Himanish Agarwal ❚
Notes on software, systems, and the science of making machines learn

Electrical Machines

[Fri Dec 3, 2021]
— Himanish

Transformers #

No Load #

  1. \[ \phi_{max} = \frac{E_1}{4.44fN_1} \quad E_1: \text{induced emf} \]
  2. \[\text{flux density} B = \frac{\phi}{A}\]
  3. \[ \frac{V_1}{V_2} = \frac{E_1}{E_2} = \frac{N_1}{N_2} = a \] \[\frac{I_1}{I_2} = \frac{1}{a}\]

Equivalent Circuit #

  1. \[\bar{z} = r + jx_l\]
  2. \[\bar{z_2}’ = a^2(z_2) \] \[\bar{z_1}’ = \frac{z_1}{a^2}\]
  3. \[\bar{Z} (HV) = \bar{z_1} + \bar{z_2}’\] \[\bar{Z} (LV) = \bar{z_2} + \bar{z_1}’\]
  4. \[Z_B = \frac{(kV)_B^2}{(MVA)_B}\]

\[\bar{z}_{pu} = \frac{\bar{z}}{Z_B}\] In pu (per unit) system, z is same on both sides.

Testing #

OC Test / No-load Test #

\[y_0 = \frac{I_{oc}}{V_{oc}} \] or \(Z_\phi = \frac{V_{oc}}{I_{oc}}\) \[G_i = \frac{P_{oc}}{V_{oc}^2}\] or \(R_c = \frac{V^2_{oc}}{P_{oc}}\) \[B_m = \sqrt{y_0^2 - G_1^2} \] or \(X_m = \frac{1}{\sqrt{1/Z_\phi^2 - 1/R_c^2}}\)

SC Test #

\[Z_{eq} = \frac{V_{sc}}{I_{sc}}\] \[R_{eq} = \frac{P_{sc}}{I_{sc}^2}\] \[X = \sqrt{Z_{eq}^2 - R_{eq}^2}\]

Full load #

\[B_m = \frac{1}{2\pi fL_m}\] \[I_m = B_mE_1\]

Efficiency and Losses #

  1. \[\eta = \frac{V_2I_2\cos \phi_2}{V_2I_2\cos \phi_2 + P_i+P_c}\]
  2. \[P_i = P_{oc}\][iron loss is constant irregardless of load] \[P_{Cu} = P_{sc}\]
  3. Load condition K = \(\frac{\text{Given load}}{\text{Full load}}\) e.g. k=0.5 for half load
  4. Output Power = \(KS\cos \phi\)
  5. Cu loss at any load = \(k^2\cdot P_{Cu,FL}\)
  6. \[\eta = \frac{\text{output}}{\text{input}} = \frac{\text{output}}{\text{output + losses}} = \frac{\text{output}}{\text{output + iron loss + Cu loss}}\]

Parallel Connection #

  1. \[|S_2| = \frac{|z_{e1}|}{|z_{e1}+z_{e2}’}|S|\]

DC Machines #

EMF and Torque #

\[\omega = \frac{2\pi N}{60}\]

Induced EMF #

  1. Per turn = \[ \frac{P\phi \omega_m}{\pi}\]

  2. Per parallel path \[E_a = \frac{P\phi N}{60} \cdot \frac{Z}{A}\] where

    1. A = P (Lap Winding)
    2. A = 2 (Wave Winding)
    3. N = rotor speed (RPM)
    4. Z = total number of rotor conductors
    5. A = number of parallel paths
    6. \(\phi =\) flux per pole

    ( Z/2 = total turns, Z/2A = total turns per parallel path)

Current #

\[I_f = V_t/R_f\] \[I_L = V_t/R_L\] \[I_a = I_L+I_f\]

Power and Losses #

  1. \( P_e (\text{mechanical/rotational [input] power}) = E_aI_a\) \[= \text{total Cu-loss} + \text{output power}\]

    • Armature Cu-loss = \(I_a^2R_a\)
    • Field Cu-loss = \(V_tI_f\)
    • Total Cu-loss = (Armature + Field) Cu-loss
    • Output Power = \(V_tI_L\)

  2. If Armature reaction, \[\phi_R = 0.96\phi\]

  3. \[E_a = V_t + I_aR_a+V_{brush}\]

    • \[V_{brush} = 2V_{per brush}\]

Torque #

  1. \[P_{shaft} = P_{rot} + P_e\]

\[T_{sh}\omega = T_{fr}\omega+T_e\omega\] where \(T_e\) is electromagnetic torque, \(T_{fr}\) is torque due to friction

  1. \[P_e = T_e\omega = E_gI_a\]
  2. \[E_g = K_g \phi \omega\] where \[ K_g (\text{emf constant}) = \frac{PZ}{2\pi A}\]

Excitation #

Synchronous Machines #

  1. \[Z_S(unsat) = \frac{V_{OC}}{I_{SC}}|_{I_f = constant}\]
  2. \[Z_S(adjusted) = \frac{V_{OC(rated)}}{I_{SC}} \]
    • \(X_S(adjusted) = \sqrt{Z_S(adjusted)^2-R_a^2}\)
    • \(R_a\) can be ignored (except when calculating efficiency) so \(X_S (adjusted) = Z_S(adjusted)\)
  3. Power factor pf = \(\cos \theta = \frac{P_{in}}{\sqrt{3}V_LI_g}\)

Induction Motors #

  1. Reading of wattmeter = Number of divison \(\times\) MF
  2. Active power in 3-phase load \(P = W_1 + W_2 \) (kW)
    1. Reactive power \(Q = \sqrt{3}(W_2-W_1)\) (kVAR)
    2. Apparent power \(S = \sqrt{P^2+Q^2}\) (kVA)
  3. Power factor of load pf = \(\cos \tan^{-1}{(Q/P)}\)