Nomenclature
\(P\)
Number of poles
\(f\)
Supply frequency (Hz)
\(N_s\)
Synchronous speed (rpm)
\(N_r\)
Rotor speed (rpm)
\(s\)
Slip
\(T_e\)
Electromagnetic torque (Nm)
\(P_{in}\)
Input power (W)
\(P_{out}\)
Output power (W)
\(P_{ag}\)
Air-gap power (W)
\(P_{mech}\)
Mechanical power (W)
\(I_s, I_r\)
Stator & rotor current (A)
\(V_s\)
Stator voltage (V)
\(R_s, R_r'\)
Resistances (Ω)
\(X_s, X_r'\)
Reactances (Ω)
\(\eta\)
Efficiency
\(PF\)
Power factor
Basic Speed & Slip Relations
1.1
Synchronous Speed
\[N_s = \frac{120f}{P}\]
Note: \(N_s\) in rpm, \(f\) in Hz, \(P\) = number of poles
1.2
Slip
\[s = \frac{N_s - N_r}{N_s} = \frac{\omega_s - \omega_r}{\omega_s}\]
Range: 0 ≤ s ≤ 1 (typically 0.01 to 0.05 at full load)
1.3
Rotor Speed
\[N_r = N_s(1-s) = \frac{120f}{P}(1-s)\]
1.4
Rotor Frequency
\[f_r = sf\]
At standstill: s=1, \(f_r = f\); At synchronous speed: s=0, \(f_r = 0\)
1.5
Angular Velocities
\[\omega_s = \frac{2\pi N_s}{60} = \frac{4\pi f}{P}\]
\[\omega_r = (1-s)\omega_s\]
Power Flow & Losses
2.1
Input Power (3-Phase)
\[P_{in} = \sqrt{3}V_L I_L \cos\phi = 3V_{ph}I_{ph}\cos\phi\]
2.2
Stator Copper Loss
\[P_{cu,s} = 3I_s^2 R_s\]
2.3
Air-Gap Power
\[P_{ag} = P_{in} - P_{cu,s} - P_{core}\]
2.4
Rotor Copper Loss
\[P_{cu,r} = 3I_r'^2 R_r' = sP_{ag}\]
Key Relation: Rotor copper loss is proportional to slip
2.5
Mechanical Power Developed
\[P_{mech} = P_{ag} - P_{cu,r} = P_{ag}(1-s) = \frac{1-s}{s}P_{cu,r}\]
2.6
Output Power
\[P_{out} = P_{mech} - P_{friction} - P_{windage} - P_{stray}\]
Power Flow Ratio
\[P_{ag} : P_{cu,r} : P_{mech} = 1 : s : (1-s)\]
Torque Relations
3.1
Electromagnetic Torque
\[T_e = \frac{P_{ag}}{\omega_s} = \frac{P_{mech}}{\omega_r}\]
3.2
Torque in Terms of Rotor Current
\[T_e = \frac{3}{\omega_s} \cdot I_r'^2 \cdot \frac{R_r'}{s}\]
3.3
Starting Torque (s = 1)
\[T_{start} = \frac{3}{\omega_s} \cdot \frac{V_s^2 R_r'}{(R_s + R_r')^2 + (X_s + X_r')^2}\]
3.4
Condition for Maximum Torque
\[s_{max} = \frac{R_r'}{\sqrt{R_{th}^2 + (X_{th} + X_r')^2}} \approx \frac{R_r'}{\sqrt{R_s^2 + (X_s + X_r')^2}}\]
3.5
Maximum Torque (Breakdown Torque)
\[T_{max} = \frac{3}{2\omega_s} \cdot \frac{V_{th}^2}{R_{th} + \sqrt{R_{th}^2 + (X_{th} + X_r')^2}}\]
Note: \(T_{max}\) is independent of \(R_r'\)
3.6
Torque-Slip Characteristic
\[\frac{T_e}{T_{max}} = \frac{2}{\frac{s}{s_{max}} + \frac{s_{max}}{s}}\]
3.7
Torque in Terms of Power
\[T_e = \frac{P_{out}}{\omega_r} = \frac{9.55 \times P_{out}(kW)}{N_r(rpm)}\]
Equivalent Circuit Parameters
4.1
Rotor Induced EMF
\[E_r = sE_s = s \cdot 4.44 f \phi_m N_s k_w\]
4.2
Rotor Impedance per Phase
\[Z_r = R_r + jX_r = R_r + jsX_{r0}\]
\(X_{r0}\) is rotor reactance at standstill
4.3
Referred Rotor Parameters
\[R_r' = K^2 R_r, \quad X_r' = K^2 X_r, \quad I_r' = \frac{I_r}{K}\]
where \(K = \frac{N_s k_{ws}}{N_r k_{wr}}\) is transformation ratio
4.4
Thévenin Equivalent Parameters
\[V_{th} = V_s \frac{X_m}{\sqrt{R_s^2 + (X_s + X_m)^2}}\]
\[R_{th} = \frac{R_s X_m^2}{R_s^2 + (X_s + X_m)^2}\]
\[X_{th} = \frac{X_s X_m}{X_s + X_m}\]
4.5
Stator Current
\[I_s = \frac{V_s}{Z_{eq}}\]
Performance Testing
5.1
No-Load Test
\[P_{nl} = \sqrt{3}V_{nl}I_{nl}\cos\phi_{nl}\]
\[P_{nl} \approx P_{core} + P_{friction} + P_{windage}\]
Determines core losses and no-load parameters
5.2
Blocked Rotor Test (s = 1)
\[P_{br} = \sqrt{3}V_{br}I_{br}\cos\phi_{br}\]
\[Z_{br} = \frac{V_{br}}{I_{br}}, \quad R_{br} = \frac{P_{br}}{3I_{br}^2}\]
\[X_{br} = \sqrt{Z_{br}^2 - R_{br}^2}\]
Determines short-circuit parameters: \(R_s + R_r'\) and \(X_s + X_r'\)
5.3
Efficiency
\[\eta = \frac{P_{out}}{P_{in}} = \frac{P_{in} - \text{Losses}}{P_{in}} \times 100\%\]
5.4
Power Factor
\[PF = \cos\phi = \frac{P_{in}}{\sqrt{3}V_LI_L}\]
Starting Methods & Ratios
6.1
Direct-on-Line Starting
\[I_{start} = 5 \text{ to } 7 \times I_{fl}\]
\[T_{start} = 0.5 \text{ to } 1.5 \times T_{fl}\]
6.2
Star-Delta Starting
\[I_{start,Y} = \frac{I_{start,\Delta}}{3}, \quad T_{start,Y} = \frac{T_{start,\Delta}}{3}\]
6.3
Auto-Transformer Starting
\[I_{start} = K^2 I_{fl}, \quad T_{start} = K^2 T_{fl}\]
where K is the transformation ratio (typically 0.5 to 0.8)
6.4
Rotor Resistance Starting
\[T_{start} \propto \frac{R_r}{s}, \quad s_{max} = \frac{R_r + R_{ext}}{\sqrt{R_s^2 + (X_s + X_r')^2}}\]
Speed Control Methods
7.1
Frequency Control
\[N_r = \frac{120f(1-s)}{P}\]
Maintain \(V/f\) ratio constant to avoid saturation
7.2
Voltage Control
\[T_e \propto V_s^2\]
7.3
Pole Changing
\[N_{s1} : N_{s2} = P_2 : P_1\]
7.4
Slip Power Recovery
\[P_{recovered} = sP_{ag}\]
Key Relations & Concepts
Operating Modes Based on Slip
- Motoring: 0 < s < 1
- Generating: s < 0 (when \(N_r > N_s\))
- Braking: s > 1 (reverse rotation)
Typical Operating Values
- Full-load slip: 2-5% (0.02-0.05)
- Starting current: 5-7 times full-load current
- Starting torque: 50-150% of full-load torque
- Maximum torque: 200-300% of full-load torque
- Efficiency: 85-95% for large motors
- Power factor: 0.8-0.9 at full load
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Circle Diagram Relations
\[\text{Diameter} = \frac{mV_s^2}{\omega_s X_{sc}}\]
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Transformation Ratios
\[a = \frac{E_s}{E_r} = \frac{N_s k_{ws}}{N_r k_{wr}}\]
Additional Important Formulas
8.1
EMF Equation
\[E_{ph} = 4.44 f \phi_m T_{ph} k_w\]
\(T_{ph}\) = turns per phase, \(k_w\) = winding factor, \(\phi_m\) = maximum flux per pole
8.2
Air-Gap Flux
\[\phi_m = \frac{E_{ph}}{4.44 f T_{ph} k_w}\]
8.3
Magnetizing Current
\[I_m = \frac{E_s}{X_m}\]
8.4
Core Loss Current
\[I_c = \frac{P_{core}}{3E_s}\]
8.5
No-Load Current Components
\[I_0 = \sqrt{I_m^2 + I_c^2}\]
\[\cos\phi_0 = \frac{I_c}{I_0}\]
8.6
Load Current
\[I_2' = \frac{sE_s}{\sqrt{R_r'^2 + (sX_r')^2}}\]
8.7
Shaft Torque
\[T_{sh} = T_e - T_{friction}\]
8.8
Percentage Slip
\[s\% = \frac{N_s - N_r}{N_s} \times 100\%\]
8.9
Gross Mechanical Power
\[P_m = T_e \omega_r = T_e \omega_s(1-s)\]
8.10
Regulation
\[\text{Regulation} = \frac{N_{nl} - N_{fl}}{N_{fl}} \times 100\%\]
Braking Methods
9.1
Regenerative Braking
\[P_{returned} = P_{mech} - P_{losses}\]
Occurs when \(N_r > N_s\) (slip is negative)
9.2
Plugging (Reverse Current Braking)
\[s = \frac{N_s - (-N_r)}{N_s} = 2 - \frac{N_r}{N_s}\]
Slip > 1, very high braking torque
9.3
Dynamic Braking Power
\[P_{braking} = 3I_r^2 R_{ext}\]
Losses & Efficiency
Total Losses Classification
- Stator copper losses: \(P_{cu,s} = 3I_s^2R_s\)
- Rotor copper losses: \(P_{cu,r} = 3I_r'^2R_r' = sP_{ag}\)
- Core losses: Hysteresis + Eddy current losses
- Friction & windage: Mechanical losses
- Stray load losses: ~0.5-1% of output
10.1
Total Losses
\[\text{Total Losses} = P_{cu,s} + P_{core} + P_{cu,r} + P_{fw} + P_{stray}\]
10.2
Condition for Maximum Efficiency
\[\text{Variable losses} = \text{Constant losses}\]
\[P_{cu,s} + P_{cu,r} = P_{core} + P_{fw}\]
10.3
Efficiency at Different Loads
\[\eta = \frac{P_{out}}{P_{out} + P_{constant} + x^2P_{variable}}\]
where x is the fraction of full load
Special Operating Conditions
11.1
At Standstill (s = 1)
\[N_r = 0, \quad f_r = f, \quad Z_r = R_r + jX_{r0}\]
11.2
At Synchronous Speed (s = 0)
\[N_r = N_s, \quad f_r = 0, \quad I_r = 0, \quad T_e = 0\]
11.3
At Maximum Torque
\[N_r = N_s(1 - s_{max})\]
\[\frac{R_r'}{s_{max}} = \sqrt{R_{th}^2 + (X_{th} + X_r')^2}\]
11.4
Crawling Speed
\[N_{crawl} = \frac{120f}{nP}\]
where n = 7, 5, 3... (harmonic order)
11.5
Cogging (Magnetic Locking)
\[\text{Avoid: } S = mP \pm 1, \pm 2\]
S = number of stator slots, m = integer
Important Performance Ratios
12.1
Starting Current Ratio
\[\frac{I_{start}}{I_{fl}} = 5 \text{ to } 7\]
12.2
Starting Torque Ratio
\[\frac{T_{start}}{T_{fl}} = 0.5 \text{ to } 1.5\]
12.3
Maximum Torque Ratio
\[\frac{T_{max}}{T_{fl}} = 2 \text{ to } 3\]
12.4
Load Power Factor
\[PF = \frac{P_{in}}{\sqrt{3}V_LI_L} = \frac{\text{True Power}}{\text{Apparent Power}}\]
Quick Reference Table
💡 Quick Revision Tips
- Remember the power flow: \(P_{in} → P_{ag} → P_{mech} → P_{out}\)
- Power ratio: \(P_{ag} : P_{cu,r} : P_{mech} = 1 : s : (1-s)\)
- \(T_{max}\) is independent of rotor resistance but \(s_{max}\) depends on it
- Lower the slip, higher the efficiency and power factor
- At synchronous speed: no torque, no rotor current
- Rotor copper loss is always proportional to slip: \(P_{cu,r} = sP_{ag}\)