Strongly confines the Magnetic Flux, , to the Interior of the Closed Ring All Turns, N 1 & N 2, of Both Coils are Linked by the Core Flux A  Area

ENGR-43_Lec-10-2_Transformers.ppt 35 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Ideal X-Former Circuit Symbol  The Ideal Xformer Eqns Faraday’s Law  The Circuit Symbol  As Two Coupled Inductors, Xformers Use the DOT Convention to Track Polarities  The Main practical Application for This Device: TRANSFORM one AC Voltage-Level to Another Iron Core Ampere’s Law

ENGR-43_Lec-10-2_Transformers.ppt 36 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Transformer Application  When a Voltage is Transfrormed, Give the INput & OUTput sides Special Names INput, v 1, Side  PRIMARY Circuit OUTput, v 2, Side  SECONDARY Circuit  The Voltage Xformer  Then the Circuit Symbol Usage as Applied to a Real Circuit  Pictorial Representation L o a d

ENGR-43_Lec-10-2_Transformers.ppt 37 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Sign (Dot) Conventions  Have TWO Choices for Polarity Definitions Symmetrical  The Form of the Governing Equations INput/OUTput (I/O)

ENGR-43_Lec-10-2_Transformers.ppt 38 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis Phasor Analysis  In Practice, The Vast Majority of Xformers are Used in AC Circuits  Ideal-Xformer Eqns Are LINEAR in i&v so PHASOR Analysis Applies  Illustration: InPut IMPEDANCE  Notice That This is an I/O Model; So the Eqns Yield

ENGR-43_Lec-10-2_Transformers.ppt 39 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis WhiteBoard Work  Let’s Work This Nice Problem

ENGR-43_Lec-10-2_Transformers.ppt 40 Bruce Mayer, PE Engineering-43: Engineering Circuit Analysis 208Vac, 80 kVA Input 208:115 Vac StepDown Xformer 208:24 Vac StepDown Xformer