Title | Resolucao capitulo 3 - Eletrônica de Potência - Daniel W. Hart Cap 3 |
---|---|
Author | Raphael Carrijo |
Course | Engenharia Elétrica |
Institution | Instituto Federal de Educação, Ciência e Tecnologia de Goiás |
Pages | 33 |
File Size | 1.6 MB |
File Type | |
Total Downloads | 37 |
Total Views | 135 |
Resolucão do livro de Eletronica de potência do Daniel W. Hart capitulo 3...
Solucionário Cap 3 - Power Electronics - D. Hart Engenharia Elétrica Universidade Federal de Ouro Preto (UFOP) 32 pag.
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 ([email protected])
CHAPTER 3 SOLUTIONS 2/20/10
3-1) a ) I0 b) I rms
V0 Vm 170 / 3.60 A. R R 15 V V 170 rms m 5.66 A. 2 R 2(15) R
c ) P I 2 R 5.662 (15) 480 W . 170 d) S Vrms I rms (5.66) 679 VA. 2 P 480 W e) pf 0.707 70.7% S 679 VA
3-2) a ) I 0 12 A.; I 0 Vo
Vm
V0 V0 I 0 R (12)(20) 240 V . R
; Vm V o 240 754 V .
754 533 V . 2 N1 240 0.45 N2 533
Vrms
N 12 b) I o I o 2 26.7 A. N1 0.45
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-3) a ) pf
V P P V V ; I rms s , rms ; VR , rms m ; V s, rms m 2 S Vs , rms I rms R 2
Vm /R 2 1 2 pf Vs , rms I rms Vm Vm 2 2 2 / R 2 b) Displacement pf cos(1 ) cos(0) 1 1 V 1 Vm I1 1 0; pf cos(1 1)DF ; DF R R 2 2 VR2, rms / R
3-4) Using Eq. 3-15, a) i( t )
Vm V sin( t ) m (sin ) et / Z Z
2 2 2 2 Z R ( L) 12 (377(0.012)) 12.8
L 1 377(0.012) tan 0.361 rad 12 R L 377(0.012) 0.377 R 12 i( t) 13.2sin( t 0.361) 4.67 e t /0.377 : 3.50 rad 201 b) I avg 4.36 A. ( numerical integration)
tan1
c) I rms 6.70 A. ( numerical integration) P I 2rmsR (6.70) 2 (12) 538 W. P 538 0.67 d) pf S (120)(6.70)
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-5) Using Eq. 3-15, a) i( t )
Vm V sin( t ) m (sin ) et / Z Z
Z R2 ( L)2 102 (377(0.015))2 11.5 L 1 377(0.015) tan 0.515 rad R 10 L 377(0.015) 0.565 R 10 i( t) 14.8sin( t 0.515) 7.27 et /0.565 : 3.657 rad 209.5 b) I avg 5.05 A. ( numerical integration)
tan1
c) I rms 7.65 A. ( numerical integration) P I 2rmsR (7.65) 2 (10) 584 W. 584 P d) pf 0.637 63.7% S (120)(7.65)
3-6) Using Eq. 3-15,
a) i( t )
Vm V sin( t ) m (sin ) e t / Z Z
Z R2 ( L)2 152 (377(0.08))2 33.7 tan1
L
1 377(0.08) tan 1.11 rad 15 R L 377(0.08) 2.01 15 R i( t) 10.1sin( t 1.11) 9.02 e t /2.01: 4.35 rad 250 b) I avg 4. 87 A. (numerical integration) 2 2 c) I rms 6.84 A. ( numerical integration) P I rmsR (6.84) (15) 701 W. P 701 d) pf 0.427 42.7% S (240)(6.84)
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-7) Using an ideal diode model, R = 48 Ω for an average current of 2 A.
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
8.0A
Current Iavg = 2 A for R = 48 ohms 4.0A (16.700m,2.0030)
Average Current
0A 0s
5ms I(R1)
10ms AVG(I(L1)) Time
15ms
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
20ms
3-8) Using Eqs. 3-22 and 3-23, Vm V sin( t ) dc Ae t / Z R V V A m sin( ) dc e / R Z
a) i( t)
Z R 2 ( L )2 102 (377(.075) 2 30.0 L 377(.075) tan 1 1.23 rad 10 R L 377(0.075) 2.83 R 10 V 100 sin 1 dc 0.299 rad 17. 1 Vm 240 2
tan 1
i( t) 11.3sin( t 1.23) 10 21.2 e t /2.83; 3.94 rad 226 I avg 3.13 A. (numerical integration), Pdc V dcI avg (100)(3.13) 313 W . 2 2 b) I rms 4.81 A. ( numerical integration) PR I rms R (4.81) (10) 231 W. P 313 231 c) pf 0.472 47.2% S (240)(4.81)
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-9) Using Eqs. 3-22 and 3-23, Vm V sin( t ) dc Ae t / Z R V V A m sin( ) dc e / Z R a) i( t)
Z
R 2 ( L) 2 12 2 (377(0.12) 2 46.8
L 377(0.12) tan 1 1.31 rad 12 R L 377(0.12) 3.77 R 12 V 48 sin 1 dc 0.287 rad 16.4 Vm 120 2
tan 1
i( t) 3.63sin( t 1.31) 4.0 7.66 e t/3.77 ; 4.06 rad 233 I avg 1.124 A. ( numerical integration), Pdc V dcI avg (48)(1.124) 54.0 W. 2 b) Irms 1.70 A. ( numerical integration) PR Irms R (1.70) 2(12) 34.5 W.
c) pf
P 54.0 34.5 0.435 43.5% S (120)(1.70)
3-10) Using Eq. 3-33, Vm V (cos cos t) dc ( t) L L V 48 sin 1 dc sin 1 0.287 rad . 120 2 Vm i( t)
i( t) 4.68 4.50cos( t) 1.23 t A.; 4.483 rad 257 1 Io i( t) d ( t) 2.00 A.; Pdc I oVdc 2.00(48) 96 W. 2
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-11)
300W
200W L = 0.25 H
100W
0W 0s
5ms AVG(W(Vdc))
10ms
15ms
Time
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
20ms
3-12) L ≈ 0.14 H for 50 W (51 W).
100W
(16.670m,51.156) 50W
L = 0.14 H
0W 0s
5ms AVG(W(Vdc))
10ms
15ms
Time
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
20ms
3-13) Using Eq. 3-34, a) V0
Vm
120 2
54.0 V .; I 0
V0 54 4.50 A . R 12
b) n Vn Zn In 0 54.02 12.00 4.50 1 84.85
25.6
3.31
2 36.01
46.8
0.77
4
91.3
0.08
7.20
The terms beyond n = 1 are insignificant.
3-14)
Run a transient response long enough to achieve steady-state results (e.g., 1000ms). The peak-topeak load current is approximately 1.48 A, somewhat larger than the 1.35 A obtained using only the first harmonic. (The inductance should be slightly larger, about 0.7 H, to compensate for the approximation of the calculation.)
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-15) a) Vm 50 3.98 A. R 4 V Vm / 2 I1 1 2 Z1 R ( L)2 I0
25 R ( L) 2
R2 ( L)2 9 ( L) 2 L
2
0.05 I0 0.199 A.
25 125 L 0.199
125 0.33 H 2 60
b) A PSpice simulation using an ideal diode model gives 0.443 A p-p in the steady state. This compares with 2(I1)=2(0.199)=0.398 A p-p.
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-16) a) V0
Vm
170
54.1 V
V0 Vdc 54.1 24 3.01 A. R 10 io 1 A. 2 I1 I1 0.5 A.
I0
Vm 2 V Z1 1 I1
V1
170 85 V 2 85 170 0.5
2 2 R ( L) L
170 450 mH . 377 b) Pdc Iavg Vdc (3.01)(24) 72.2 W. L
2 c) PR Irms R; Irms
I
2 n, rms
(3.01)2 (0.5 / 2) 2 3.12 A.
PR (3.12) 2 (10) 97.4 W .
3-17) a) τ = RC = 10310-3=1 s; τ/T = 60. With τ >> T, the exponential decay is very small and the output voltage has little variation. b) Exact equations:
tan1 ( RC ) tan 1 (377) 1.5573 rad 90.15 Vm sin 200sin(90.15 ) 199.9993 sin sin e (2 )/ RC 0 1.391 rad 79.72 Vo Vm (1 sin ) 3.21 V .
c) Approximation of Eq. 3-51: Vo
Vm 200 3.33 V . fRC (60)(10 3 )(10 3)
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-18) a) R = 100 Ω: τ = RC (100)10-3 = 0.1 s; τ/T = 6.
tan 1 ( RC ) tan 1 (37.7) 1.5973 rad 91.52 Vm sin 200 sin(91.52 ) 199.93 sin sin e (2 )/ RC ) 0 1.0338 rad 59.23 Vo Vm (1 sin ) 28.16 V . (exact ) Vo
Vm 200 33.3 V . (approximation ) 3 fRC (60)(100)(10 )
b) R = 10 Ω: τ = RC (10)10-3 = 0.01 s; τ/T = .6.
tan 1 ( RC ) tan 1 (3.77) 1.830 rad 104.9 Vm sin 200sin(104.9) 193.3 sin sin e (2 )/ RC ) 0 0.2883 rad 16.5 Vo Vm (1 sin ) 143.2 V . (exact ) Vo
Vm 200 333V . (approximation ) fRC (60)(10)(10 3 )
In (a) with τ/T=6, the approximation is much more reasonable than (b) where τ/T=0.6.
3-19) a) With C = 4000 µF, RC = 4 s., and the approximation of Eq. 3-51 should be reasonable. Vo
Vm 120 2 0.707 V . fRC (60)(4)
b) With C = 20 µF, RC = 0.02, which is on the order of one source period. Therefore, the approximation will not be reasonable and exact equations must be used.
tan 1 ( RC ) tan 1 ((377)(1000)(20(10) 6 ) 1.703 rad 97.6 ) 0.5324 rad 30.5 ( numerically from Eq. 3 43) Vo Vm Vm sin 83.6 V .
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-20) a) With C = 4000 µF, RC = 2 s., and the approximation of Eq. 3-51 should be reasonable. Vo
120 2 Vm 1.41 V . fRC (60)(2.0)
b) With C = 20 µF, RC = 0.01, which is on the order of one source period. Therefore, the approximation will not be reasonable and exact equations must be used.
tan 1 ( RC ) tan 1 ((377)(500)(20(10) 6 ) 1.83 rad 104.9) 0.2883 rad 16.5 ( numerically from Eq. 3 43) Vo Vm Vm sin 121 V . 3-21) From Eq. 3-51 C
Vm 120 2 1,886 F fR Vo 60(750)(2)
sin 1 1
Vo 2 1 sin 1 1.417 rad 81.2 Vm 120 2
sin I D, peak Vm C cos 18.7 A . R V I D, avg m 0.226 A. R
3-22) Assuming Vo is constant and equal to Vm, P
Vo2 Vm2 V 2 (120 2) 2 R m 576 R R P 50
From Eq. 3-51 C
Vm 120 2 3, 270 F fR Vo 60(576)(1.5)
sin 1 1
Vo 1.5 1 sin 1 1.438 rad 82.4 Vm 120 2
sin I D, peak Vm C cos 28.1 A. R V I D, avg m 0.295 A. R
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-23) Using the definition of power factor and Vrms from Eq. 3-53, pf
Vm 2
2 2 P Vrms Vrms V /R /R rms S (Vs ,rms )( Is ,rms ) (Vs, rms )( Vrms / R) Vs, rms
sin 2 sin 2 1 1 sin 2 2 1 2 2 2 4 Vm / 2 2
1
3-24) Vm 120 2 (1 cos ) (1 cos 45 ) 46.1 V. 2 2 V2 V sin 2 b) P rms ; Vrms m 1 2 2 R a) Vo
120 2 0.785 sin(2(0.785)) 80.9 V . 1 2 2 80.92 P 65.5 W . 100 P 65.5 80.9 c) S Vs, rmsI rms (120) 0.674 67.4% 97.1 VA; pf S 97.1 100
3-25) Vm (1 cos ) 2 2Vo 2 (75) 1 cos 1 1 65.5 or 1.143 rad cos 1 240 2 Vm V2 b) P o ,rms R V 1.143 sin(2(1.143)) sin 2 240 2 Vo ,rms m 1 147.6 V . 1 2 2 2 2 147.62 P 726 W . 30 P 726 147.6 c) S V s, rmsI rms (240) 0.615 61.5% 1181 VA; pf S 1181 30 a ) vo I o R (2.5)(30) 75 V
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-26) a) i( t) 5.42sin( t 0.646) 1.33 et /0.754 A. 25 0.524 rad, 3.79 rad 217 ( numerically)
b) I o
1 i( t) d( t) 1.80 A. 2
c) I rms
1 2
i ( t) d ( t) 2.80 2
2 A.; Po PR I rms R (2.80) 225 193 W.
3-27) a) i( t ) 3.46sin( t 0.615) 6.38 e t/0.707 A. 60 1.047 rad, 3.748 rad 215 ( numerically)
b) I o
1 i( t) d( t) 0.893 A. 2
c) I rms
1 2 i ( t) d ( t) 1.50 A.; Po PR I 2rms R (1.50) 240 90.3 W. 2
3-28) α ≈ 46°. Do a parametric sweep for alpha. Use the default (Dbreak) diode, and use Ron = 0.01 for the switch. Alpha of 46 degrees results in approximately 2 A in the load.
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-29) α ≈ 60.5°. Do a parametric sweep for alpha. Use the default (Dbreak) diode, and use Ron = 0.01 for the switch. Alpha of 60.5 degrees results in approximately 1.8 A in the load.
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-30) From Eq. 3-61, a) i( t) 4.29 sin( t 1.263) 4.0 7.43 e t/3.142 A., 0.873 t 3.95 rad
Io
1 i (t )d (t ) 1.04 A., Pdc I oV dc (1.04)(48) 50.1 W. 2
b) I rms c) pf
1 2
i ( t) d( t) 1.67 2
2 A.; PR I rms R (1.67) 212 33.5 W.
P 50.1 33.5 0.417 41.7% S (120)(1.67)
3-31) From Eq. 3-61, a) i( t) 2.95sin( t 0.515) 0.96 3.44 e t/0.565 A., 1.047 t 3.32 rad
Io
1 i(t) d (t) 0.454 A., Pdc I oV dc (0.454)(96) 43.6 W. 2
b) I rms c) pf
1 2
i ( t) d( t) 0.830 2
2 A.; PR Irms R (0.830) 2100 69.0 W.
P 43.6 69.0 0.565 56.5% S (240)(0.830)
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-32) α ≈ 75°. Alpha = 75 degrees gives 35 W in the dc voltage source. An Ron = 0.01 for the switch and n = 0.001 for the diode (ideal model).
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-33) From Eq. 3-61, a) i( t ) 5.99sin( t 1.50) 24.0 29.3 e t /14.1 A., 0.873 t 4.24 rad
Io
1 i (t )d (t ) 1.91 A., Pdc I oVdc (1.91)(48) 91.6 W. 2 1 2
b) I rms
i ( t) d( t) 2.93 2
2 A.; PR I rms R (2.93) 22 17.1 W.
3-34) α ≈ 81° 3-35) di(t ) Vm sin t Vdc dt di (t ) 1 [Vm sin t Vdc ] or dt L di (t ) 1 [V m sin t V dc] d (t ) L L
t
i ( t )
1 (Vm sin t V dc ) d ( t ) L
Vm V (cos cos t ) dc ( t) L L i ( t ) 4.34 7.58 cos t 1.82 t A., 1.309 t 4.249
Io
1 i (t )d ( t ) 1.9 1 A. 2
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-36) v0 = vs when S1 on, v0=0 when D2 on
Io
Vo V 1 V sin(t) d( t) m (1 cos ) , Vo R 2 m 2
Io
Vm (1 cos ) 2 R
3-37) I X u cos 1 1 L s ; X s Ls 377(1.5)(10) 3 0.566 V m 5(0.452) u cos 1 1 10.47 120 2 Vm X L X s 120 2 5(.566) 1 53.57 V . 1 2Vm 2 2(120) V (compared to m 54.0 V .)
Vo
PSpice: Use a current source for the constant load current:
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
D1 to D2
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
D2 to D1
3-38) u = 20°. Run the simulation long enough for steady-state results. From the Probe output, the commutation angle from D1 to D2 is about 20 degrees, and from D2 to D1 is about 18 degrees. Note that the time axis is changed to angle in degrees here.
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-39) Run the simulation long enough for steady-state results. From the Probe output, the commutation angle from D1 to D2 is about 16.5 degrees, and from D2 to D1 is about 14.7 degrees. Note that the time axis is changed to angle in degrees here.
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-40) At ωt = π, D2 turns on, D1 is on because of the current in LS (see Fig. 3-17). KVL; v LS Vm sin t Ls
diD1 did 1 LS dt d (t )
diD1 V t m sin( t) d( t) i D1( ) d t Ls Vm [ 1 cos( u)] IL Ls V V cos( u ) cos u 0 m (1 cos u) I L m cos u I L Ls Ls at t u, iD1 0
I X u cos 1 1 L s Vm
Document shared on www.docsity.com Downloaded by: raphael-carrijo-10 (raphaelc2008@hotmail com)
3-41) At ωt = α, t
V 1 Vm sin( t) d( t) 0 m [cos cos t] Ls Ls V iD 2 ( wt ) IL is IL m [cos cos t] Ls V iD 2 ( u) 0 IL m [cos cos( u)] L s IL L s cos cos( u) Vm is ( t)
I X u cos 1 cos L s Vm
3-42) A good solution is to use a controlled half-wave rectifier with an inductor in series with the 48-V source and resistance (Fig. 3-15). The switch will change the delay angle of the SCR to produce the two required power levels. The values of the delay angle depend on the value selected for the inductor. This solution avoids adding resistance, thereby avoiding introducing power losses. 3-43) Several circuit can accomplish this objective, including the half-wave rectifier of Fig. 3-2a and half-wave rectifier with a freewheeling diode of Fig. 3-7, each with resistance added. Another solution is to use the controlled half-wave rectifier of Fig. 3-14a but with no resistance. The analysis of that circuit is like that of Fig. 3-6 but without Vdc. The resulting value of α is 75°, obtain from a PSpice simulation. That solution is good because no resistance is needed, and losses are not introduced. 3-44 and 3-45) The controlled half-wave rectifier o...