Introduction

Almost everything in electrical engineering rests on two ideas: converting energy between electrical and mechanical forms, and moving electrical energy between voltage levels. DC machines do the first with direct current. Transformers do the second with alternating current. They look very different, but both are built on the same physics: magnetic fields, induced voltages, and forces on current-carrying conductors.

This article starts with that shared physics, then covers DC machines (construction, operation, types, characteristics, control, testing) and transformers (construction, theory, equivalent circuits, tests, three-phase connections, special types). Worked examples and a formula summary come at the end.

---

Part 1: The Physics Behind Both Devices

1.1 Magnetic fields and magnetic circuits

A current-carrying conductor produces a magnetic field. Winding it into a coil around an iron core concentrates the field. Key quantities:

• Flux (Φ), in webers (Wb): the total magnetic field passing through an area.
• Flux density (B), in tesla (T): flux per unit area, B = Φ/A.
• Magnetomotive force (MMF), in ampere-turns: MMF = N·I.
• Reluctance (S): opposition to flux, S = l/(μA). This is the magnetic analogue of resistance.

A magnetic circuit obeys Φ = MMF / S, analogous to Ohm’s law. Ferromagnetic materials (silicon steel, cast steel) have high permeability (μ), so they carry flux with very little MMF, which is why every machine and transformer has an iron path.

1.2 Faraday’s and Lenz’s laws

Faraday’s law: a changing flux linkage induces an EMF:

e = −N · dΦ/dt

Lenz’s law: the induced EMF opposes the change that created it (the minus sign). This one principle explains back-EMF in motors, the opposing flux in a loaded transformer, and much else.

There are two ways to change flux linkage:

1. Statically induced EMF: the flux varies with time while the coil stays still. This is the transformer.
2. Dynamically induced (motional) EMF: a conductor moves through a steady field. For a conductor of length l moving at velocity v perpendicular to field B: e = B·l·v. This is the DC generator.

1.3 The force on a current-carrying conductor

A conductor carrying current I in field B experiences a force:

F = B · I · l

(with direction from Fleming’s left-hand rule). This is the DC motor principle. Fleming’s right-hand rule gives the direction of induced EMF in a generator.

1.4 Core losses

Iron cores in changing fields lose energy in two ways:

• Hysteresis loss: energy spent re-aligning magnetic domains each cycle. It is proportional to frequency and to B^1.6–2 (Steinmetz).
• Eddy-current loss: circulating currents induced in the core itself. It is proportional to f²B²t², where t is the lamination thickness. This is why cores are built from thin insulated laminations (0.27–0.5 mm), and why higher-resistivity silicon steel is used.

Together these form the iron (core) loss, which appears in both DC machines (in the armature, which sees alternating flux as it rotates) and transformers (in the core).

---

Part 2: DC Machines

2.1 What is a DC machine?

A DC machine is a rotating electromechanical converter that works with direct current at its terminals.

• As a generator, it converts mechanical energy to DC electrical energy.
• As a motor, it converts DC electrical energy to mechanical energy.

The same physical machine can do either. That reversibility follows directly from the physics above.

2.2 Construction

Stator (stationary parts)

• Yoke (frame): the outer cylindrical shell, made of cast iron or rolled steel. It gives mechanical support and carries the return path for flux.
• Main poles: laminated steel poles bolted to the yoke. The flared end (pole shoe) spreads flux evenly over the air gap.
• Field winding: coils around the poles that produce the main flux when energised. (Small machines may use permanent magnets instead.)
• Interpoles (commutating poles): small poles between main poles, wound in series with the armature, that improve commutation.
• Compensating windings: in large or heavy-duty machines, windings in slots in the pole faces that cancel armature reaction.

Rotor (armature)

• Armature core: a laminated steel cylinder with slots on the surface, mounted on the shaft.
• Armature winding: copper coils placed in the slots, where EMF is induced (generator) or torque is produced (motor).
• Commutator: a cylinder of wedge-shaped, mutually insulated copper segments, one pair per coil, mounted on the shaft.
• Brushes and brush gear: carbon or graphite blocks that press against the commutator and connect the rotating armature to the external circuit.

Air gap: the small clearance between poles and armature. It is kept small to minimise reluctance.

2.3 How a DC generator works

Consider a single loop rotating in a magnetic field. The EMF induced in the loop is naturally alternating: each conductor alternately passes under north and south poles. The commutator acts as a mechanical rectifier. Each time the loop’s EMF reverses, the commutator segments swap which brush they touch, so the external circuit always sees current in one direction.

With many coils and many commutator segments, the output ripple becomes very small, giving a nearly steady DC voltage.

2.4 How a DC motor works

Supply DC to the armature through the brushes. Current flows in the armature conductors, which sit in the field flux. Each conductor feels a force F = BIl, and the forces on opposite sides of a coil create a torque. The commutator reverses the current in each coil as it passes the neutral zone, so the torque always acts in the same direction and the rotor turns continuously.

Back EMF (Eb): as the armature rotates, its conductors cut flux and induce an EMF. By Lenz’s law this opposes the applied voltage, so it is called the back or counter EMF. It is the central concept in motor behaviour:

V = Eb + Ia·Ra (motor)
E = V + Ia·Ra (generator)

Back EMF acts as the motor’s self-regulator: if the load increases and the motor slows, Eb drops, more current flows, and torque rises to meet the load.

2.5 Armature windings

Armature coils can be connected in two main ways:

• Lap winding: the number of parallel paths A = P (number of poles). It suits high current, low voltage machines.
• Wave winding: A = 2 regardless of poles. It suits high voltage, low current machines.

2.6 The EMF and torque equations

EMF equation (both modes):

E = (P · Φ · Z · N) / (60 · A) volts

where P = poles, Φ = flux per pole (Wb), Z = total armature conductors, N = speed (rpm), A = parallel paths. For a given machine this reduces to E = Kₐ·Φ·ω.

Torque equation:

T = (P · Φ · Z · Iₐ) / (2π · A) N·m, or T = Kₐ·Φ·Iₐ

Power developed: P_dev = Eb·Ia = T·ω

Two consequences follow directly:

• Speed ∝ Eb/Φ (so weakening the field raises the speed).
• Torque ∝ Φ·Ia (torque comes from flux and armature current).

2.7 Armature reaction

When the armature carries current, it produces its own MMF at right angles (cross-magnetising) to the main field. Effects:

1. Distortion of the main flux: it strengthens under one pole tip and weakens under the other.
2. Shifting of the magnetic neutral axis, so brushes must be moved or commutation suffers.
3. Net weakening of flux at high saturation, which reduces generated voltage or raises motor speed.

Remedies: interpoles, compensating windings, brush shifting (small machines), and pole-face designs with higher reluctance at the tips.

2.8 Commutation

Commutation is the reversal of current in a coil as it passes under a brush. Ideally the current falls linearly from +I to −I during the short time the coil is shorted by the brush. In practice, coil inductance (reactance voltage) opposes the reversal, causing sparking.

Improvement methods:

• Resistance commutation: high-resistance carbon brushes.
• EMF commutation: interpoles that generate a voltage in the shorted coil to assist the current reversal.

Interpole polarity: in a generator, the interpole has the same polarity as the next main pole in the direction of rotation. In a motor, it has the same polarity as the previous main pole.

2.9 Types of DC machines by excitation

• Separately excited: field supplied from an independent source.
• Self-excited, in which the machine supplies its own field:
  • Series: field winding (few turns of thick wire) in series with the armature.
  • Shunt: field winding (many turns of thin wire) in parallel with the armature.
  • Compound: both series and shunt fields.
    • Cumulative: series flux aids shunt flux.
    • Differential: series flux opposes shunt flux.
    • Long shunt / short shunt: depends on whether the shunt field is connected across the armature plus series field, or across the armature alone.

2.10 DC generator characteristics

Open-circuit characteristic (OCC)/magnetisation curve: generated EMF vs. field current at constant speed. It rises linearly and then bends over as the iron saturates. A small residual EMF exists at zero field current because of retentivity.

Voltage build-up in a self-excited shunt generator requires:

1. Residual magnetism in the poles.
2. Field winding connected so its MMF aids the residual flux.
3. Field circuit resistance below the critical resistance (the tangent to the OCC).
4. Speed above the critical speed.

External characteristic (V vs. load current):

• Separately excited: slight droop from armature resistance drop and armature reaction.
• Shunt: greater droop, because falling terminal voltage also reduces field current. Beyond a maximum current the voltage collapses.
• Series: voltage rises with load (up to saturation), so it is rarely used as a generator.
• Cumulative compound: the most useful. It can be over-compounded (voltage rises with load, compensating line drop), flat-compounded, or under-compounded.
• Differential compound: voltage falls sharply, suiting constant-current applications such as arc welding.

2.11 DC motor characteristics

The three characteristics of interest are torque vs. armature current (T–Ia), speed vs. armature current (N–Ia), and speed vs. torque (N–T).

Shunt motor

• Flux is nearly constant, so T ∝ Ia (a straight line).
• Speed drops only slightly with load (typically 3–5%), so it is a near-constant-speed motor.
• Applications: lathes, fans, blowers, pumps, machine tools.

Series motor

• Flux ∝ Ia (below saturation), so T ∝ Ia²: very high starting torque.
• Speed is inversely related to load: high speed at light load, low speed at heavy load.
• Never run a series motor unloaded: speed can rise dangerously high (“runaway”).
• Applications: traction (trains, trams, electric vehicles historically), cranes, hoists, starter motors.

Compound motor

• Cumulative: good starting torque plus reasonably steady speed. It is used for presses, shears, and elevators.
• Differential: unstable and rarely used.

Motor	Starting torque	Speed regulation	Typical use
Shunt	Moderate	Very good (near-constant)	Machine tools, pumps
Series	Very high	Poor (varies widely)	Traction, cranes
Cumulative compound	High	Moderate	Presses, elevators
Permanent magnet	Moderate	Good	Small drives, automotive, robotics

2.12 Starting a DC motor

At standstill, Eb = 0, so the armature current would be Ia = V/Ra. With Ra typically well under an ohm, that is a huge surge (10–20× rated or more), enough to damage the commutator and windings. Hence a starter is needed: a variable resistance in series with the armature that is progressively cut out as the motor accelerates.

• Three-point starter: connects the shunt field in series with the holding coil (the no-volt coil), so it releases if the supply fails and protects against overload. Its weakness is that when the field is weakened for speed control, the holding coil current also drops.
• Four-point starter: the holding coil is on a separate branch, which fixes that problem.

Modern drives replace resistor starters with electronically controlled voltage ramps.

2.13 Speed control of DC motors

From N ∝ (V − Ia·Ra)/Φ, there are three ways to control speed:

1. Armature voltage control (constant-torque region, speed below base speed):
   • Series resistance in the armature circuit (simple, but wasteful).
   • Variable armature voltage (Ward-Leonard system, or thyristor/chopper drives).
2. Field flux control (constant-power region, speed above base speed): a rheostat in the shunt field circuit. Reducing flux raises speed but reduces available torque.
3. Ward-Leonard system: a motor-generator set that gives smooth, wide-range, four-quadrant control. It was historically dominant for lifts, rolling mills, and mine hoists. It has largely been replaced by solid-state converters.

Modern approach: controlled rectifiers (thyristors) from an AC source, or DC choppers (MOSFET/IGBT) from a DC source, adjust the armature voltage, giving efficient control with fast response.

2.14 Electric braking

• Rheostatic (dynamic) braking: the motor is disconnected from the supply and acts as a generator, dumping energy into a resistor.
• Regenerative braking: the motor acts as a generator and returns energy to the supply. Used in electric trains and EVs.
• Plugging (reverse-current braking): reverse the armature supply connections while the machine is still turning. This gives fast stopping but high currents and heat.

2.15 Losses and efficiency

Loss category	Description	Varies with
Copper losses	Armature (Ia²Ra), field (If²Rf), interpole/series winding	Load (armature), constant (shunt field)
Iron losses	Hysteresis + eddy currents in armature core	Speed and flux (nearly constant)
Mechanical losses	Bearing friction, brush friction, windage	Speed
Stray load losses	Non-uniform current distribution, etc.	Load

Maximum efficiency occurs when variable losses equal constant losses. Typical efficiencies range from 75–80% (small machines) to over 95% (large machines).

2.16 Testing DC machines

• Brake test: direct loading of small motors. It gives accurate efficiency but is impractical for large machines.
• Swinburne’s (no-load) test: run the machine as a motor at no-load and measure input. This gives constant losses, from which efficiency at any load can be calculated. Limits: cannot be used for series machines, and it ignores the change in armature reaction and iron loss under load.
• Hopkinson’s (back-to-back) test: two identical machines are coupled mechanically and electrically, one motoring and one generating, so the supply provides only the losses. It gives efficiency at full load with low power consumption, but needs two identical machines.
• Retardation (running-down) test: measure how quickly the armature decelerates, to separate iron and mechanical losses.

2.17 Applications and current relevance

DC machines remain important where simple, precise, wide-range speed control matters: electric traction, steel rolling mills, elevators, paper mills, cranes, robotics, automotive starters, wipers, power windows, and small servo drives. Permanent-magnet DC (PMDC) and brushless DC (BLDC) motors dominate small and medium applications like drones, EVs, HVAC, and computer fans. A BLDC motor replaces the mechanical commutator with electronic switching, eliminating brush wear and sparking, though it uses the same basic torque physics.

Disadvantages of brushed DC machines: commutator and brush maintenance, sparking (unsuitable for explosive atmospheres), speed limits due to commutation, and higher cost than AC induction motors.

---

Part 3: Transformers

3.1 What is a transformer?

A transformer is a static device that transfers electrical energy between two or more circuits by electromagnetic induction, without changing frequency, usually with a change in voltage and current levels. It has no moving parts, so its efficiency is very high (often 95–99.5%).

Why transformers matter: power is generated at moderate voltage (11–25 kV), stepped up to 132–765 kV for transmission (higher voltage means lower current, hence lower I²R loss), and stepped down again for distribution and use. Without transformers, long-distance AC power systems would not be practical.

3.2 Working principle

Two coils (primary and secondary) share a common magnetic core. An alternating voltage V₁ applied to the primary drives an alternating flux Φ in the core. This flux links the secondary, inducing an EMF in it by Faraday’s law. This is mutual induction.

The flux is the same in both windings (ideally), so voltage per turn is the same, and:

V₁/V₂ = N₁/N₂ = a (turns ratio)

• Step-up transformer: N₂ > N₁ (a < 1).
• Step-down transformer: N₂ < N₁ (a > 1).

3.3 The EMF equation

For sinusoidal flux Φ = Φm sin(ωt), the RMS induced EMF per winding is:

E = 4.44 · f · N · Φm = 4.44 · f · N · Bm · Aᶜ

where Aᶜ is the core cross-sectional area. This equation drives the whole design: for a given voltage and frequency, more turns means less flux (smaller core) and vice versa. Higher frequency also allows a smaller core, which is why aircraft use 400 Hz and switch-mode power supplies use kHz-range transformers.

3.4 The ideal transformer

Assumptions: no winding resistance, no leakage flux, infinitely permeable core, no core loss. Then:

• V₁/V₂ = N₁/N₂ = I₂/I₁
• V₁I₁ = V₂I₂ (input VA = output VA)
• Impedance transforms with the square of the turns ratio: Z₁ = a²·Z₂

The last point is the basis of impedance matching.

3.5 Construction

Core

• Built of thin silicon steel laminations (typically 3–4% silicon, sometimes cold-rolled grain-oriented (CRGO) steel or amorphous metal) to reduce eddy currents and hysteresis loss.
• Laminations are stacked with interleaved joints (overlapping) to minimise reluctance at joints.

Two core arrangements

• Core type: windings surround the core limbs. There is one magnetic path, and it is common in high-voltage power transformers.
• Shell type: the core surrounds the windings. It has multiple flux paths, better short-circuit strength, and is used in some large power and low-voltage, high-current applications.

Windings

• Copper (or aluminium) conductors with paper or enamel insulation.
• Concentric (cylindrical) windings: LV is nearest the core (needs less insulation to the core), HV is on the outside.
• Sandwich (disc) windings: alternate LV and HV sections, giving lower leakage reactance.

Tank and accessories (oil-immersed power transformers)

• Transformer oil for insulation and cooling.
• Conservator: an expansion tank that accommodates oil volume changes.
• Breather: contains silica gel to dry the air entering the conservator.
• Buchholz relay: a gas-actuated protective relay between tank and conservator that detects internal faults (gas from insulation or oil breakdown).
• Bushings: insulated terminals bringing conductors through the tank.
• Tap changer: adjusts turns ratio. It may be off-circuit (de-energised) or on-load (OLTC).
• Radiators/fans/pumps: cooling.
• Pressure relief device, temperature indicators, marshalling box.

Dry-type transformers use air (or cast resin) for cooling and insulation and are common indoors, since they avoid oil-fire risk.

3.6 Practical (real) transformer

Real transformers differ from ideal ones in four ways:

1. Winding resistances R₁ and R₂ (copper loss, voltage drop).
2. Leakage flux, which links only one winding and behaves like a series inductance, giving leakage reactances X₁ and X₂.
3. Core loss, modelled as a resistance R_c (or R₀) in parallel with the magnetising branch.
4. Finite permeability, requiring a magnetising current to establish the flux, modelled as a magnetising reactance X_m.

No-load operation: the primary draws a small no-load current I₀ (typically 2–5% of rated), which has two components:

• Core-loss component I_w (in phase with V₁)
• Magnetising component I_μ (lagging V₁ by 90°)

so I₀ = √(I_w² + I_μ²), and no-load power factor cos φ₀ = I_w / I₀. It is low (0.1–0.2 lagging) and the current waveform is distorted by saturation (contains harmonics, especially the third).

On-load operation: when a load draws I₂, the secondary MMF (N₂I₂) tends to reduce the flux. The primary draws an additional current I₂′ = I₂·(N₂/N₁) to cancel that MMF, so the core flux stays essentially constant from no-load to full load. This is why core loss is nearly constant.

3.7 Equivalent circuit

Referring all quantities to the primary side:

• R₂′ = a²R₂, X₂′ = a²X₂
• R₀₁ = R₁ + R₂′, X₀₁ = X₁ + X₂′, Z₀₁ = √(R₀₁² + X₀₁²)

Approximate equivalent circuit: shunt branch (R_c ∥ X_m) at the input, then a series impedance R₀₁ + jX₀₁, then the (referred) load. This is accurate enough for most power-system calculations.

3.8 Determining parameters: the OC and SC tests

Open-circuit (no-load) test

• Rated voltage applied to the LV side (safer and more convenient), HV side open.
• Readings: V₀, I₀, W₀.
• W₀ ≈ iron loss (copper loss is negligible at such small current).
• Gives R_c = V₀²/W₀ and X_m, from the split of I₀.

Short-circuit test

• One winding (usually LV) shorted; a low voltage (typically 3–10% of rated) is applied to the other side until rated current flows.
• Readings: V_sc, I_sc, W_sc.
• W_sc ≈ full-load copper loss (core loss is negligible at such low flux).
• Gives Z₀₁ = V_sc/I_sc, R₀₁ = W_sc/I_sc², X₀₁ = √(Z₀₁² − R₀₁²).

The advantage of these two tests is that the transformer’s full-load efficiency and regulation can be found without loading it, using little power (only the losses).

3.9 Voltage regulation

Voltage regulation is the change in secondary terminal voltage from no-load to full load at constant primary voltage:

Regulation (%) = (V₂,nl − V₂,fl) / V₂,nl × 100

Approximately:

Regulation ≈ (I·R₀₂·cos φ ± I·X₀₂·sin φ) / V₂ × 100

• + sign for lagging power factor (voltage drops)
• − sign for leading power factor (voltage can actually rise: the Ferranti-like effect)

Regulation is zero-to-negative at a specific leading power factor. Small power transformers typically have 1–3% regulation, while large ones are around 5–12% (often intentionally, to limit short-circuit currents).

3.10 Losses and efficiency

• Iron loss (P_i): hysteresis + eddy current. It depends on voltage and frequency, so it is essentially constant from no-load to full load.
• Copper loss (P_cu): I²R in windings. It varies with the square of load.
• Stray load losses and dielectric losses are small.

Efficiency at fraction x of full load, power factor cos φ:

η = (x·S·cos φ) / (x·S·cos φ + P_i + x²·P_cu,fl)

Maximum efficiency occurs when variable loss equals constant loss:

x = √(P_i / P_cu,fl)

Distribution transformers, which are energised 24 hours a day but loaded only part of the time, are designed with low iron loss because their all-day (energy) efficiency matters more than their peak efficiency. All-day efficiency = energy out over 24 hours / energy in over 24 hours.

3.11 Autotransformer

A single winding with a tap, part of which is common to both primary and secondary. Some power is transferred conductively and the rest magnetically.

• Advantages: less copper (saving ≈ (1 − V₂/V₁) of the two-winding copper), smaller, lower losses, better regulation, higher kVA rating for the same size.
• Disadvantages: no electrical isolation between sides, and a fault in the common winding can put high voltage on the LV side.
• Uses: motor starting, voltage adjustment (variac), interconnecting two power-system voltage levels (e.g. 400 kV/220 kV) when the ratio is close to 1.

3.12 Three-phase transformers

Three-phase power can be transformed with a bank of three single-phase transformers or a single three-limb core three-phase unit (more compact and cheaper).

Winding connections (Y = star, D or Δ = delta):

• Star-star (Yy): neutral available. It has problems with unbalanced loads and third harmonics unless a tertiary delta winding is added.
• Delta-delta (Dd): no phase shift, tolerates unbalanced loads, and continues to work (open-delta, at 58% capacity) if one unit fails.
• Delta-star (Dy) and star-delta (Yd): the most common. They provide a neutral on the secondary and suppress third harmonics. Dyn is standard for distribution transformers, and Yd is standard for generator step-up.
• Zig-zag: used for earthing transformers and to handle unbalanced loads.

Vector group and phase displacement: the notation uses the “clock” convention, where the LV phasor’s angle relative to the HV phasor is given as a clock hour (each hour = 30°). For example, Dyn11 means HV delta, LV star with neutral, LV leading HV by 30° (the “11 o’clock” position).

3.13 Parallel operation

Transformers share load properly in parallel only if they have:

1. The same voltage ratio (else circulating current flows even at no-load).
2. The same polarity.
3. The same phase sequence and phase displacement (vector group; three-phase only).
4. Equal per-unit impedances, so load shares in proportion to kVA rating.
5. Similar X/R ratios (for best power-factor matching).

3.14 Instrument transformers

Used for measurement and protection at high voltage or current:

• Current transformer (CT): primary in series with the line, secondary feeding ammeters, meters, and relays (typically 1 A or 5 A). Never open-circuit the secondary of an energised CT: it produces dangerously high voltages and can saturate the core.
• Potential (voltage) transformer (PT/VT): connected across the line, giving standardised secondary voltage (e.g. 110 V). Never short-circuit the secondary.

Key metrics: ratio error and phase-angle error (accuracy class).

3.15 Other special transformers

• Isolation transformer: 1:1, providing galvanic separation for safety and noise.
• Pulse and audio transformers, and high-frequency transformers in switch-mode power supplies (ferrite cores).
• Furnace, rectifier, traction, and welding transformers.
• Phase-shifting transformers: control power flow between networks.
• Grounding (earthing) transformers.

3.16 Important operational phenomena

Inrush current: when a transformer is energised, the core flux can start with an offset (depending on the switching instant and residual flux), driving the core deep into saturation. The magnetising current can spike to 5–10 times or more of rated current and decay over seconds. It is rich in second harmonics, which protection relays use to distinguish inrush from a true fault.

Cooling classes: ONAN (oil natural, air natural), ONAF (oil natural, air forced), OFAF (oil forced, air forced), ODAF (oil directed, air forced), and AN/AF for dry-type. Temperature rise limits insulation ageing, and insulation life roughly halves for every ~6–8 °C above design temperature.

Protection: Buchholz relay (internal faults), differential protection (winding faults), overcurrent and earth-fault relays, overfluxing protection (V/Hz), pressure relief, and winding/oil temperature alarms.

Short-circuit forces: fault currents produce large mechanical forces on windings, which is why the mechanical design of the windings and clamping is as important as the electrical design.

Per-unit system: transformer impedances are usually expressed as a percentage or per-unit of rated values, so they are the same whether viewed from the primary or secondary side. This simplifies power-system analysis.

---

Part 4: Worked Examples

Example 1: DC shunt motor

A 220 V shunt motor has armature resistance 0.5 Ω and draws 20 A armature current at 1000 rpm. Find the back EMF, developed torque, and starting current without a starter.

• Eb = V − Ia·Ra = 220 − (20 × 0.5) = 210 V
• Developed power = Eb·Ia = 210 × 20 = 4200 W
• ω = 2π × 1000/60 = 104.7 rad/s
• T = 4200 / 104.7 ≈ 40.1 N·m
• Starting current = V/Ra = 220/0.5 = 440 A (22× the running value), so a starter is essential.

If the load torque stays the same and the field flux is reduced by 10%, then torque ∝ Φ·Ia means Ia rises to about 22.2 A. The new Eb ≈ 220 − 22.2×0.5 = 208.9 V, so N ≈ 1000 × (208.9/210) / 0.9 ≈ 1105 rpm, illustrating that field weakening raises speed.

Example 2: Single-phase transformer

A 50 kVA, 2400/240 V, 50 Hz transformer gives: OC test (LV side) W₀ = 400 W; SC test (HV side) at rated current: V_sc = 60 V, W_sc = 600 W.

• Rated HV current = 50,000/2400 = 20.83 A
• Z₀₁ = 60/20.83 = 2.88 Ω; R₀₁ = 600/20.83² = 1.383 Ω; X₀₁ = √(2.88² − 1.383²) ≈ 2.53 Ω

Full-load efficiency at 0.8 pf lagging:
Output = 50,000 × 0.8 = 40,000 W; total losses = 400 + 600 = 1000 W
η = 40,000/41,000 = 97.56%

Regulation at 0.8 pf lagging:
= 20.83 × (1.383 × 0.8 + 2.53 × 0.6) / 2400 × 100 ≈ 2.28%

Maximum efficiency:
x = √(400/600) = 0.816, so the load is 0.816 × 50 = 40.8 kVA.
At unity pf: output ≈ 40,825 W, losses = 2 × 400 = 800 W
η_max = 40,825/41,625 = 98.08%

---

Part 5: Comparison and Connections

Aspect	DC Machine	Transformer
Motion	Rotating	Static
Energy conversion	Electrical ↔ Mechanical	Electrical ↔ Electrical
Induction type	Motional (dynamic) EMF	Mutual (static) EMF
Efficiency	~75–95%	~95–99.5%
Maintenance	Brushes, commutator, bearings	Very low (oil, insulation monitoring)
Operates on	DC (with internal commutation)	AC only

Both share the same ingredients: a magnetic circuit, windings, an EMF equation with Φ and frequency/speed, copper and iron losses, and efficiency maximised when variable and constant losses balance. Understanding one strengthens understanding of the other, and both lead into AC machines (induction and synchronous), where the transformer’s equivalent circuit reappears directly (an induction motor is essentially a rotating transformer).

---

Quick Formula Sheet

DC machines

• E = PΦZN/(60A); T = PΦZIa/(2πA)
• Motor: V = Eb + IaRa; Generator: E = V + IaRa
• N ∝ Eb/Φ; T ∝ ΦIa
• Pdev = EbIa = Tω
• Lap: A = P; Wave: A = 2

Transformers

• V₁/V₂ = N₁/N₂ = I₂/I₁ = a
• E = 4.44 f N Φm
• R₀₁ = R₁ + a²R₂; X₀₁ = X₁ + a²X₂
• Regulation ≈ (IR cos φ ± IX sin φ)/V × 100
• η = xS cos φ / (xS cos φ + P_i + x²P_cu)
• Max efficiency at x = √(P_i/P_cu)
• Autotransformer copper saving ≈ 1 − V₂/V₁

---

Conclusion

DC machines showed how electrical and mechanical energy interconvert, and their controllability shaped early industrial drives and traction. Transformers made the modern AC grid possible. DC machines have given way in many applications to AC drives and brushless motors, but their principles (back-EMF, torque-flux-current relations, speed control by voltage and flux) still underpin how all electric motors are analysed. Transformers remain essential in every part of the power system, from gigawatt-scale grid substations to the tiny transformer in a phone charger.