EMF Equation of Transformer — Derivation, Formula & Numerical Examples
Introduction
The EMF equation of a transformer is a fundamental expression that relates the induced electromotive force (EMF) in a winding to the flux, frequency, and number of turns. When a sinusoidal AC voltage is applied to the primary winding, an alternating magnetic flux is established in the iron core. This flux links both primary and secondary windings, inducing an EMF in each according to Faraday's law of electromagnetic induction.
Understanding this equation is essential for transformer design, as it determines the voltage rating, core cross-section, and number of turns required for a given application.
Derivation of EMF Equation
Let us define the following quantities:
- Φₘ = Maximum value of magnetic flux in the core (Weber)
- f = Supply frequency (Hz)
- N₁ = Number of turns in the primary winding
- N₂ = Number of turns in the secondary winding
- T = Time period of one cycle = 1/f
The flux in the core varies sinusoidally and can be expressed as:
By Faraday's law, the instantaneous EMF induced in a winding of N turns is:
Substituting Φ = Φₘ sin(ωt):
e = −N × ω × Φₘ × cos(ωt)
e = N × ω × Φₘ × sin(ωt − π/2)
The maximum (peak) value of induced EMF is:
Final EMF Equation (E = 4.44fΦₘN)
For a sinusoidal waveform, the RMS value is related to the peak value by:
E = (N × 2πf × Φₘ) / √2
E = (2π / √2) × f × Φₘ × N
E = 4.44 × f × Φₘ × N
This is the EMF equation of a transformer. The constant 4.44 comes from (2π/√2) = 4.4429. The induced EMF depends on three factors: supply frequency, maximum flux in the core, and number of turns.
EMF Equation for Primary & Secondary
Since the same mutual flux Φₘ links both windings:
E₂ = 4.44 × f × Φₘ × N₂ (Secondary EMF)
Where Φₘ = Bₘ × A (maximum flux density × core cross-sectional area).
Voltage Transformation Ratio
Dividing the two equations:
Solved Numerical Examples
Example 1: A single-phase transformer has 480 primary turns and 90 secondary turns. The maximum flux in the core is 60 mWb. Find E₁ and E₂ at 50 Hz.
E₂ = 4.44 × 50 × 0.06 × 90 = 1,199 V
Example 2: A 25 kVA, 2000/200 V, 50 Hz transformer has a core area of 120 cm². Find the maximum flux density and number of turns on each winding.
Φₘ = Bₘ × A = 1.2 × 120×10⁻⁴ = 0.0144 Wb
N₁ = E₁ / (4.44 × f × Φₘ) = 2000 / (4.44 × 50 × 0.0144) = 625 turns
N₂ = E₂ / (4.44 × f × Φₘ) = 200 / (4.44 × 50 × 0.0144) = 63 turns
Example 3: A transformer operates at 60 Hz with Φₘ = 0.05 Wb. If the secondary has 100 turns, find the secondary EMF.
Key Points to Remember
Practical Applications of the EMF Equation
- Core Design: Determines the required cross-sectional area of the core (A = Φₘ/Bₘ) for a given voltage rating.
- Turns Calculation: Engineers use E = 4.44fΦₘN to calculate the exact number of turns needed for desired voltage levels.
- Frequency Conversion: When transformers operate at different frequencies (50 Hz vs 60 Hz), the EMF equation helps recalculate flux and turns.
- Volts per Turn: E/N = 4.44fΦₘ is constant for both windings — a useful design parameter.
- Saturation Check: If applied voltage increases beyond design, Φₘ increases causing core saturation and excessive magnetizing current.
Frequently Asked Questions
Q1: What is the EMF equation of a transformer?
The EMF equation of a transformer is E = 4.44 × f × Φₘ × N, where f is frequency in Hz, Φₘ is maximum flux in Weber, and N is the number of turns. It gives the RMS value of induced EMF in any winding.
Q2: Why is the constant 4.44 used in the transformer EMF equation?
The constant 4.44 comes from the mathematical derivation: 2π/√2 = 4.4429 ≈ 4.44. The 2π converts frequency to angular velocity, and dividing by √2 converts peak EMF to RMS value for a sinusoidal waveform.
Q3: What happens if frequency increases while voltage remains constant?
From E = 4.44fΦₘN, if E is constant and f increases, then Φₘ must decrease. This means the core operates at lower flux density, reducing iron losses. This is why aircraft transformers use 400 Hz — smaller cores with less iron loss.
Q4: Does the EMF equation apply to both ideal and practical transformers?
Yes, the EMF equation applies to both. In a practical transformer, the applied voltage V₁ differs slightly from E₁ due to primary winding resistance and leakage reactance drops (V₁ = E₁ + I₁R₁ + jI₁X₁).
Q5: How is the EMF equation different for a square wave supply?
For a square wave, the form factor is 1.0 (instead of 1.11 for sine wave), so the equation becomes E = 4.0 × f × Φₘ × N. The constant changes from 4.44 to 4.0.
No comments:
Post a Comment