AC Motor Control with Magnetic Contactors and VFDs

1. H1Synchronous speed & slip
   $$n_s = \frac{120 f}{p} \qquad s = \frac{n_s - n}{n_s}$$
2. H2Motor rated current
   $$I_n = \frac{P_{out}}{\sqrt{3}\, V_L\, \eta\, \cos\varphi}$$
3. H3Star–delta reduction
   $$I_{L,Y} = \tfrac{1}{3} I_{L,\Delta} \qquad T_Y = \tfrac{1}{3} T_\Delta \qquad I_{r,OL} = \frac{I_n}{\sqrt{3}} \approx 0.58\, I_n$$
4. H4V/f and affinity laws
   $$\frac{V}{f} = \frac{400}{50} = 8\ \text{V/Hz} \qquad \frac{P_2}{P_1} = \left(\frac{n_2}{n_1}\right)^3$$
5. M1Synchronous speed
   $$n_s = \frac{120\, f}{p}\ \ \text{rpm} \qquad (p = \text{number of poles})$$
6. M2Slip
   $$s = \frac{n_s - n}{n_s} \qquad n = n_s(1 - s)$$
7. M3Rotor frequency
   $$f_r = s\, f$$
8. M4Torque from power and speed
   $$T = \frac{P}{\omega} = \frac{60\,P}{2\pi n} \;\;\Rightarrow\;\; T\,[\text{N}{\cdot}\text{m}] = \frac{9550\, P\,[\text{kW}]}{n\,[\text{rpm}]}$$
9. M5Power split in the rotor
   $$P_{ag} = 3 I_2'^2 \frac{R_2'}{s} \qquad P_{cu2} = s\,P_{ag} \qquad P_{mech} = (1 - s)\,P_{ag} \qquad T = \frac{P_{ag}}{\omega_s}$$
10. M6Rated current
    $$I_n = \frac{P_{out}}{\sqrt{3}\,V_L\,\eta\,\cos\varphi} \qquad P_{in} = \frac{P_{out}}{\eta}$$
11. M7Starting current
    $$I_{st} = \left(\frac{I_a}{I_n}\right) I_n \qquad \frac{I_a}{I_n} \approx 6\text{–}8$$
12. M8Torque versus voltage
    $$T(s) \propto V^2 \qquad \frac{T_2}{T_1} = \left(\frac{V_2}{V_1}\right)^2 \qquad I(s) \propto V$$
13. S1Line and phase voltage
    $$V_L = \sqrt{3}\,V_{ph} \qquad 230 \times \sqrt3 = 398 \approx 400\ \text{V} \qquad V_{peak} = \sqrt2\, V_{rms}$$
14. S2Three-phase power
    $$P = \sqrt{3}\,V_L I_L \cos\varphi \qquad S = \sqrt{3}\,V_L I_L$$
15. S3Reversing rotation
    $$(L1, L2, L3) \to (L3, L2, L1) \;\Rightarrow\; n_s \to -n_s$$
16. S4Voltage unbalance (NEMA working definition)
    $$\%VU = \frac{\max |V_{L} - \bar V_L|}{\bar V_L} \times 100 \qquad \bar V_L = \frac{V_{12} + V_{23} + V_{31}}{3}$$
17. S5Current unbalance estimate
    $$\%I_{unb} \approx \frac{I_{st}}{I_n}\,\%VU \approx (6\text{–}10)\,\%VU$$
18. S660 Hz motor on 50 Hz
    $$\Phi \propto \frac{V}{f} \qquad V_{50} = V_{60}\cdot\frac{50}{60} \qquad n_{50} = \frac{5}{6} n_{60} \qquad P_{50} \approx \frac{5}{6} P_{60}$$
19. C1Star (Y)
    $$V_{ph} = \frac{V_L}{\sqrt{3}} \qquad I_L = I_{ph}$$
20. C2Delta (Δ)
    $$V_{ph} = V_L \qquad I_L = \sqrt{3}\, I_{ph}$$
21. C3Power in either connection
    $$P = 3\,V_{ph}\,I_{ph}\cos\varphi = \sqrt{3}\,V_L\,I_L\cos\varphi$$
22. C4Same windings, same supply: Y vs Δ
    $$\frac{V_{ph,Y}}{V_{ph,\Delta}} = \frac{1}{\sqrt3} \;\Rightarrow\; \frac{I_{L,Y}}{I_{L,\Delta}} = \frac{1}{3} \qquad \frac{T_Y}{T_\Delta} = \frac{1}{3}$$
23. C5Direction of rotation
    $$L1\!\to\!U1,\ L2\!\to\!V1,\ L3\!\to\!W1 \;\Rightarrow\; \text{CW (viewed from shaft end)} \qquad \text{swap 2 lines} \Rightarrow \text{CCW}$$
24. D1Coil operating range
    $$0.85\,U_c \le U_{pick\text{-}up} \le 1.1\,U_c \qquad U_{drop\text{-}out} \approx (0.2\text{–}0.75)\,U_c$$
25. D2Contactor selection
    $$I_e(\text{AC-3},\,400\ \text{V}) \ge I_n \qquad \text{for AC-4 duty derate to about } 0.3\text{–}0.5\,I_e(\text{AC-3})$$
26. D3Overload setting
    $$I_r = I_n \;(\text{DOL, reversing, soft starter}) \qquad I_r = \frac{I_n}{\sqrt3} \approx 0.58\,I_n \;(\text{star–delta, in the phase-current path})$$
27. D4Trip class check
    $$t_{trip}(7.2\,I_r) \le \text{class [s]} \qquad t_{start} < 0.8 \times t_{trip}(I_{st}/I_r)$$
28. D5Breaking capacity
    $$I_{cu} \ge I_{k,max}\ \text{at the point of installation} \qquad I_{k} \approx \frac{S_{tr}}{\sqrt3\,V_L\,u_k} \;(\text{at the transformer terminals})$$
29. D6No nuisance trip at start
    $$I_{m} > (1.5\text{–}2)\,I_{st} \approx (10\text{–}13)\,I_n \qquad (\text{allowing for the asymmetric first half-cycle inrush})$$
30. D7Control transformer sizing
    $$S_{tr} \ge \sum S_{hold} + S_{inrush,\,max} \qquad \text{check voltage dip at pick-up} \le 5\text{–}10\,\%$$
31. G1Series = AND
    $$K = A \cdot B$$
32. G2Parallel = OR
    $$K = A + B$$
33. G3NC contact = NOT
    $$K = \overline{A}$$
34. G4Self-holding (seal-in)
    $$K1 = \overline{S0}\cdot\overline{F2}\cdot(S1 + K1)$$
35. L1DOL starting current
    $$I_{st} = k_{st}\,I_n \qquad k_{st} = I_a/I_n \approx 6\text{–}8$$
36. L2Voltage dip at the transformer
    $$\Delta U\% \approx \frac{S_{st}}{S_{sc} + S_{st}}\times 100 \qquad S_{st} = \sqrt3\,V_L\,I_{st} \qquad S_{sc} = \frac{S_{tr}}{u_k}$$
37. L3Cable voltage drop
    $$\Delta V = \sqrt3\, I\, L\,(R\cos\varphi + X\sin\varphi) \qquad \cos\varphi_{start} \approx 0.3$$
38. L4Seal-in logic
    $$K1_{\text{new}} = \overline{F2}\cdot\overline{S0}\cdot(S1 + K1_{\text{old}})$$
39. R1Phase swap
    $$\text{K1: } (L1, L2, L3)\to(U1, V1, W1) \qquad \text{K2: } (L3, L2, L1)\to(U1, V1, W1)$$
40. R2Fault if both close
    $$I_{k2} = \frac{V_{L1L3}}{2 Z_{src}} = \frac{\sqrt3}{2} I_{k3} \approx 0.87\,I_{k3}$$
41. R3Interlocked logic
    $$K1 = \overline{S0}\,\overline{F2}\,(S1 + K1)\,\overline{S2}\,\overline{K2} \qquad K2 = \overline{S0}\,\overline{F2}\,(S2 + K2)\,\overline{S1}\,\overline{K1}$$
42. R4Slip during plugging
    $$s = \frac{-n_s - n}{-n_s} = 1 + \frac{n}{n_s} \approx 2 \quad (n \approx n_s)$$
43. R5Rotor heat per operation (load with no torque)
    $$E_{start} = \tfrac12 J\omega_s^2 \qquad E_{plug\,(\omega_s \to 0)} = \tfrac32 J\omega_s^2 \qquad E_{reverse\,(\omega_s \to -\omega_s)} = 4\cdot\tfrac12 J\omega_s^2$$
44. Y1Starting current in star
    $$I_{L,Y} = \frac{I_{L,\Delta}}{3} = \frac{k_{st}\,I_n}{3}$$
45. Y2Starting torque in star
    $$T_Y(n) = \frac{T_\Delta(n)}{3} \qquad T_{st,Y} = \frac{T_a/T_n}{3}\,T_n \approx 0.7\text{–}0.8\,T_n$$
46. Y3Main and delta contactors
    $$I_e(\text{KM1}) = I_e(\text{KM3}) \ge \frac{I_n}{\sqrt3} \approx 0.58\,I_n \quad (\text{AC-3})$$
47. Y4Star contactor
    $$I_e(\text{KM2}) \ge \frac{I_n}{3} \approx 0.33\,I_n \quad (\text{AC-3})$$
48. Y5Overload setting (in the phase-current path)
    $$I_r = \frac{I_n}{\sqrt3} \approx 0.58\,I_n$$
49. Y6Changeover dead time
    $$t_{dead} \approx 40\text{–}100\ \text{ms} \;>\; t_{arc,KM2} + t_{close,KM3}\ \text{margin}$$
50. Y7Acceleration time
    $$J\frac{d\omega}{dt} = T_m - T_L \;\Rightarrow\; t_{acc} \approx \frac{J\,\Delta\omega}{\overline{T_m - T_L}}$$
51. Y8Condition before changing to delta
    $$T_Y(n) > T_L(n)\ \ \forall\, n < n_{ch} \qquad n_{ch} \approx (0.85\text{–}0.9)\,n_n$$
52. Y9Cable sizing for six leads
    $$I_z \ge \frac{I_n}{\sqrt3} \ \text{per conductor} \qquad \Delta V \text{ from the phase current and the actual length}$$
53. E1Winding voltage in Y and Δ
    $$V_{w,Y} = \frac{V_L}{\sqrt3} \qquad V_{w,\Delta} = V_L \qquad I_{L,\Delta} = \sqrt3\,I_{w,\Delta}$$
54. E2Line-current ratio at the same speed
    $$\frac{I_{L,Y}}{I_{L,\Delta}} = \frac{V_L/(\sqrt3\,Z_w)}{\sqrt3\,V_L/Z_w} = \frac13 \qquad Z_w = \text{winding impedance at that slip}$$
55. E3Torque ratio (torque ∝ V²)
    $$\frac{T_Y(s)}{T_\Delta(s)} = \left(\frac{V_{w,Y}}{V_{w,\Delta}}\right)^{2} = \frac13 \qquad \frac{T_{k,Y}}{T_k} = \frac13,\ \ s_k \text{ unchanged}$$
56. E4Input power and starting kVA
    $$P_{in} = \sqrt3\,V_L I_L\cos\varphi \qquad S_{st} = \sqrt3\,V_L\,I_{st} \qquad S_{st,Y} = \tfrac13\,S_{st,\Delta}$$
57. E5Kloss approximation
    $$\frac{T}{T_k} = \frac{2}{s/s_k + s_k/s} \qquad T_{k,Y} = \frac{T_k}{3}$$
58. E6Slip in star near rated speed
    $$T \approx \frac{3V_w^2}{\omega_s R_2'}\,s \;\Rightarrow\; s_Y \approx 3\,s_\Delta \ \text{(same load torque)}$$
59. E7Star equilibrium and feasibility
    $$T_Y(n_Y) = T_L(n_Y),\ \ \frac{d(T_Y - T_L)}{dn}\Big|_{n_Y} < 0 \qquad T_Y(0) > T_L(0),\ \ n_Y \ge 0.9\,n_N$$
60. E8Run-up time in star
    $$J\frac{d\omega}{dt} = T_Y(\omega) - T_L(\omega) \;\Rightarrow\; t(\omega_1) = J\int_0^{\omega_1}\frac{d\omega}{T_Y(\omega) - T_L(\omega)}$$
61. E9Mechanical time constant
    $$\tau_m = \frac{J\,\omega_s}{T_n} \qquad \omega_s = \frac{2\pi n_s}{60} \qquad T_n = \frac{9550\,P_n}{n_N}$$
62. E10Timer setting rule
    $$t_{set} = \max\!\big(1.2\,t_{98}(V_n),\ t_{95}(0.9V_n)\big) = c\,\tau_m$$
63. E11Average-torque estimate
    $$t \approx \frac{J\,\Delta\omega}{\overline{T_Y - T_L}}$$
64. E12No-load run-up with Kloss (closed form)
    $$t = \frac{J\omega_s}{T_k}\left[\frac{1-s_e^2}{4s_k} + \frac{s_k}{2}\ln\frac1{s_e}\right] \qquad t_Y = 3\,t_\Delta$$
65. E13Residual voltage after opening
    $$\underline e(t) \approx \underline e_0\,e^{-t/T_0'}\,e^{\,j\int_0^t (p\,\omega_r - \omega_s)\,d\tau} \qquad T_0' = \frac{L_r}{R_r}$$
66. E14Phase drift during the dead time
    $$\delta(t_d) \approx s_{ch}\,\omega_s\,t_d + \frac{p\,T_L}{2J}\,t_d^{2} \quad \text{(electrical rad)}$$
67. E15Transient driving voltage
    $$|\Delta\underline V| = |\underline V_\Delta - \underline e| = \sqrt{V_\Delta^2 + e^2 - 2V_\Delta e\cos\theta} \qquad \theta = \pm30^\circ - \varphi_0 - \delta$$
68. E16Changeover peak estimate
    $$I_{pk} \approx k_a\,\frac{|\Delta\underline V|}{V_\Delta}\,I_{LR} \qquad k_a \approx 1.35\text{–}1.45$$
69. E17Transition resistor
    $$R_t = r\,Z_b \qquad Z_b = \frac{V_L}{I_n/\sqrt3} \qquad r \approx 0.1\text{–}0.3$$
70. E18Rotor heat, no load
    $$E_r = \tfrac12 J\omega_s^2\,(s_1^2 - s_2^2) \quad \Rightarrow \quad E_r(1\to0) = \tfrac12 J\omega_s^2 \ \text{independent of voltage}$$
71. E19Rotor heat with load
    $$E_r = J\omega_s^2\int_{s_2}^{s_1}\frac{s\,T_m(s)}{T_m(s) - T_L(s)}\,ds$$
72. E20I²t and overload thermal budget
    $$Q = \int i_{rms}^2\,dt \qquad B = \frac{\int (I/I_r)^2\,dt}{7.2^2\,t_{class}}$$
73. E21Voltage dip at the PCC
    $$\Delta V\% \approx \frac{S_m}{S_{sc} + S_m}\times100 \qquad S_{sc} = \frac{S_T}{u_k}\ \ \text{or}\ \ \frac{S_G}{x_d''} \qquad S_m = \sqrt3\,V_L I$$
74. E22Losses in Y versus Δ
    $$P_{loss} = P_{Cu1} + P_{Cu2} + P_{Fe} + P_{fw} \qquad P_{Fe,Y} \approx \tfrac13 P_{Fe,\Delta} \qquad s_Y \approx 3\,s_\Delta$$
75. E23Saving and break-even
    $$\Delta P = P_{in,\Delta} - P_{in,Y} \qquad \Delta P(x_{be}) = 0 \qquad E_{year} = \Delta P\cdot h \qquad \text{cost} = E_{year}\times\text{tariff}$$
76. E24Torque margin in star
    $$\frac{T_{k,Y}}{T_L} = \frac{T_k}{3\,T_L} \ge 2.5 \;\Rightarrow\; T_L \le \frac{T_k}{7.5} \approx 0.4\,T_n \ (T_k = 3T_n)$$
77. T1Reduced-voltage starting
    $$\frac{V}{V_n} = k \;\Rightarrow\; I_{st} = k\,I_{st,DOL} \qquad T_{st} = k^2\,T_{st,DOL}$$
78. T2Torque at a current limit
    $$T_{st} = T_{st,DOL}\left(\frac{I_{lim}}{I_{st,DOL}}\right)^2$$
79. T3RMS voltage vs firing angle (resistive approximation)
    $$\frac{V_{rms}}{V} = \sqrt{1 - \frac{\alpha}{\pi} + \frac{\sin 2\alpha}{2\pi}}$$
80. $a$ soft starters & autotransformers i_{motor} = a\,i_{st,dol} qquad i_{line} = a^2\,i_{st,dol} qquad t_{st} = a^2\,t_{st,dol} ">T4Autotransformer starter at tap $a$
    $$I_{motor} = a\,I_{st,DOL} \qquad I_{line} = a^2\,I_{st,DOL} \qquad T_{st} = a^2\,T_{st,DOL}$$
81. T5Supply dip for any method
    $$\Delta U\% \approx \frac{\sqrt3\,V_L\,I_{line,st}}{S_{tr}/u_k + \sqrt3\,V_L\,I_{line,st}}\times 100$$
82. A1Sequence interlock
    $$K2 = K1 \cdot T_{K3T} \cdot \overline{S0} \qquad K1 = \overline{S0}\,(S1 + K1)$$
83. A2Maximum starts per hour
    $$t_{cycle} = \frac{V}{Q_p - q} + \frac{V}{q} \;\ge\; \frac{4V}{Q_p}\ \ (q = Q_p/2) \qquad N_{max} = \frac{Q_p}{4V}$$
84. A3Equal running hours
    $$h_1 \approx h_2 \approx \frac{h_{total}}{2} \qquad \text{the standby pump must run at least every 1–2 weeks}$$
85. A4Receiver size for compressor cycling
    $$t_{off} = \frac{V_{tank}\,\Delta p}{p_0\,q_{use}} \qquad t_{on} = \frac{V_{tank}\,\Delta p}{p_0\,(Q_c - q_{use})}$$
86. A5Pole-changing speeds
    $$n_{s,low} = \frac{120f}{2p} \qquad n_{s,high} = \frac{120f}{p} = 2\,n_{s,low}$$
87. V1DC bus voltage
    $$V_{dc} \approx 1.35\,V_L = 540\ \text{V (loaded)} \qquad V_{dc,0} = \sqrt2\,V_L = 566\ \text{V (no load)}$$
88. V2Maximum output voltage
    $$V_{LL,rms} = 0.612\,m\,V_{dc}\ (\text{SPWM}, m \le 1) \qquad V_{LL,max} \approx 0.707\,V_{dc}\ (\text{SVPWM})$$
89. V3Constant V/f
    $$V(f) = V_n\frac{f}{f_n} + V_{boost}\left(1 - \frac{f}{f_n}\right) \qquad \frac{V_n}{f_n} = \frac{400}{50} = 8\ \text{V/Hz}$$
90. V4Speed with a VFD
    $$n = \frac{120 f}{p} - \Delta n \qquad \Delta n \approx s_n\,n_{s,n}\ \ (\text{constant at rated torque})$$
91. V5Field weakening (f > f_n)
    $$T_{cont} \approx T_n\,\frac{f_n}{f} \qquad T_{max} \approx T_{max,n}\left(\frac{f_n}{f}\right)^2 \qquad P \approx P_n$$
92. V6Braking resistor
    $$E = \tfrac12 J(\omega_1^2 - \omega_2^2) \qquad P_{peak} \approx \frac{J\,\omega_1\,\Delta\omega}{t_{dec}} \qquad R \le \frac{V_{br}^2}{P_{peak}},\ R \ge R_{min}$$
93. P1Minimum ramp time without tripping
    $$t_{acc,min} \approx \frac{J\,\omega_n}{T_{lim} - \overline{T_L}} \qquad T_{lim} \approx 1.5\,T_n\ \ (\text{set by the drive current limit})$$
94. P2Analog reference scaling
    $$f = f_{min} + (f_{max} - f_{min})\,\frac{x - x_0}{x_1 - x_0} \qquad x: 0\text{–}10\ \text{V or } 4\text{–}20\ \text{mA}$$
95. P3Affinity law — flow
    $$\frac{Q_2}{Q_1} = \frac{n_2}{n_1}$$
96. P4Affinity law — head
    $$\frac{H_2}{H_1} = \left(\frac{n_2}{n_1}\right)^2$$
97. P5Affinity law — power
    $$\frac{P_2}{P_1} = \left(\frac{n_2}{n_1}\right)^3$$
98. P6System with static head
    $$H_{sys} = H_{st} + kQ^2 \qquad \frac{n}{n_1} \approx \sqrt{h_0 + (1 - h_0)q^2} \quad (h_0 = H_{st}/H_1,\ q = Q/Q_1)$$
99. P7Energy saved per year
    $$\Delta E = (P_{throttle} - P_{VFD})\,h \qquad \text{money} = \Delta E \times \text{tariff per kWh}$$
100. P8Simple payback
     $$t_{payback} = \frac{\text{installed VFD cost}}{\text{savings per year}}$$
101. K1Shaft power of a pump or a drive
     $$P_{pump} = \frac{\rho g Q H}{\eta_p} \qquad P = T\,\omega = \frac{T\,n}{9550}\ \text{(kW)}$$
102. K2Motor size margin
     $$P_{motor} \ge (1.10\text{–}1.15)\,P_{load,max} \;\rightarrow\; \text{next standard size}$$
103. K3Cable ampacity after derating
     $$I_z\,k_{temp}\,k_{group} \ge I_n \qquad I_z \ge I_r\ (\text{overload protection setting})$$
104. K4Running voltage drop
     $$\Delta V\% = \frac{\sqrt3\,I_n\,L\,(R\cos\varphi + X\sin\varphi)}{V_L}\times100 \le 3\text{–}5\,\% \qquad R = \frac{\rho}{A},\ \rho_{Cu,70^\circ C} \approx 0.0225\ \Omega\,\text{mm}^2/\text{m}$$
105. K5Starting voltage drop
     $$\Delta V_{st}\% \approx \Delta V_{tr}\% + \frac{\sqrt3\,I_{st}\,L\,(R\cdot0.3 + X\cdot0.95)}{V_L}\times100 \le 10\text{–}15\,\%$$
106. K6Earth fault at the cable end (TN system)
     $$I_{f} = \frac{U_0}{Z_s} \ge I_a \qquad Z_s \approx Z_{src} + (1 + m)\,\frac{\rho L}{A_{ph}}\ \ (m = A_{ph}/A_{PE})$$
107. K7Enclosure temperature rise and fan airflow
     $$\Delta T = \frac{P_{loss}}{k\,A_e}\ \ (k \approx 5.5\ \text{W/m}^2\text{K painted steel}) \qquad \dot V\,[\text{m}^3/\text{h}] \approx \frac{3.1\,P_{loss}\,[\text{W}]}{\Delta T_{allow}}$$
108. K8Insulation resistance acceptance
     $$R_{iso} \ge 1\ \text{M}\Omega\ @\ 500\ \text{V DC (IEC 60204-1)} \qquad R_{40^\circ C} \ge 5\ \text{M}\Omega\ \text{windings} < 1\ \text{kV (IEEE 43)} \qquad PI = \frac{R_{10\,min}}{R_{1\,min}} \ge 2$$

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