Electrical Three-Phase Power System

A single phase electric power system is a fundamental alternating current (AC) system.
Voltage and current fluctuate in magnitude and direction cyclically, typically at frequencies of 50 or 60 Hz.
A four poles single phase system is indicated, highlighting its configuration.

The voltage can be expressed as:
v(t) = Vmax sin(ωt)
- Vmax represents the maximum amplitude of the wave.
- ω indicates the angular frequency of the wave.

The average voltage over one period (T) for a sine wave is zero:
vave = (1/T) ∫ from 0 to T Vmax sin(ωt) dt = 0
The root mean square (RMS) value is calculated using the formula:
vrms = (1/T) ∫ from 0 to T (Vmax sin(ωt))^2 dt = Vmax/√2

Complex impedance can be represented in polar form as:
Z = x + jy = A∠ϕ = A(cos ϕ + j sin ϕ)
- A indicates the magnitude, while ϕ represents the phase angle.

Aims to relate current in a closed loop with the magnetic field produced.
Magneto-motive force (MMF) is given by the sum of Ni, where N is the number of turns and i is the current.
Under uniform area (Ac) and high permeability magnetic materials, the relationship simplifies to MMF = Hl.

A changing magnetic field induces an electric field in space.
The line integral of electric field intensity (E) around a closed loop equals the rate of magnetic flux through the surface defined by that loop.
Induced voltage (EMF) is defined as e(t) = dφ/dt = N(dλ/dt), representing voltage due to changing flux linkage.

Magnetic field intensity (H) has units of A·turn/m and represents the effort required to create a magnetic field.
Magnetic flux density (B) is measured in tesla (T) and indicates the amount of magnetic flux per unit area in a core.

The relationship between flux density (B) and magnetic field intensity (H) is expressed as B = μH, where μ = μ₀μᵣ.
Magnetic permeability (μ) indicates how easily a material can support the formation of a magnetic field.

Ferromagnetic: Composed of iron, nickel, cobalt, and their alloys; these materials can retain magnetization.
Paramagnetic: Include platinum and aluminum; they do not retain magnetization without an external magnetic field.
Diamagnetic: Composed of materials like carbon and copper; they resist an applied magnetic field.

Ferromagnetic materials can become permanently magnetized once exposed to an external magnetic field.
Two categories of ferromagnetic materials: soft (easily magnetized and demagnetized) and hard (permanent magnets).

Total magnetic flux (φ) is calculated as the surface integral of the normal component of B, expressed as φ = ∫B·da.
Assuming uniform B across Ac, it simplifies to φ = BcAc.

Reluctance (R) quantifies a material's opposition to magnetic flux and is given by R = lc/(μcAc).
Permeance represents the ease of magnetic flux and is the inverse of reluctance, measured in A·turns/Wb.

Air gaps in magnetic circuits affect flux distribution and may require consideration of fringing effects if the gap area (Ag) is larger than the core area (Ac).

Magnets attract objects and have aligned poles with Earth's magnetic field.
Magnetic field lines extend from the north to the south pole of a magnet, reflecting the orientation of magnetic domains.

A current-carrying wire generates a magnetic field, influencing transformer action, motor action, and generator action.
Magnetic circuits typically consist of current, wire coils, and ferromagnetic materials.

Fundamental principles governing electromagnetism include Gauss’s laws for electricity and magnetism, Faraday's law, and Ampere's law, encapsulating the behavior of electric and magnetic fields.

Excitation current is present in transformers and comprises primary and secondary sides.
Induced EMF determined by Faraday's law: ( e_1 = N_1 \frac{d\phi}{dt} ).
Kirchhoff's Voltage Law (KVL) gives equation ( v_1 = R_1 i_\phi + e_1 ).
With negligible resistance, ( e_1 = wN_1\phi_{\text{max}} \cos(wt) ), indicating EMF leads magnetic flux by 90 degrees.
RMS voltage ( E_{1rms} = \frac{E_{\text{peak}}}{\sqrt{2}} ) correlates to highest flux under sine waveform conditions.

Excitation current displays a non-sinusoidal waveform, influenced by magnetic material properties.
Two components of excitation current:
- Magnetizing Current (I_M): Required for flux generation, lags EMF.
- Core Loss Current (I_c): Compensates for losses due to hysteresis and eddy currents, in phase with EMF.
Core losses expressed as ( P_c = E_1 I_c = E_1 I_\phi \cos(\theta_c) ).

Assumes zero winding resistances, no core losses, and high permeability.
Voltage and current transformation formula: ( \frac{V_1}{V_2} = \frac{N_1}{N_2} ) and ( \frac{I_1}{I_2} = \frac{N_2}{N_1} ).
Apparent power and induced parameters change according to turns ratio ( a ).

Input power ( P_{in} = P_1 = V_1 I_1 \cos(\theta_1) ) and output power ( P_{out} = P_2 = V_2 I_2 \cos(\theta_2) ).
In ideal transformers, ( P_1 = P_2 ) and reactive power ( Q_1 = Q_2 ).

Calculation based on a single-phase transformer example with specifications:
- Rated voltages: 120/12.6 V, 25 VA, 60 Hz.
- Rated primary current ( I_{1,\text{rat}} = 0.2 , A ) and secondary current ( I_{2,\text{rat}} = 1.98 , A ).
Induced impedance transformation shows changes in voltage, current, and impedance due to turns ratio.

Losses include copper losses (resistive), eddy current losses, and hysteresis losses.
Leakage flux characterizing transformer efficiency.

An equivalent circuit includes elements for resistive losses, leakage flux, and the decomposition of input current into excitation and load current.
Excitation current consists of core loss current (resistive, in-phase) and magnetization current (reactive, lagging).

Open Circuit Test: Focuses on core losses, typically performed on HV side to determine ( R_c ) and ( X_M ).
Short Circuit Test: Assesses equivalent resistance and reactance, performed on the LV side to find ( R_{eq} = R_1 + R_2 ) and ( X_{eq} = X_1 + X_2 ).
Excitation current in good conductors can be approximated to be 2-3% of input current and is often neglected in calculations.