7.3 Transformer Principles, Construction & Calculations

Transformer Principles, Construction & Calculations

The electrical transformer is arguably the most critical piece of equipment within alternating current (AC) power systems. It is a static electromagnetic device with no moving parts that transfers electrical power between two or more circuits at the same operating frequency (50 Hz in the UK) through mutual electromagnetic induction.

Transformers make modern long-distance electrical transmission economically feasible by stepping generator voltages up to hundreds of thousands of volts (minimizing transmission currents and line losses), before progressively stepping voltages down to safe utilization levels for domestic and commercial consumers.

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Working Principle of the Double-Wound Single-Phase Transformer

A basic single-phase double-wound transformer consists of two electrically independent, insulated copper windings wound on a common high-permeability laminated ferromagnetic steel core:

1. Primary Winding: Connected to the incoming alternating voltage source ($V_p$). It carries an alternating primary current ($I_p$) that sets up an alternating magnetic flux ($\Phi$) throughout the core.
2. Magnetic Core: Provides an unbroken, low-reluctance magnetic circuit that couples and confines the alternating flux, linking the turns of both windings.
3. Secondary Winding: Links with the alternating core flux. By Faraday's Law of Electromagnetic Induction, the continuous reversal of flux induces an alternating electromotive force ($V_s$) across the secondary winding, driving secondary load current ($I_s$) into connected equipment.

          Primary Input (Vp, Ip)               Secondary Output (Vs, Is)
                 |                                         |
                 v                                         v
           ||--[ Winding ]--||    Laminated Core    ||--[ Winding ]--||
           ||  (Np turns)   ||======================||  (Ns turns)   ||
           ||               ||    Mutual Flux Φ     ||               ||
           ||--[ Winding ]--||======================||--[ Winding ]--||
                                (Galvanic Isolation)

Galvanic Isolation

In a standard double-wound transformer, there is zero direct electrical connection between the primary and secondary conductors. Electrical energy is converted into magnetic energy within the core and reconverted into electrical energy in the secondary winding. This complete physical separation is termed galvanic isolation (or electrical separation). It prevents transient DC voltages and high-voltage supply faults from directly propagating to the secondary side, forming the basis of specialized safety systems such as Separated Extra-Low Voltage (SELV) and isolated construction site supplies.

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The Transformer Equation and Turns Ratio

Because the exact same alternating magnetic flux ($\Phi$) links every turn of both the primary and secondary windings through the shared magnetic core, the induced voltage per turn is identical in both windings:

$\frac{V_p}{N_p} = \frac{V_s}{N_s}$

Rearranging this relationship gives the fundamental Transformer Turns Ratio Equation:

$\frac{V_p}{V_s} = \frac{N_p}{N_s} = \frac{I_s}{I_p} = k$

Where:

• $V_p$ = Primary terminal voltage in volts ($\text{V}$)
• $V_s$ = Secondary terminal voltage in volts ($\text{V}$)
• $N_p$ = Number of turns on primary winding
• $N_s$ = Number of turns on secondary winding
• $I_p$ = Primary full-load current in amperes ($\text{A}$)
• $I_s$ = Secondary full-load current in amperes ($\text{A}$)
• $k$ = Transformation ratio

Step-Up versus Step-Down Transformers

Transformers are classified according to how they modify voltage levels:

• Step-Up Transformer ($N_s > N_p$):
  • Secondary turns exceed primary turns.
  • Secondary voltage is greater than primary voltage ($V_s > V_p$).
  • Secondary current is lower than primary current ($I_s < I_p$).
  • Example: Power station generator transformers stepping up from $25\text{ kV}$ to $400\text{ kV}$ for high-voltage grid transmission.
• Step-Down Transformer ($N_s < N_p$):
  • Secondary turns are fewer than primary turns.
  • Secondary voltage is lower than primary voltage ($V_s < V_p$).
  • Secondary current is higher than primary current ($I_s > I_p$).
  • Example: Substation distribution transformers stepping $11\text{ kV}$ down to $400\text{ V} / 230\text{ V}$, or domestic bell transformers stepping $230\text{ V}$ down to $12\text{ V}$.

The Inverse Current Relationship and Power Conservation

Assuming an ideal transformer with 100% efficiency, input electrical power equals output electrical power:

$S_{\text{input}} = S_{\text{output}} \implies V_p \times I_p = V_s \times I_s$

This conservation principle explains the inverse current relationship ($\frac{V_p}{V_s} = \frac{I_s}{I_p}$):

• When a transformer steps voltage up by a factor of 10, current is simultaneously stepped down by a factor of 10.
• When a transformer steps voltage down by a factor of 10, current is simultaneously stepped up by a factor of 10.

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Transformer Power Rating: Volt-Amperes (VA and kVA)

The capacity of an electrical transformer is universally quoted in Volt-Amperes (VA) or kilo-Volt-Amperes (kVA), representing Apparent Power ($S$):

$S = V_p \times I_p = V_s \times I_s$

$\text{kVA} = \frac{V \times I}{1,000}$

Why Transformers Are Rated in kVA, Not Kilowatts (kW)

In electrical engineering exams, candidates are frequently asked why transformers are rated in kVA rather than kW:

1. Unknown Load Power Factor: Active (true) power in Watts depends on the circuit power factor ($P = V \times I \times \cos\phi$). A transformer manufacturer builds the machine without knowing what type of load the end user will connect (e.g. resistive heaters with $\cos\phi = 1.0$, induction motors with $\cos\phi = 0.75$, or electronic power supplies).
2. Thermal Heating Limits: Internal winding heating is dictated by Joule losses ($I^2R$), which depend solely on current ($I$), regardless of phase displacement. Core magnetic saturation and iron loss depend solely on voltage ($V$). Therefore, the physical thermal rating of the transformer depends strictly on the product of $V$ and $I$ (Volt-Amperes), not real power (Watts).

Step-by-Step Worked Calculations

Worked Example 1: Turns, Voltage, and Current Calculations

Problem: A single-phase double-wound step-down transformer is connected to a $230\text{ V}$, $50\text{ Hz}$ supply. The primary winding has $600\text{ turns}$ and the secondary winding has $30\text{ turns}$. If the secondary feeds a resistive heating load drawing $20\text{ A}$, determine:

1. The secondary terminal voltage ($V_s$)
2. The primary full-load current ($I_p$)
3. The transformer apparent power rating in $\text{kVA}$

Solution:

1. Calculate secondary voltage: $\frac{V_p}{V_s} = \frac{N_p}{N_s} \implies V_s = V_p \times \frac{N_s}{N_p} = 230\text{ V} \times \frac{30}{600} = 230 \times 0.05 = 11.5\text{ V}$
2. Calculate primary current using inverse current ratio: $\frac{I_s}{I_p} = \frac{N_p}{N_s} \implies I_p = I_s \times \frac{N_s}{N_p} = 20\text{ A} \times \frac{30}{600} = 20 \times 0.05 = 1.0\text{ A}$
3. Calculate apparent power rating: $S = V_s \times I_s = 11.5\text{ V} \times 20\text{ A} = 230\text{ VA} = 0.23\text{ kVA}$ (Check with primary: $S = V_p \times I_p = 230\text{ V} \times 1.0\text{ A} = 230\text{ VA}$—identical values confirm calculation accuracy.)

Worked Example 2: Determining Primary Turn Count

Problem: A $15\text{ kVA}$ site transformer steps down $400\text{ V}$ single-phase supply to $110\text{ V}$ Centre-Tapped-to-Earth. The secondary winding contains $88\text{ turns}$. Calculate the number of turns required on the primary winding.

Solution:

1. Transpose the turns ratio formula for $N_p$: $\frac{N_p}{N_s} = \frac{V_p}{V_s} \implies N_p = N_s \times \frac{V_p}{V_s}$
2. Substitute values: $N_p = 88 \times \frac{400\text{ V}}{110\text{ V}} = 88 \times 3.6364 = 320\text{ turns}$

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Transformer Construction: Cores and Windings

Practical transformers are engineered to maximize mutual magnetic coupling while minimizing energy losses and mechanical hum.

Core-Type versus Shell-Type Construction

1. Core-Type Transformer: Windings are arranged in concentric cylindrical layers surrounding two vertical limbs of a rectangular magnetic core. The low-voltage winding is placed closest to the iron core to ease insulation requirements, with the high-voltage winding wound over it. Core-type units are common in high-voltage transmission equipment.
2. Shell-Type Transformer: The magnetic iron core completely encloses and surrounds the copper windings like an external shell. The core provides two parallel return paths for magnetic flux, yielding superior magnetic confinement and reduced flux leakage. Shell-type designs are common in commercial low-voltage distribution transformers and power supply units.

Core Material and Lamination Geometry

Transformer cores are constructed from thin stampings of high-permeability silicon alloy steel (cold-rolled grain-oriented, CRGO), coated with insulating varnish to minimize eddy current and hysteresis losses. Stampings are produced in standardized shapes:

• 'E' and 'I' Laminations: The central limb of the 'E' stamping carries twice the magnetic flux of the outer limbs. The 'I' strip bridges the top.
• Interleaved Assembly: Successive layers of laminations are reversed and alternated during core stacking to eliminate continuous air gaps at butt joints, significantly reducing magnetic reluctance.

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Transformer Losses and Efficiency

Because transformers have no rotating parts, they experience zero mechanical friction or windage losses. Total internal energy losses consist purely of electrical and magnetic losses:

1. Copper Losses ($P_{\text{cu}}$) — Variable Losses

Caused by the electrical resistance of the primary and secondary copper windings carrying current ($I^2R$ Joule heating).

• Dependent entirely on load current: At zero load ($I = 0$), copper loss is negligible. At full load, copper loss reaches maximum ($I_{\text{fl}}^2 R$).
• At half-load ($50\% I$), copper loss drops to $(\frac{1}{2})^2 = \frac{1}{4}$ (25%) of its full-load value.

2. Iron / Core Losses ($P_i$) — Constant Losses

Caused by magnetic hysteresis and eddy currents within the steel core laminations.

• Independent of load: Because applied voltage and frequency remain constant, core flux density remains virtually identical from zero load to full load. Iron losses remain constant 24 hours a day whenever the transformer is energized.

Efficiency Formula ($\eta$)

$\eta = \frac{\text{Output Power}}{\text{Input Power}} \times 100\% = \frac{P_{\text{out}}}{P_{\text{out}} + P_{\text{losses}}} \times 100\% = \frac{V_s I_s \cos\phi}{V_s I_s \cos\phi + P_{\text{cu}} + P_i} \times 100\%$

Because there are no rotating mechanical parts, modern power transformers achieve extraordinary efficiencies ranging between $95\%$ and $99\%$.

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Autotransformers: Construction, Advantages & Critical Safety Hazards

An autotransformer differs fundamentally from a double-wound transformer: it consists of only one continuous winding on a laminated iron core, which acts as both primary and secondary.

                    Primary Supply (230 V)
             L ------------------------------.
                                             |
                                         [ Winding ]
                                             |-------> Tapped Secondary Output (110 V)
                                         [ Winding ]
                                             |
             N ------------------------------+-------> Secondary Common Neutral

Operational Advantages

• Reduced Copper and Size: Because part of the winding carries the difference between primary and secondary currents ($I_s - I_p$), lighter copper conductors and smaller cores can be used.
• Higher Efficiency & Lower Cost: Lower internal resistance minimizes $I^2R$ copper losses, yielding lower purchase and running costs.
• Used commonly in motor soft-starters (Korndorfer starters) and laboratory variable AC power supplies (Variacs).

Critical Safety Hazard: Lack of Galvanic Isolation

• Broken Common Neutral Risk: If the shared neutral connection becomes loose, high-resistance, or severed, the full primary phase voltage (e.g. $230\text{ V}$) appears directly on the low-voltage secondary terminals, creating an immediate, lethal electric shock risk to operators expecting extra-low or reduced voltage.
• Prohibitions in BS 7671: Because the source for these measures must be a safety isolating (double-wound) transformer or equivalent, an autotransformer cannot be used where electrical separation is required for protection:
  • Separated Extra-Low Voltage (SELV) circuits (e.g. bathroom lighting, pool equipment).
  • Protective Extra-Low Voltage (PELV) circuits.
  • Bathroom shaver supply units conforming to BS EN 61558-2-5 (which mandate a double-wound isolating transformer with an earthed metal screen).
  • Construction site temporary supplies to BS 7375 / BS 7671 Section 704.

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Practical Electrotechnical Applications in UK Installations