Three Phase Systems

Phasor representation

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

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

Star & Delta

In star connection:

$$V_\text{line} = \sqrt{3} V_\text{phase} \angle 30^\circ \;\; \text{and} \;\; I_\text{line} = I_\text{phase}$$

In delta connection:

$$V_\text{line} = V_\text{phase} \;\; \text{and} \;\; I_\text{line} = \sqrt{3} V_\text{phase} \angle 30^\circ$$

Definitions

Attenuation & Phase Shift

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

$$E_\text{in} = E_m \sin{(\omega t)}\;\; \text{and} \;\;E_\text{out} = A E_m \sin{(\omega t + \alpha)}$$

Here $A$ is the attenuation or magnification factor and $\alpha$ is the phase shift.

Power

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

$$P_\text{ph} = V_\text{ph} \cdot I^*_\text{ph} = P_\text{active} + j P_\text{reactive}$$

Here $I^*_\text{ph}$ 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, $\delta$-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.

$$\text{Efficiency} = \frac{\text{Output Power}}{\text{Input Electrical Power}}$$

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 ($\mathbf{a}$)
  Has same phase sequence as the original 3-phase system
• Negative sequence ($\mathbf{b}$)
  Has reverse phase sequence as the original 3-phase system
• Zero sequence ($\mathbf{c}$)
  Has equal magnitude and phase angle in all 3-phases

$$\begin{bmatrix} A \\ B \\ C \\ \end{bmatrix} = \begin{bmatrix} A_0 \\ B_0 \\ C_0 \\ \end{bmatrix} + \begin{bmatrix} A_1 \\ B_1 \\ C_1 \\ \end{bmatrix} + \begin{bmatrix} A_2 \\ B_2 \\ C_2 \\ \end{bmatrix}$$

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

$$\begin{bmatrix} A \\ B \\ C \\ \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha^2 & \alpha \\ 1 & \alpha & \alpha^2 \\ \end{bmatrix} \begin{bmatrix} A_0 \\ B_0 \\ C_0 \\ \end{bmatrix}$$

Symmetrical component matrix

$$[\Lambda]= \begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha^2 & \alpha \\ 1 & \alpha & \alpha^2 \\ \end{bmatrix}$$ $$[\Lambda]^{-1}= \frac{1}{3} \begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha & \alpha^2 \\ 1 & \alpha^2 & \alpha \\ \end{bmatrix} = \frac{1}{3} \cdot [\Lambda]^{*}$$

Power

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

Impedances

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

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

Here:

• $Z_m$ is the impedance caused by mutual coupling between phases.
• $Z_s$ is the impedance in a single phase.

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

$$[Z_s] = [\Lambda]^{-1} Z_p [\Lambda]$$