שנאים — 33.101 · המרת אנרגיה

E = 4.44·f·N·Φ_max | k = U₁U₂ = N₁N₂ = I₂I₁ | Φ_max = U₁4.44·f·N₁ | U₁·I₁ = U₂·I₂ | U₁N₁ = U₂N₂

1000

100

4000 V

50 Hz

10.00

400 V

4.00 V

18.018 mWb

100.0 %

20.0 ms

25.0 A

250.0 A

100.0 kVA

k = U₁U₂ = 4000400 = 10

N₂ = N₁k = 100010 = 100

Φ_max = U₁4.44·f·N₁ = 40004.44·50·1000 = 0.018018 Wb = 18.018 mWb

I_1n = S_nU₁ = 1000004000 = 25 A I_2n = S_nU₂ = 100000400 = 250 A I_2nI_1n = 25025 = 10 = k

U₁·I_1n = 4000·25 = 100000 VA U₂·I_2n = 400·250 = 100000 VA U₁N₁ = 40001000 = 4 V , U₂N₂ = 400100 = 4 V

Φ_max(I₂ = 0) = 40004.44·50·1000 = 18.018 mWb Φ_max(I₂ = 250 A) = 40004.44·50·1000 = 18.018 mWb

4000 V

500 W

4.0 %

1.000 A

0.1250

82.82 °

0.1250 A

0.9922 A

32000 Ω

4031.6 Ω

2.88 W

0.58 %

18.018 mWb

400 V

I_1n = S_nU₁ = 1000004000 = 25 A I₀ = i₀·I_1n = 0.04·25 = 1.0 A

cos φ₀ = P₀U₁·I₀ = 5004000·1.0 = 0.125 φ₀ = 82.82°

I_a = I₀·cos φ₀ = 1.0·0.125 = 0.125 A sin φ₀ = √1 − 0.125² = 0.9922 I_μ = I₀·sin φ₀ = 1.0·0.9922 = 0.9922 A √0.125² + 0.9922² = 1.000 A

R_fe = U₁I_a = 40000.125 = 32000 Ω X_m = U₁I_μ = 40000.9922 = 4031.6 Ω

R_k = P_kI_1n² = 180025² = 2.88 Ω P_cu0 = I₀²·R_k = 1.0²·2.88 = 2.88 W P_cu0P_k = i₀² = 0.04² = 1625 P_cu0P₀ = 2.88500 = 0.0058

100 kVA

4.0 %

1800 W

4000 V

25.0 A

250 A

0 V

0.0 A

0 W

160 V

6.40 Ω

2.88 Ω

5.7154 Ω

1.80 %

3.5721 %

63.26 °

625 A

25.0 ×

0.160 %

I_1n = S_nU_1n = 1000004000 = 25 A

U_k = u_k%·U_1n100 = 4·4000100 = 160 V

Z_k = U_kI_1n = 16025 = 6.4 Ω

R_k = P_kI_1n² = 180025² = 1800625 = 2.88 Ω P_fe(U_k) = P₀·u_k²100² = 500·0.0016 = 0.8 W

X_k = √Z_k² − R_k² = √6.4² − 2.88² = √32.6656 = 5.7154 Ω

u_r% = P_kS_n·100 = 1800100000·100 = 1.8 % u_x% = √u_k² − u_r² = √4² − 1.8² = √12.76 = 3.5721 %

cos φ_k = R_kZ_k = 2.886.4 = 0.45 cos φ_k = u_ru_k = 1.84 = 0.45 : φ_k = 63.26°

ΔU% = β · ( u_r·cos φ + u_x·sin φ )
β = SS_n = I₂I_2n   |   u_r% = P_kS_n · 100   |   u_x% = √u_k² − u_r²
U₂ = U₂₀ · ( 1 − ΔU%100 )   |   ΔU_[V] = U₂₀ · ΔU%100
ΔU% = 0 ⇔ tan φ₀ = u_ru_x   |   cos φ₀ = u_xu_k = 3.57214 = 0.8930

0.70

0.80

‎—‎

0.800

0.6000

2.5083 %

10.03 V

389.97 V

3.5833 %

1.8 %

3.5721 %

0.8930

26.74 °

u_r% = P_kS_n · 100 = 1800100 000 · 100 = 1.8 % u_x% = √4² − 1.8² = √16 − 3.24 = √12.76 = 3.5721 %

sin φ = √1 − 0.8² = √0.36 = +0.6

ΔU% = 0.7 · ( 1.8 · 0.8 + 3.5721 · 0.6 ) = 0.7 · ( 1.44 + 2.1433 ) = 0.7 · 3.5833 = 2.5083 %

ΔU_[V] = 400 · 2.5083100 = 10.03 V U₂ = 400 · ( 1 − 2.5083100 ) = 400 − 10.03 = 389.97 V

sin φ = − √1 − 0.9² = − √0.19 = −0.4359 ΔU% = 1 · ( 1.8 · 0.9 + 3.5721 · (−0.4359) ) = 1.62 − 1.5570 = 0.063 % U₂ = 400 · ( 1 − 0.063100 ) = 400 − 0.25 = 399.75 V

η = β · S_n · cos φβ · S_n · cos φ + P₀ + β² · P_k   |   P_fe = P₀ = const · P_cu = β² · P_k
dηdβ = 0 ⟺ β² · P_k = P₀   |   β_opt = √P₀P_k
η_max = β_opt · S_n · cos φβ_opt · S_n · cos φ + 2 · P₀   |   P_cuP_fe = β² · P_kP₀ = 1

P₀ 500 W

P_k 1800 W

cos φ 0.80

β 0.70

56.00 kW

500 W

882 W

1382 W

57.38 kW

97.59 %

0.5270

97.68 %

1000 W

P₂ = β · S_n · cos φ = 0.7 · 100 · 0.8 = 56 kW = 56 000 W

P_fe = P₀ = 500 W P_cu = β² · P_k = 0.7² · 1800 = 0.49 · 1800 = 882 W ΣP = 500 + 882 = 1382 W

η = 56 00056 000 + 1382 = 56 00057 382 = 0.9759 = 97.59 %

β_opt = √P₀P_k = √5001800 = √0.2778 = 0.5270 P_cu(β_opt) = 0.5270² · 1800 = 0.2778 · 1800 = 500 W = P_fe ΣP = 500 + 500 = 1000 W

P₂ = 0.5270 · 100 000 · 0.8 = 42 163.7 W η_max = 42 163.742 163.7 + 2 · 500 = 42 163.743 163.7 = 0.9768 = 97.68 %

P₂ = 1 · 100 · 0.9 = 90 kW = 90 000 W ΣP = 500 + 1² · 1800 = 2300 W η = 90 00092 300 = 0.9751 = 97.51 % < 97.68 %

20000 V

86.6

400 kVA

Dy

20000 V

230.9 V

400.0 V

50.00

86.60

11.547 A

6.667 A

577 A

U_1ph = U_1L = 20000 V

U_2ph = U_2L√3 = 400√3 = 230.94 V

k = U_1phU_2ph = 20000230.94 = 86.6

a = U_1LU_2L = 20000400 = 50 k = a · √3 = 50 · 1.732 = 86.6

I_1L = S_n√3·U_1L = 4000001.732 · 20000 = 11.547 A I_1ph = I_1L√3 = 11.5471.732 = 6.667 A

I_2L = S_n√3·U_2L = 4000001.732 · 400 = 577.35 A I_2ph = I_2L = 577.35 A

100 kVA

4.0 %

160 kVA

6.0 %

200 kVA

96.77 kVA

103.23 kVA

96.8 %

64.5 %

260 kVA

206.67 kVA

53.33 kVA

a₁ = S_n1u_k1 = 1004 = 25.000 a₂ = S_n2u_k2 = 1606 = 26.667 a₁ + a₂ = 25.000 + 26.667 = 51.667

S₁ = S·a₁a₁ + a₂ = 200·25.00051.667 = 96.77 kVA S₂ = S·a₂a₁ + a₂ = 200·26.66751.667 = 103.23 kVA 96.77 + 103.23 = 200.00 kVA

β₁ = S₁S_n1 = 96.77100 = 0.968 = 96.8 % β₂ = S₂S_n2 = 103.23160 = 0.645 = 64.5 % β₁β₂ = u_k2u_k1 = 64 = 1.5

β₁ = 1 : S_max = S_n1·a₁ + a₂a₁ = 100·51.66725.000 = 206.67 kVA S_max = u_k,min·(a₁ + a₂) = 4·51.667 = 206.67 kVA S_n1 + S_n2 − S_max = 260 − 206.67 = 53.33 kVA 53.33260 = 0.205 = 20.5 %

a₁ = 1004 = 25 , a₂ = 1604 = 40 , a₁ + a₂ = 65 S_max = 4·65 = 260 kVA = S_n1 + S_n2

k = U₁U₂ = N₁N₂ | Φ_m = U₁4.44·f·N₁ | Z_k = U_kI_1n | Δu% = β(u_rcosφ + u_xsinφ) | η = βS_ncosφβS_ncosφ + P₀ + β²P_k

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