Lecture Set 7: and

S.D. Sudhoff Spring 2021 About this Lecture Set • Reading – Power Magnetic Devices: A Multi-Objective Design Approach, Appendix E, (Available through IEEE Xplore or blackboard) – Electromechanical Motion Devices, 2nd Edition, Section 1.5 • Goal – Review phasors and analyze a single- in order to set the stage for the induction machine

2 Lecture 64

Review of Analysis

3 Review of Phasor Analysis • Basis of phasor analysis:

d eaeau au du

ejj cos sin

4 Review of Phasor Analysis • Consider the dy d22 y dx d x ay a a ... bx b b  ... 01dt 2dt22 01 dt 2 dt • This may also be written as NMdynm dx ab nmnm nm00dt dt • Form of input (and output)

fA2cos(f  t f )

5 Review of Phasor Analysis • Phase-Shifted Problem NMdynm dx abp  p nmnm nm00dt dt • Form of input (and output)

fAtp 2sin(ff  )

6 Review of Phasor Analysis • Composite variables

f fjfp

• Clearly

f  real(f )

f p  imag(f )

7 Review of Phasor Analysis

• We can show

f  2 fe jt

• where

f Aej f  f fAf  f

   f  angle(f ) Af  f

8 Review of Phasor Analysis

9 Review of Phasor Analysis • We can show

M m bm j m1 yx N n  ajn () n1

yye real( 2 jt )

yA2cos(yy t )

10 Review of Phasor Analysis

11 Review of Phasor Analysis

12 Review of Phasor Analysis

13 Review of Phasor Analysis

14 Lecture 65

Phasor Representation of Circuit Elements

15 Review of Phasor Analysis • Consider a resistor

16 Review of Phasor Analysis • Consider a capacitor

17 Review of Phasor Analysis • Consider an inductor

18 Review of Phasor Analysis • Consider a series RLC circuit

19 Review of Phasor Analysis

20 Review of Phasor Analysis • Numerical example

21 Review of Phasor Analysis • Numerical example

22 Lecture 66

Transformers

23 A Single Phase Transformer

24 Construction

25 Analysis

• Voltage Equations

d v  r i  1 1 1 1 dt d v  r i  2 2 2 2 dt

 v1  r1 0  i1  d  1            v2  0 r2  i2  dt  2 

26 Analysis

• Flux linkage equations

1111122L iLi

2211222L iLi

2 2 N1 N1 L11  R   Ll1  Lm1 l1 Rm 2 2 N2 N2 L22  R   L12  Lm2 l2 Rm

N1N2 L12  L21  Rm

27 Analysis

• Flux Terms

28 Analysis

• Flux Terms

29 Analysis

• Flux Terms

30 Analysis

• Flux Terms

31 T-Equivalent Circuit Model

32 T-Equivalent Model

• Definition of • Voltage equations referred variables v1  r1 0  i1  d  1           '  N2 v2  0 r2  i2  dt 2  i2  i2 N1 2  N1  N1 r2    r2 v2  v2  N2  N2  • Flux linkage equations N1 2  2 N2  1  Ll1i1  Lm1(i1  i2 ) 2  Ll2i2  Lm1(i1  i2 )

2  N  L   1  L l2 N l2  2  33 Derivation of T-Equivalent Circuit

34 Derivation of T-Equivalent Circuit

35 Derivation of T-Equivalent Circuit

36 Derivation of T-Equivalent Circuit

37 Derivation of T-Equivalent Circuit

38 Derivation of T-Equivalent Circuit

39 An Example

40 An Example

41 An Example

42 Lecture 67

Transformer Examples

43 Parameter ID Example • DC Test:

– Z1dc = 0.2 

• Short Circuit Test: o – Z1sc= 2.63 at 80.4 

44 Parameter ID Example • Short Circuit Test (continued)

45 Parameter ID Example • Open Circuit Test o – Z1oc = 46.2 at 84.7 

• Turns Ratio Test

– |V1 |= 110 V

– |V2| = 10.7 V

46 Parameter ID Example

47 Parameter ID Example

48