Transformers, Generators and Induction Motors

Tip

Exam Focus

Confirmed heavy hitters: transformer construction + EMF equation $E=4.44fN\phi_m$ (the $4.44=4\times1.11$); DC generator construction + $E_g=\dfrac{P\phi ZN}{60A}$ (lap $A=P$, wave $A=2$) — a confirmed 10-mark Eng. Sewa question; DC motor back-EMF $E_b=V-I_aR_a$, $T\propto\phi I_a$; alternator $f=PN_s/120$; induction-motor slip $s=(N_s-N)/N_s$. Fleming: Right = geneRator, Left = motor.

Transformer

A transformer is a static device that transfers AC power between two circuits at the same frequency by mutual induction, usually changing the voltage level, with no electrical connection between windings.

Core-type single-phase transformer: primary and secondary windings on a laminated core linked by mutual flux $\phi$.

Part	Construction and function
Magnetic core	Laminated CRGO silicon-steel limbs/yokes: low-reluctance flux path; thin laminations cut eddy currents
Primary $N_1$	Turns fed by the AC source; sets up the alternating mutual flux
Secondary $N_2$	Links the same flux and supplies the isolated load
Tank, oil, radiators	Insulate and remove heat in oil-filled power units
Conservator, breather, Buchholz	Allow oil expansion, exclude moisture, detect gas faults

Transformer construction (core-type: windings around limbs; shell-type: core around the windings).

Caution

Key Formula — EMF equation and turns ratio

Flux rises $0\to\phi_m$ in $T/4=1/4f$, so avg. EMF/turn $=4f\phi_m$ and (form factor $1.11$) RMS EMF/turn $=4.44f\phi_m$:

$$E_1=4.44\,fN_1\phi_m,\quad E_2=4.44\,fN_2\phi_m,\quad \frac{E_2}{E_1}=\frac{N_2}{N_1}=\frac{V_2}{V_1}=\frac{I_1}{I_2}=K.$$

$K>1$ step-up, $K<1$ step-down; ideally $V_1I_1=V_2I_2$ (the $4.44=4\times1.11$).

Loss	Depends on	Reduced by
Hysteresis	Voltage/frequency (nearly constant)	Low-loss CRGO steel
Eddy current	Voltage/frequency (nearly constant)	Thin insulated laminations
Copper $I^2R$	Square of load	Larger conductor area, cooling

Transformer losses (iron loss $=$ hysteresis $+$ eddy).

Caution

Key Formula — Efficiency and voltage regulation

$$\eta=\frac{P_{out}}{P_{out}+P_{core}+P_{cu}}\times100\%,\qquad \%VR=\frac{V_{NL}-V_{FL}}{V_{FL}}\times100\%.$$

$\eta$ is maximum when copper loss $=$ core loss (95–99%, no rotating parts). An open-circuit test gives core loss; a short-circuit test gives full-load copper loss and series impedance. Types: power, distribution, autotransformer (single winding, saves copper), instrument (CT, PT) and isolation ($1{:}1$) transformers.

DC Generator

Converts mechanical energy to DC by Faraday’s law ($e=Blv$); the commutator acts as a mechanical rectifier turning the internal AC into unidirectional output. Direction: Fleming’s right-hand rule.

Single-loop DC generator: a coil rotates in the pole flux; the split commutator and brushes deliver DC to the load.

Part	Function
Yoke	Mechanical frame and low-reluctance flux return path
Pole core / shoe	Carry field coils; spread flux across the air gap
Field winding	Produces the stationary main field
Armature (laminated)	Rotating slotted core; laminations cut eddy loss
Commutator	Mica-insulated copper segments; rectify armature AC
Brushes	Sliding carbon contacts collecting the DC output

DC generator (and motor) construction.

Caution

Key Formula — DC generator EMF

$$E_g=\frac{P\,\phi\,Z\,N}{60\,A},\qquad \text{lap: }A=P,\quad\text{wave: }A=2,\qquad E_g\propto\phi N.$$

$P$ poles, $\phi$ flux/pole (Wb), $Z$ conductors, $N$ speed (rpm), $A$ parallel paths. Lap ($A=P$) gives many low-voltage/high-current paths; wave ($A=2$) gives high-voltage/low-current.

Type	Field connection	Feature / use
Separately excited	External DC source	Wide voltage control; testing
Shunt	Field $\parallel$ armature	Near-constant $V$; battery charging
Series	Field in series	$V$ rises with load; boosters
Compound	Shunt $+$ series	Flat regulation; general supply

DC generator types by excitation.

Armature reaction: the armature flux distorts and weakens the main field and shifts the neutral axis; commutation: is the current reversal in a coil at the brush. Both are improved by interpoles and compensating windings (with correct brush position).

DC Motor

A current-carrying conductor in a field feels $F=BIl$ (Fleming’s left-hand rule). Rotation generates a back-EMF opposing the supply, making the motor self-regulating.

DC motor: field $\times$ armature current gives conductor forces $F$ and torque; the rotating armature develops a back-EMF $E_b$.

Caution

Key Formula — Back-EMF, torque, speed

$$E_b=\frac{P\phi ZN}{60A}=V-I_aR_a,\qquad I_a=\frac{V-E_b}{R_a},$$$$P_m=E_bI_a,\quad T=\frac{P_m}{\omega_m}=\frac{PZ}{2\pi A}\phi I_a \;\Rightarrow\; T\propto\phi I_a,\qquad N\propto\frac{V-I_aR_a}{\phi}.$$

At starting $E_b=0$, so $I_a=V/R_a$ is dangerously large — a starter or electronic current limiter is required. Self-regulation: more load $\Rightarrow$ slower $\Rightarrow$ smaller $E_b\Rightarrow$ larger $I_a\Rightarrow$ more torque.

Motor	Characteristic	Application
Shunt	Near-constant flux and speed	Fans, lathes, pumps
Series	Very high starting torque (never unloaded)	Cranes, traction, hoists
Compound	Good starting torque $+$ better speed regulation	Elevators, presses

DC motor types.

Speed control: flux control (field rheostat), armature-resistance control, and armature-voltage control (Ward–Leonard / chopper).

AC Machines

Both AC machines use the rotating field set up by balanced three-phase stator currents, turning at the synchronous speed $N_s=120f/P$.

Caution

Key Formula — Synchronous speed, slip, alternator EMF

$$N_s=\frac{120f}{P},\quad s=\frac{N_s-N}{N_s},\quad f_2=sf,\quad f=\frac{PN_s}{120},\quad E_{ph}=4.44\,f\,\phi\,T_{ph}\,k_w.$$

A three-phase induction motor induces current in its cage/wound rotor; the rotor runs below $N_s$ with slip $s$ (typically 2–5%, rotor-current frequency $f_2=sf$) and is self-starting ($s=1$ at standstill). A synchronous motor locks to $N_s$ (zero steady slip), is not self-starting, and when over-excited supplies leading VAR (synchronous condenser). An alternator generates 3$\phi$ AC with a DC-excited rotor, its frequency locked to speed by $f=PN_s/120$.

Both machines use a rotating stator field; the induction rotor slips below $N_s$, the synchronous rotor locks to $N_s$.

Feature	Induction motor	Synchronous motor
Rotor excitation	Induced (cage rotor)	DC or permanent magnet
Running speed	Below $N_s$ (slips with load)	Exactly $N_s$ to pull-out
Starting	Self-starting (3$\phi$)	Needs a starting arrangement
Power factor	Lagging	Adjustable (lag/unity/lead)
Use	Pumps, fans, compressors	Constant-speed drives, PF correction

Induction versus synchronous motor.

Feature	DC machine	Induction machine
Supply	DC	AC
Commutator	Yes (maintenance)	No
Speed control	Easy, wide range	Needs a VFD
Robustness / cost	Lower / higher	Very rugged / lower

DC versus AC machines.

Note

Example — Machine numericals

(a) 4-pole lap generator, $Z=480$, $\phi=20\,mWb$, 1200 rpm: $A=P=4$, $E_g=\dfrac{4(0.02)(480)(1200)}{60\times4}=192\,V$.
(b) 50 Hz, $N_1=500$, $\phi_m=3\,mWb$: $E_1=4.44\times50\times500\times0.003=333\,V$.
(c) 4-pole, 50 Hz induction motor at 1440 rpm: $N_s=1500$, $s=(1500-1440)/1500=4\%$.