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Sahithyan's S2
Sahithyan's S2 — Theory of Electricity

Three Phase Systems

Phasor representation

Suppose V=Vmsin(ωt+α)V = V_m \sin{(\omega t + \alpha)} is a phasor.

V˙=jωV\dot{V} = j\omega \cdot V V=1jωV\int V = \frac{1}{j\omega} \cdot V

Star & Delta

In star connection:

Vline=3Vphase30    and    Iline=IphaseV_\text{line} = \sqrt{3} V_\text{phase} \angle 30^\circ \;\; \text{and} \;\; I_\text{line} = I_\text{phase}

In delta connection:

Vline=Vphase    and    Iline=3Vphase30V_\text{line} = V_\text{phase} \;\; \text{and} \;\; I_\text{line} = \sqrt{3} V_\text{phase} \angle 30^\circ

Definitions

Attenuation & Phase Shift

Suppose a linear, passive network has the input and output as:

Ein=Emsin(ωt)    and    Eout=AEmsin(ωt+α)E_\text{in} = E_m \sin{(\omega t)}\;\; \text{and} \;\;E_\text{out} = A E_m \sin{(\omega t + \alpha)}

Here AA is the attenuation or magnification factor and α\alpha is the phase shift.

Power

Power on a 1-phase circuit can be easily found using 1-phase power formula:

Pph=VphIph=Pactive+jPreactiveP_\text{ph} = V_\text{ph} \cdot I^*_\text{ph} = P_\text{active} + j P_\text{reactive}

Here IphI^*_\text{ph} is the complex conjugate of the current phasor.

Standard terminology

Examples:

  • A 3-ph, 415 V, 50 Hz, 100 kVA transformer
  • A 3-ph, 33 kV, 50 Hz, 1 MVA, 3-wire transmission line
  • A 3-ph, δ\delta-connected, 415 V, 3.2 kW, 0.85 pf motor

Here:

  • Voltage specified is always line voltage
  • Active power or apparent power is always the total 3-ph quantity
  • If apparent power is given, maximum current capacity of the device can be determined
  • 4-wire system has the neutral wire connected between the star-points of supply and load.
  • For motors, the power specified is the output mechanical power. The operating power factor of the motor is specified at its rated power.
Efficiency=Output PowerInput Electrical Power\text{Efficiency} = \frac{\text{Output Power}}{\text{Input Electrical Power}}

Symmetrical Components

A technique used to handle unbalanced voltages or current sources. Any unbalanced system of three-phase circuits can be decomposed into three symmetrical components:

  • Positive sequence (a\mathbf{a})
    Has same phase sequence as the original 3-phase system
  • Negative sequence (b\mathbf{b})
    Has reverse phase sequence as the original 3-phase system
  • Zero sequence (c\mathbf{c})
    Has equal magnitude and phase angle in all 3-phases
[ABC]=[A0B0C0]+[A1B1C1]+[A2B2C2]\begin{bmatrix} A \\ B \\ C \\ \end{bmatrix} = \begin{bmatrix} A_0 \\ B_0 \\ C_0 \\ \end{bmatrix} + \begin{bmatrix} A_1 \\ B_1 \\ C_1 \\ \end{bmatrix} + \begin{bmatrix} A_2 \\ B_2 \\ C_2 \\ \end{bmatrix}

The above equation can be simplified as below. Here α=1.0    120\alpha = 1.0\; \angle\; 120^\circ.

[ABC]=[1111α2α1αα2][A0B0C0]\begin{bmatrix} A \\ B \\ C \\ \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha^2 & \alpha \\ 1 & \alpha & \alpha^2 \\ \end{bmatrix} \begin{bmatrix} A_0 \\ B_0 \\ C_0 \\ \end{bmatrix}

Symmetrical component matrix

[Λ]=[1111α2α1αα2][\Lambda]= \begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha^2 & \alpha \\ 1 & \alpha & \alpha^2 \\ \end{bmatrix} [Λ]1=13[1111αα21α2α]=13[Λ][\Lambda]^{-1}= \frac{1}{3} \begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha & \alpha^2 \\ 1 & \alpha^2 & \alpha \\ \end{bmatrix} = \frac{1}{3} \cdot [\Lambda]^{*}

Power

S=3VA,0IA,0+3VA,1IA,1+3VA,2IA,2+S = 3V_\text{A,0} I^*_\text{A,0}+ 3V_\text{A,1} I^*_\text{A,1}+ 3V_\text{A,2} I^*_\text{A,2}+ S=3[VA,0VA,1VA,2][IA,0IA,1IA,2]S = 3 \begin{bmatrix} V_\text{A,0} & V_\text{A,1} & V_\text{A,2} \\ \end{bmatrix} \begin{bmatrix} I^*_\text{A,0} \\ I^*_\text{A,1} \\ I^*_\text{A,2} \\ \end{bmatrix} S=3[VSy]T[ISy]S = 3 [V_\text{Sy}]^{T} [I_\text{Sy}]^*

Impedances

Phase impedance matrix [Zp][Z_p] is defined as:

[Zp]=[ZsZmZmZmZsZmZmZmZs][Z_p] = \begin{bmatrix} Z_s & Z_m & Z_m \\ Z_m & Z_s & Z_m \\ Z_m & Z_m & Z_s \\ \end{bmatrix}

Here:

  • ZmZ_m is the impedance caused by mutual coupling between phases.
  • ZsZ_s is the impedance in a single phase.

Sequence impedance matrix [Zs][Z_s] is defined as:

[Zs]=[Λ]1Zp[Λ][Z_s] = [\Lambda]^{-1} Z_p [\Lambda]