Visualizing Transformer Behavior: Drawing Phasor Diagrams and Equivalent Circuits

Phasor diagrams are graphical representations that help analyze the behavior of electrical quantities in AC circuits. When applied to transformers, phasor diagrams demonstrate the relationship between current, voltage, and power in both load and no-load conditions. 

 

Image used courtesy of Adobe Stock

 

Equivalent circuits can be used to represent transformers in a simplified manner. The equivalent circuit for a no-load condition includes the magnetizing branch, which accounts for the magnetizing current and core losses. In the load condition, the equivalent circuit consists of the series impedance branch (including resistance and reactance), representing the voltage drop across the transformer windings (primary and secondary) and the load impedance.

 

No-Load Transformer Conditions

When the output terminals of a transformer are not connected, it is known as a no-load condition. During this state, no current flows through the secondary windings. However, an alternating voltage applied to the input terminals induces a small primary current, which generates the magnetic flux in the core. This current is commonly referred to as the magnetizing current. Hysteresis loss occurs when the magnetic flux in a core continuously increases to its peak level in one direction, reverses to a peak in the opposite direction, and the core absorbs energy. Eddy currents are also induced in the transformer core due to the alternating magnetic flux, which results in additional energy dissipation in the core.

To minimize hysteresis loss, it is crucial to use magnetic material with a narrow hysteresis loop, and the core of the transformer is built of laminations to minimize eddy current loss. However, the laminated construction introduces air gaps in the core, as shown in Figure 1, and the magnetizing current must create a flux in both the core and the air gaps. In addition to these losses, there is also an I²R loss in the transformer primary windings that must be supplied. As a result, the primary no-load current comprises the magnetizing current and the current required to provide core losses.

 

Figure 1. Energy losses occur in a transformer core because of eddy currents and air gaps. The primary no-load current must supply these core losses as well as create a magnetic flux in the core. Image used courtesy of Amna Ahmad

 

No-Load Phasor Diagram

The transformer's circuit and an approximate phasor diagram under no-load conditions are illustrated in Figure 2. Since the transformer windings are inductive, the input voltage V_P leads the magnetizing current (I_mag) by 90°.

 

(a) Circuit of a transformer with output open-circuited

 

(b) Approximate phasor diagram

Figure 2. Circuit and phasor diagram for a transformer on no-load. The core flux (ɸ) is in phase with the magnetizing current (I_mag). The windings are inductive, so I_mag lags the primary voltage (V_P) by 90°. Image used courtesy of Amna Ahmad

 

The magnetizing current causes the flux (ɸ) to fluctuate, resulting in the flux phasor being in phase with Imag. I_C is the portion of the primary current responsible for providing core losses and a minor power loss in the primary winding. The total power loss is equal to (I_C×V_P), and as a result, I_C is in phase with V_P. The no-load primary current (I_o) is the result of the phasor sum of I_mag and I_C. Given that I_C is typically much smaller than I_mag, the no-load power factor (cos ɸ₁) is quite small. The voltages induced in the secondary and primary windings, E_S and E_P, respectively, lag the flux by 90°, and thus the E_S and E_P phasors are drawn opposite to the V_P phasor. In Figure 2, E_S and E_P are shown as equal voltages (i.e., assuming a 1:1 transformer). An approximation in the phasor diagram of Figure 2 occurs because E_P has been taken as exactly equal and opposite to V_P. In fact, E_P is equal and opposite to the phasor sum of V_P, and the winding voltage drops due to l_o. This becomes more evident when a transformer on-load is considered.

 

No-Load Equivalent Circuit

Figure 3 shows the equivalent circuit for the transformer when it is under no-load condition. In this circuit, an ideal transformer is used to replace the actual transformer, and it has a parallel resistance R_o and inductive reactance X_o on its primary side. R_o is used to represent core losses, and the current I_C that supplies the core losses flows through it. X_o represents a loss-free coil that carries the magnetizing current I_mag. Hence, R_a, X_o, and the ideal transformer together simulate the actual transformer's no-load state.

 

Figure 3. The no-load equivalent circuit of a transformer is made up of an ideal (no loss) transformer with its primary in parallel with a resistance (R_o) representing core losses and an inductive reactance (X_o) to represent the winding inductance. Image used courtesy of Amna Ahmad

 

Transformer On Load

When a load is connected to the secondary winding of a transformer, a current flows through the secondary circuit. As illustrated in Figure 4, the secondary current tends to generate a flux ɸ₂ in the transformer core. To supply the secondary current, the current must flow in the primary winding. The primary current generates a flux ɸ₁ which is exactly equal and opposite to ɸ₂. Thus, ɸ₁ and ɸ₂ cancel each other out, and the core flux remains at the level set up by the magnetizing current.

In Figure 4, it is illustrated that when currents pass through the primary and secondary windings of a transformer, not all of the flux created by the currents flows through the iron core. Some of the flux passes through the air surrounding each coil, which is known as leakage flux. Since the magnetic path through the iron core has a much lower reluctance than the air path around each coil, the amount of leakage flux is usually small. However, the leakage flux still links with the winding turns in each coil, resulting in the generation of EMFs that oppose the current flow through each coil. This is similar to an unwanted inductance connected in series with each winding, which is referred to as leakage inductance.

 

Figure 4. The currents flowing in the primary and secondary windings of a transformer set up core fluxes (ɸ₁ and ɸ₂). Leakage flux also occurs around each winding. Image used courtesy of Amna Ahmad

 

Complete Equivalent Circuit

Figure 5 shows the complete equivalent circuit of a transformer. It includes the leakage inductances of the primary and secondary windings represented by inductive reactance X_P and X_S, respectively, as well as the winding resistances R_P and R_S. R_S and X_S are connected in series with the secondary winding, causing a voltage drop when a load current is present. Similarly, R_P and X_P are connected in series with the primary winding, resulting in voltage drops when a primary current flows. R_o and X_o remain in parallel with the primary to mimic the no-load losses and magnetizing current, respectively.

 

Figure 5. The complete equivalent circuit for a transformer is simply the no-load equivalent circuit (Figure 3) with components R_P, X_P, R_S, and X_S included to represent the winding resistances and leakage inductances. Image used courtesy of Amna Ahmad

 

Phasor Diagram for Secondary Circuit

The phasor diagram for the secondary circuit of a transformer under load is shown in Figure 6. Referring to Figures 5 and 6(a), V_o is the voltage at the transformer output terminals, and I_S is the secondary (load) current. For a load with a lagging phase angle (ɸ_O), the I_S phasor is shown lagging the V_o phasor by angle ɸ_o. The secondary current flows through R_S and X_S and produces voltage drops: I_SR_S and I_SX_S. The resistive voltage drop (I_SR_S) is in phase with I_S, and the inductive voltage drop (I_SX_S) leads the current by 90°, as shown in Figure 6(a).

 

(a) Partial phasor diagram for a secondary circuit with an inductive (lagging ɸ) load

 

 

(b) E_S is the phasor sum of V_o, I_SX_S, and I_SR_S

Figure 6. Phasor diagram for the secondary of a transformer under load. Secondary current I_S lags the output voltage Vo by the load phase angle ɸo. (I_SX_S) and (I_SR_S) are the voltage drops across X_S and R_S. The secondary induced voltage E_S is the phasor sum of Vo, (I_SX_S), and (I_SR_S). Image used courtesy of Amna Ahmad

 

In Figure 6(b), the phasors of V_O, I_S, I_SX_S, and I_SR_S are reproduced from Figure 6(a). The secondary induced voltage E_S (also see Figure 5) is the phasor sum of V_o, I_SR_S, and I_SX_S.

\[E_{S}=\sqrt{(V_{O}\,cos\,cos\,\Phi_{O}+I_{S}R_{S})^{2}+(V_{O}\,sin\,sin\,\Phi_{O}+I_{S}X_{S})^{2}}(1)\]

and the phase angle between E_S and I_S is

\[\Phi_{S}=\Big( \frac{V_{O}sinsin\Phi_{O}+I_{S}X_{S}}{V_{O}+I_{S}X_{S}}\Big)(2)\]

Since the secondary induced voltage always lags 90° behind the core flux ɸ, the core flux phasor may be drawn 90° ahead of E_S.

 

Phasor Diagram for the Primary

The phasor diagram for the transformer primary may be constructed in a similar way to that just discussed for the secondary. In this case, it is necessary to commence with the voltage E_P and current I_P right at the primary winding (see Figure 5). E_P and IP' may be calculated from the knowledge of the turn’s ratio and the secondary current and voltage. The phase angle between E_P and IP' is ɸ_S, the phase angle of the secondary circuit.

Referring to Figure 7(a), the -E_P phasor is first drawn horizontally, and the IP' current phasor is drawn at an angle of ɸ_S lagging -E_P. Note that because E_P is the voltage induced in the primary winding by the changing core flux, it is equal and opposite to the component of the applied voltage at the (ideal)transformer primary winding. Consequently, the applied voltage component is -E_P.

 

(a) IP' lags – E_P by ɸ_S

 

(b) I_P is the phasor sum of IP', I_mag, and I_C

 

(c) V_i is the phasor sum of -E_P, I_PX_P, and I_PR_P

Figure 7. Phasor diagram for the primary of a transformer under load. The primary current (IP') lags the primary voltage (E_P) by the secondary phase angle (ɸ_S). No-load currents I_C and I_mag are also included, and the total primary current (I_P) is the phasor sum of IP', I_C, and I_mag. The supply voltage (V_i) is the phasor sum of -E_P, (I_PX_P), and (I_PR_P). Image used courtesy of Amna Ahmad

 

As well as IP', the no-load current I_o (composed of I_C and I_mag) must be supplied. The actual current drawn from the supply is I_P, which is the phasor sum of IP', I_C, and I_mag as shown in Figures 5 and 7(b).

\[I_{P}\sqrt{(I^{'}_{P}\,cos\,cos\,\Phi_{S}+I_{C})^{2}}+(I^{'}_{P}+I_{mag})^{2}(3)\]

And

\[\Phi_{1}=\Big(\frac{I^{'}_{P}sin\,sin\,\Phi_{S}+I_{mag}}{I^{'}_{P}+I_{C}}\Big)(4)\]

 

The primary current (I_P) produces a voltage drop along R_P and X_P (Figure 5). The I_PR_P phasor is in phase with I_P, and the I_PX_P phasor leads I_P by 90°, as shown in Figure 7(c). The phasor sums of I_PR_P, I_PX_P, and -E_P gives the supply voltage V_i. As illustrated, the primary input phase angle is ɸP, the angle between V_i and I_P. 

Given the secondary load and the parameters of the transformer equivalent circuit, the primary input voltage and current can be calculated using the phasor diagram.

 

Takeaways of Transformer Phasor Diagrams and Equivalent Circuits

The phasor diagram and equivalent circuit of a transformer provide valuable information about the electrical behavior of the transformer under both load and no-load conditions. These concepts help analyze voltage regulation, power losses, and transformer efficiency. Therefore, it is crucial to have a solid understanding of the phasor diagram and equivalent circuit to ensure the safe and efficient operation of transformers.