Lecture Set 7: Phasors and Transformers
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-phase transformer in order to set the stage for the induction machine
2 Lecture 64
Review of Phasor Analysis
3 Review of Phasor Analysis • Basis of phasor analysis:
d eaeau au du
ejj cos sin
4 Review of Phasor Analysis • Consider the differential equation 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 nm00dt dt • Form of input (and output)
fA2cos(f t f )
5 Review of Phasor Analysis • Phase-Shifted Problem NMdynm dx abp p nmnm nm00dt 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 fAf f
f angle(f ) Af f
8 Review of Phasor Analysis
9 Review of Phasor Analysis • We can show
M m bm j m1 yx N n ajn () n1
yye real( 2 jt )
yA2cos(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
1111122L iLi
2211222L 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 Ll2i2 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