Transformers and rotating machines notes — Unit 4

A transformer is a static (non-moving) electrical device that transfers alternating current (AC) electrical energy from one circuit to another circuit at the same frequency, but usually at changed values of voltage and current.

⇒Principle of Operation

It works on the principle of Faraday's Law of Electromagnetic Induction, specifically Mutual Induction between two magnetically coupled coils.

• Step-up transformer: Increases voltage, decreases current.
• Step-down transformer: Decreases voltage, increases current.

A transformer consists of two primary components: the magnetic core and the electrical windings.

• Magnetic Core: Made of highly permeable material (Silicon Steel) to provide a low-reluctance path for the magnetic flux. It is laminated (made of thin sheets coated with varnish) to severely reduce eddy current losses.
• Windings: Two electrically isolated coils of insulated copper wire. The coil connected to the AC supply is the Primary Winding, and the one connected to the load is the Secondary Winding.

Based on core arrangement, transformers are broadly classified as Core-Type (windings surround the core) and Shell-Type (core surrounds the windings).

• 1. An alternating voltage $V_1$ is applied to the primary winding (which has $N_1$ turns).
• 2. This causes an alternating current $I_1$ to flow, generating an alternating magnetic flux ($\Phi$) in the core.
• 3. This flux completely links with both the primary and the secondary winding ($N_2$ turns).
• 4. The changing flux induces a mutually induced EMF ($E_2$) in the secondary winding.
• 5. If a load is connected, a secondary current $I_2$ flows, delivering power.

Crucially, a self-induced EMF ($E_1$) is also generated in the primary, which opposes the supply voltage $V_1$ (Lenz's Law).

Let the alternating flux be $\Phi = \Phi_m \sin(\omega t)$.

By Faraday's law, the induced EMF in a coil of $N$ turns is $e = -N(d\Phi/dt)$.

e = -N * d/dt [Φ_m sin(ωt)]
e = -N * Φ_m * ω * cos(ωt)
e = N * Φ_m * ω * sin(ωt - 90°)

The maximum value of EMF is $E_{max} = N \Phi_m \omega = N \Phi_m (2\pi f)$.

The RMS value of EMF is $E = E_{max} / \sqrt{2}$.

E = (2π / √2) f N Φ_m
E = 4.44 f N Φ_m

Applying the EMF equation to the primary ($E_1$) and secondary ($E_2$):

E₁ = 4.44 f N₁ Φ_m
E₂ = 4.44 f N₂ Φ_m

Dividing the two equations gives the Transformation Ratio ($K$):

K = E₂ / E₁ = N₂ / N₁

For an ideal transformer, Input Power = Output Power ($V_1 I_1 = V_2 I_2$).

K = V₂/V₁ = N₂/N₁ = I₁/I₂

Notice that current is inversely proportional to voltage.

An ideal transformer has no winding resistance, no magnetic leakage flux, and no core losses.

When the secondary is open (No-Load), the primary draws a very small current called the Magnetizing Current ($I_\mu$) just to establish the flux.

⇒Phasor Diagram

• Flux $\Phi$ is taken as the reference (0°).
• Since $e = -N(d\Phi/dt)$, the induced EMFs $E_1$ and $E_2$ lag the flux by 90°.
• The applied voltage $V_1$ perfectly opposes $E_1$, so $V_1 = -E_1$ (points at +90°).
• The magnetizing current $I_\mu$ produces the flux, so it is strictly in-phase with $\Phi$.

A real transformer has hysteresis and eddy current losses in the core, meaning it consumes some active power even on no-load. The total no-load current is $I_0$.

$I_0$ splits into two components:

• Magnetizing Component ($I_\mu = I_0 \sin\phi_0$): Creates the flux. Lags $V_1$ by 90°.
• Core Loss Component ($I_w = I_0 \cos\phi_0$): Supplies the iron losses. In-phase with $V_1$.

I₀ = √(I_w² + I_μ²)
No-load Power P₀ = V₁ I₀ cosφ₀

The no-load current $I_0$ is very small, typically 2% to 5% of full-load current.

When a load is connected to the secondary, a current $I_2$ flows. By Lenz's law, $I_2$ creates its own flux ($\Phi_2$) that opposes the main flux $\Phi$.

This briefly weakens the main flux, reducing $E_1$, which allows more current to flow from the supply. The primary draws an extra current $I_1'$ to create flux to exactly cancel out $\Phi_2$ and restore the main flux.

N₁ I₁' = N₂ I₂
I₁' = I₂ (N₂/N₁) = K·I₂

Total primary current under load: $\mathbf{I}_1 = \mathbf{I}_0 + \mathbf{I}_1'$ (phasor sum).

To make calculations easy, we represent the magnetic coupling of a transformer with an equivalent electrical circuit on one side (referring everything to the primary side).

Resistance and Reactance can be "shifted" from secondary to primary by dividing by $K^2$.

R₂' = R₂ / K²
X₂' = X₂ / K²

The total equivalent resistance referred to the primary is $R_{01} = R_1 + R_2'$. The equivalent reactance is $X_{01} = X_1 + X_2'$.

This removes the physical separation and lets us solve it like a standard AC series circuit.

Since it is a static device, there are no friction or windage losses. The losses are strictly electrical and magnetic.

⇒1. Core (Iron) Losses ($P_i$)

Occur in the magnetic core due to the alternating flux. Consists of Hysteresis loss and Eddy Current loss. Because flux remains constant from no-load to full-load, Core loss is CONSTANT.

⇒2. Copper Losses ($P_{cu}$)

Occur in the primary and secondary windings due to their ohmic resistance ($I_1^2 R_1 + I_2^2 R_2$). Copper loss depends heavily on the load current. Copper loss is VARIABLE.

Copper loss at fractional load x = x² * P_{cu(full load)}

Efficiency is the ratio of Output Power to Input Power.

η = Output / (Output + Losses)
η = (V₂ I₂ cosφ) / (V₂ I₂ cosφ + P_i + P_{cu})

⇒Condition for Maximum Efficiency

By taking the derivative of efficiency with respect to load current ($I_2$) and setting it to zero, we find the critical condition:

Copper Loss = Iron Loss
P_{cu} = P_i

A transformer operates at maximum efficiency at the specific load where its variable copper loss exactly equals its constant iron loss.

When a transformer is loaded, the secondary terminal voltage ($V_2$) drops due to voltage drops in the internal resistance and reactance of the windings ($I_2 R_{02}$ and $I_2 X_{02}$).

Voltage Regulation is the percentage change in secondary voltage from No-Load to Full-Load.

% Regulation = (E₂ - V₂) / E₂ * 100
Or using equivalent parameters:
% Reg = (I₂ R_{02} cosφ ± I₂ X_{02} sinφ) / E₂ * 100
(+ for lagging load, - for leading load)

An ideal transformer has 0% regulation (voltage never drops). Thus, a lower percentage is better.

Instead of fully loading a massive transformer (which wastes huge amounts of energy), we perform two tests to find its parameters and efficiency.

⇒Open Circuit (OC) Test

Conducted on the Low Voltage (LV) side with the HV side open. Voltage applied is rated. Current drawn is $I_0$ (very small). The wattmeter reads only the Core Loss ($P_i$). It determines $R_0$ and $X_m$.

⇒Short Circuit (SC) Test

Conducted on the High Voltage (HV) side with LV shorted. A tiny voltage (5-10% of rated) is applied until full load current flows. Core loss is negligible at such low voltage. The wattmeter reads only the Full Load Copper Loss ($P_{cu}$). It determines $R_{01}$ and $X_{01}$.

A DC machine can operate as either a Generator (converts mechanical energy to DC electrical energy) or a Motor (converts DC electrical energy to mechanical energy). The construction is identical.

⇒Main Parts:

• Stator/Yoke: The stationary outer frame that provides physical support and a path for magnetic flux.
• Field Poles: Electromagnets on the stator that create the main magnetic field.
• Armature (Rotor): The rotating cylindrical cylinder containing copper coils where EMF is induced.
• Commutator: A mechanical rectifier. It converts the alternating AC induced in the armature coils into unidirectional DC at the external brushes.

Principle: Faraday's law of electromagnetic induction. When an armature coil is physically rotated in a magnetic field, it cuts flux, inducing a dynamically generated AC EMF, which the commutator converts to DC.

⇒EMF Equation

E_g = (Φ Z N P) / (60 A)

Where:
Φ = Flux per pole
Z = Total number of armature conductors
N = Speed in RPM
P = Number of poles
A = Number of parallel paths (A=2 for Wave winding, A=P for Lap winding)

Principle: When a current-carrying conductor is placed in a magnetic field, it experiences a mechanical Lorentz force. The direction is given by Fleming's Left Hand Rule.

⇒Back EMF ($E_b$)

Once the motor starts rotating, its conductors cut the magnetic flux. By Faraday's law, an EMF is induced. By Lenz's law, this induced EMF opposes the applied voltage $V$.

V = E_b + I_a R_a
I_a = (V - E_b) / R_a

Back EMF makes the DC motor self-regulating. When load increases, speed drops slightly, $E_b$ drops, so armature current $I_a$ automatically increases to provide more torque.

The most robust and widely used motor in industry. It has two main parts: a stationary Stator and a rotating Rotor (usually a Squirrel Cage design).

⇒Rotating Magnetic Field (RMF)

When a 3-phase supply is given to the 3-phase stator windings (separated by 120°), it sets up a magnetic field of constant magnitude ($1.5 \Phi_m$) that physically rotates around the stator at Synchronous Speed ($N_s$).

N_s = (120 f) / P
Where f = supply frequency, P = number of poles.

The RMF sweeps across the stationary short-circuited rotor bars. By Faraday's law, an EMF is induced in the rotor. A heavy rotor current flows.

This rotor current interacts with the stator's RMF to produce a mechanical torque.

By Lenz's law, the rotor will try to catch up to the RMF to reduce the relative speed (which is the cause of the induction). The rotor begins to spin in the same direction as the RMF.

The difference between the synchronous speed ($N_s$) and the actual rotor speed ($N_r$) is called Slip Speed. It is usually expressed as a percentage called Slip ($s$).

s = (N_s - N_r) / N_s
% slip = s * 100

⇒Rotor Frequency ($f'$)

The frequency of the induced EMF in the rotor depends on the relative speed (slip speed).

f' = s * f
(Where f is stator supply frequency)

At standstill, slip $s=1$, so rotor frequency equals supply frequency. At running speeds (slip 2-5%), rotor frequency is very low (e.g., 2 Hz).

⇒Unit 4 Mastery

• Derive the EMF equation of a transformer.
• Draw the phasor diagram of a practical transformer on No-Load.
• Prove the condition for maximum efficiency in a transformer.
• What is the significance of OC and SC tests?
• Explain the function of the Commutator in a DC machine.
• Why does a 3-phase induction motor never run at synchronous speed?
• Define Slip and write the formula relating rotor frequency to supply frequency.