Energy Systems 244 - Week 3 to 4 - 3.2 to 3.7

Draw a sketch of an ideal transformer

State the assumptions made about ideal transformers

• Windings have zero resistance

  • I² losses in the windings = 0

• Core permeability, $\mu_c$ is infinite

  • zero core reluctance

• There is no leakage flux

  • The entire flux $\Phi_c$ is confined to the core and links both windings

• There are no core losses

Describe what is meant by permeability

• The ability of a material to concentrate magnetic flux lines more easily

Describe Ampere’s Law

• The magnetic flux density around a closed loop is proportional to the total electric current passing through that loop

• $$\int B dl = \mu_c I_{enclosed}$$

  • B = Magnetic flux density

  • $\mu_c$ = permeability of the core

  • $I_{enclosed}$ = net current enclosed by the loop

• $$H_c = \frac{B}{\mu_c}$$

  • $$\int {H_{tan}} dl = I_{enclosed}$$

    • $$I_{enclosed} = N * I$$

      • N = number of turns

      • I = current flowing through each turn
        I_{enclosed} = N \dot

Use Ampere’s laws to derive an ideal transformer relationship

• $H_c l_c = N_1I_1 - N_2I_2$ (1)

• $B_c = \mu_cH_c$ [Wb/m²] (2)

• $\Phi_c = B_cA_c$ [Wb] (3)

• Sub (2) and (3) into (1):

  • $N_1I_1-N_2I_2=\frac{l_c}{\mu_cA_c}\Phi_{c}$ (4)

• Core reluctance $R_c = \frac{l_c}{\mu_cA_c}$ (5)

• sub (5) into (4)

  • $N_1I_1 - N_2I_2 = R_c\Phi_c$ (6)

• since $\mu_c$ is infinite, $R_c$ tends to zero thus:

  • $$N_1I_1 - N_2I_2 = 0$$

  • $N_1I_1 = N_2I_2$ (7)

• $\therefore \frac{N_1}{N_2} = \frac{I_1}{I_2}$ (8)

Describe Faraday’s Law

• Voltage induced is proportional to the rate of change of flux $\phi$

  • e(t) = $N \frac{d\phi}{dt}$(1)

• 
  

Use Faraday’s law to derive ideal transformer relationships

• We now do the following:

  • assume the rate of change is a sinusoidal-steady-state flux with constant frequency, $\omega$

  • Represent e(t) and $\phi$ as phasors, E and $\Phi$

• thus, (1) becomes:

  • E = $N(j\omega)\Phi$ (2)

• For an ideal transformer, the entire flux is confined to the core, thus:

  • $E_1 = N_1(j\omega)\Phi_c$ (3)

  • $E_2 = N_2(j\omega)\Phi_c$ (4)

• Dividing (3) by (4) gives us:

  • $$\frac{V_1}{V_2} = \frac{N_1j\omega}{N_2j\omega}$$

  • $\frac{V_1}{V_2} = \frac{N_1}{N_2}$ (5)

Use both Faraday’s Law and Ampere’s Law in order derive relationship equations for:

• EMF

• Current

• Impedance

• we say

  • $$a_t = \frac{N_1}{N_2}$$

• $$\therefore$$

  • $E_1 = \frac{N_1}{N_2}\cdot E_2$ ….(1)

  • $I_2 = \frac{N_2}{N_1} \cdot I_2 = \frac{I_2}{a_t}$ …(2)

• Thus:

  • $S_1 = S_2$ …(3)

  • $Z’_2 = a²Z_2$ …(4)

Describe the phase shifting transformer:

• Definition

• $$a_t$$

• EMF

• Current

• Complex power

• Impedance

• single phase

• $$a_t = e^{j\phi}$$

• $$E_1 = a_tE_2 = e^{j\phi} \cdot E_2$$

• $$I_1 = \frac{I_2}{a*_t} = e^{j\phi} \cdot I_2$$

• $$S_1 = S_2$$

• $$Z’_2 = Z_2$$

State the properties of a practical transformer

• Windings have resistances

• Core permeability is finite

• Magnetic flux is not all confined to the core

• There are real and reactive losses in the core

Draw a sketch of a practical transformer

For a practical transformer, describe the winding resistances

• Represent the resistance of the conductors used to turn the respective windings

• accounts for real power loss I²R in the windings

For a practical transformer, describe the leakage reactances

• there are leakage fluxes that come from the windings

• they cause a voltage drop

• accounts for reactive power loss I²X

For a practical transformer, describe the magnetising susceptance

• since core permeability$\mu_c$ is finite,

  • core reluctance ≠ 0

• $N_1I_1 - N_2I_2 = R_c\Phi_c$ …(1)

• $E_1 = N_1(j\omega)\Phi_c$ …(2)

• divide (1) by $N_1$ and sub (2):

  • $$I_1 - \frac{N_2}{N_1}I_2 = \frac{R_c}{N_1}\Phi_c$$

  • $$= \frac{R_c}{N_1}(\frac{E_1}{j\omega N_1²})E_1$$

  • $$= -j(\frac{R_c}{\omega N_1²})E_1$$

• Accounts for reactive losses in core

For a practical transformer, describe the Core loss conductance

• There is an additional (shunt) branch along with the magnet susceptance branch with resistance $G_c$

• It carries core loss current $I_c$

  • $I_c$ is in phase with $E_1$

• Accounts for real power losses

  • eddy current losses

  • hysteris loss

For a practical transformer, describe the exciting current

• $$I_e$$

• $$I_e = I_m + I_c$$

  • $$I_m = magnetising susceptance$$

  • $I_c =$ Core loss conductance

Describe what is meant by nameplate data

• the rated voltages and power

Describe what is meant by the open circuit test

• rated voltage is applied to primary with secondary open

• measure the primary current and losses

• Determine shunt admittance of winding 1

  • $$Y_m = G_c -jB_m$$

  • neglect series impedance

Describe what is meant by the short circuit test

• short the secondary

• apply voltage to the primary and obtain rated current flow

• measure voltage and losses

• determine series impedance referred to winding 1

  • $$Z_{eq1} = R_{eq1} + jX_{eq1}$$

  • neglect shunt admittance

Describe the Per unit system

• The transformer equivalent circuit can be simplified

  • the ideal transformer element is eliminated

State the per unit system formula

• $$per-unit\space quantity = \frac{actual\space quantity}{base\space value\space of\space quantity}$$

• it is dimensionless

• base value of quantity is always a real value

Describe the per unit system

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Describe the balanced 3 phase circuit

Describe the 3 phase transformer connection and phase shift

Describe the steps for converting a 3-phase transformer

Describe the advantages of the delta winding

Describe the convention for selecting base values of a balanced 3-phase 2-winding transformers

Sketch the Per-unit equivalent circuits positive and negative sequences of the following single line diagrams:

Describe 3-winding transformers

Describe the advantages and disadvantages of a autotransformers