City & Guilds Electrical Installation (Level 3) Electrical Science and Principles Syllabus

State Lenz's law and explain what the negative sign in Faraday's law means.

Lenz's law: an induced current always flows in a direction that opposes the change producing it. The negative sign in $e = -N\dfrac{d\Phi}{dt}$ expresses this opposition (conservation of energy).

Give the EMF equation for a conductor moving through a magnetic field.

$e = B\,l\,v\,\sin\theta$, where $B$ is flux density, $l$ the conductor length, $v$ the velocity and $\theta$ the angle between motion and field. Maximum when motion is perpendicular to the field ($\theta = 90^{\circ}$).

Distinguish self-inductance from mutual inductance.

Self-inductance: a changing current in a coil induces an EMF in the same coil, $e = -L\dfrac{di}{dt}$. Mutual inductance: a changing current in one coil induces an EMF in a neighbouring coil, $e_{2} = -M\dfrac{di_{1}}{dt}$. Both measured in henries ($\text{H}$).

State Fleming's right-hand and left-hand rules and their uses.

Right-hand rule (generator) gives the direction of induced EMF/current; left-hand rule (motor) gives the direction of force on a current-carrying conductor. In both, thumb = motion/force, first finger = field, second finger = current.

Give the equation for the force on a current-carrying conductor in a magnetic field.

$F = B\,I\,l\,\sin\theta$, where $F$ is force in newtons, $B$ flux density, $I$ current, $l$ length and $\theta$ the angle between conductor and field. This force underlies motor action.

Explain the basic principle of motor action versus generator action.

Motor action: a current-carrying conductor in a magnetic field experiences a force ($F = BIl$), converting electrical to mechanical energy. Generator action: a conductor moved through a field has an EMF induced ($e = Blv$), converting mechanical to electrical energy.

Define the period, frequency and their relationship for an AC waveform.

Period $T$ is the time for one complete cycle (seconds); frequency $f$ is cycles per second (hertz). They are reciprocals: $f = \dfrac{1}{T}$. The UK mains frequency is $50\,\text{Hz}$.

Define peak, peak-to-peak, average and RMS values of a sinusoidal AC waveform.

Peak $V_{m}$ is the maximum value; peak-to-peak $= 2V_{m}$; average (over half cycle) $= 0.637\,V_{m}$; RMS (effective) $= 0.707\,V_{m} = \dfrac{V_{m}}{\sqrt{2}}$.

Define form factor and peak (crest) factor for a sine wave and give their values.

Form factor $= \dfrac{\text{RMS}}{\text{average}} = \dfrac{0.707}{0.637} = 1.11$. Peak factor $= \dfrac{\text{peak}}{\text{RMS}} = \dfrac{1}{0.707} = 1.414$.

Write the instantaneous equation of a sinusoidal voltage and relate angular frequency to $f$.

$v = V_{m}\sin(\omega t + \phi)$, where $V_{m}$ is the peak value, $\phi$ the phase angle and $\omega = 2\pi f$ is the angular frequency in $\text{rad/s}$.

Define inductive reactance and capacitive reactance with their formulas.

Inductive reactance $X_{L} = 2\pi f L$ (ohms), increasing with frequency. Capacitive reactance $X_{C} = \dfrac{1}{2\pi f C}$ (ohms), decreasing with frequency.

How do voltage and current phase relationships differ for pure resistance, pure inductance and pure capacitance?

Resistance: $V$ and $I$ in phase. Pure inductance: current lags voltage by $90^{\circ}$. Pure capacitance: current leads voltage by $90^{\circ}$. (Mnemonic: CIVIL.)

Define impedance and give its formula for a series RL or RC circuit.

Impedance $Z$ is the total opposition to AC current (ohms): $Z = \dfrac{V}{I}$. For series $RL$: $Z = \sqrt{R^{2} + X_{L}^{2}}$; for series $RC$: $Z = \sqrt{R^{2} + X_{C}^{2}}$.

Write the impedance formula for a series RLC circuit and the phase angle.

$Z = \sqrt{R^{2} + (X_{L} - X_{C})^{2}}$, with phase angle $\phi = \tan^{-1}\!\left(\dfrac{X_{L} - X_{C}}{R}\right)$.

Define power factor and give two equivalent expressions for it.

Power factor is the cosine of the phase angle between voltage and current: $\text{p.f.} = \cos\phi = \dfrac{R}{Z} = \dfrac{P\,(\text{true power})}{S\,(\text{apparent power})}$. It ranges from 0 to 1.

Distinguish true power, apparent power and reactive power with their units.

True (active) power $P = VI\cos\phi$ in watts ($\text{W}$); apparent power $S = VI$ in volt-amperes ($\text{VA}$); reactive power $Q = VI\sin\phi$ in volt-amperes reactive ($\text{VAr}$). They form the power triangle: $S^{2} = P^{2} + Q^{2}$.

Why is a low power factor undesirable and how is it commonly corrected?

A low (lagging) power factor draws more current for the same true power, causing higher cable/transformer losses, larger voltage drop and possible utility charges. It is corrected by connecting capacitors in parallel to supply leading reactive power and bring $\cos\phi$ closer to unity.

What is series resonance and at what frequency does it occur?

Series resonance occurs when $X_{L} = X_{C}$, so the reactances cancel and the circuit is purely resistive ($Z = R$, minimum). The resonant frequency is $f_{r} = \dfrac{1}{2\pi\sqrt{LC}}$.

Describe the characteristics of a series resonant (acceptor) circuit at resonance.

At resonance impedance is minimum (equal to $R$), current is maximum, power factor is unity, and large voltages can appear across $L$ and $C$ (voltage magnification by the Q-factor). It is called an acceptor circuit.

How does a parallel resonant (rejector) circuit behave at resonance?

A parallel resonant circuit presents maximum (dynamic) impedance, minimum line current, and unity power factor at resonance. Large circulating current flows in the LC loop. It is called a rejector circuit. Resonant frequency $f_{r} = \dfrac{1}{2\pi\sqrt{LC}}$ (ideal case).

Define the Q-factor of a series resonant circuit.

$Q = \dfrac{X_{L}}{R} = \dfrac{2\pi f_{r} L}{R} = \dfrac{1}{R}\sqrt{\dfrac{L}{C}}$. It is the voltage magnification factor and indicates the sharpness of the resonance (selectivity).

Explain how a three-phase supply is generated.

Three coils (windings) are spaced $120^{\circ}$ apart on the stator of an alternator. As the rotor's magnetic field rotates, it induces three EMFs of equal magnitude and frequency, each displaced by $120^{\circ}$ ($\tfrac{1}{3}$ of a cycle) from the others.

What is the phase sequence of a three-phase system and why does it matter?

Phase sequence is the order in which the phase voltages reach their peaks, conventionally L1–L2–L3 (red, yellow, blue / brown, black, grey). It determines the direction of rotation of three-phase motors; swapping any two lines reverses rotation.

For a balanced star (Y) connection, give the relationship between line and phase values.

Line voltage $V_{L} = \sqrt{3}\,V_{ph}$, and line current equals phase current: $I_{L} = I_{ph}$. A neutral point is available.

For a balanced delta ($\Delta$) connection, give the relationship between line and phase values.

Line voltage equals phase voltage: $V_{L} = V_{ph}$, and line current $I_{L} = \sqrt{3}\,I_{ph}$. There is no neutral point.

Why does the UK use a $400\,\text{V}/230\,\text{V}$ four-wire system, and how are these values related?

The system is a star connection where $400\,\text{V}$ is the line-to-line voltage and $230\,\text{V}$ is the line-to-neutral (phase) voltage: $400 = \sqrt{3} \times 230$. The neutral allows single-phase loads to be supplied between any line and neutral.

Distinguish balanced and unbalanced three-phase loads and state the effect on the neutral.

A balanced load has equal impedance/current in all three phases, so in a star system the neutral current is zero. An unbalanced load has unequal phase currents, producing a resultant (out-of-balance) current that flows in the neutral conductor.

Give the formula for total power in a balanced three-phase load using line quantities.

$P = \sqrt{3}\,V_{L}\,I_{L}\cos\phi$ (watts). Apparent power $S = \sqrt{3}\,V_{L}\,I_{L}$ (VA) and reactive power $Q = \sqrt{3}\,V_{L}\,I_{L}\sin\phi$ (VAr). This holds for both star and delta.

Express balanced three-phase power in terms of phase quantities.

$P = 3\,V_{ph}\,I_{ph}\cos\phi$. This is equivalent to $\sqrt{3}\,V_{L}\,I_{L}\cos\phi$ for both star and delta connections.

State the transformer EMF equation.

$E = 4.44\,f\,N\,\Phi_{m}$, where $E$ is the RMS EMF, $f$ the frequency, $N$ the number of turns and $\Phi_{m}$ the maximum flux. It applies to both primary and secondary windings.

Give the transformer turns/voltage/current ratio relationships for an ideal transformer.

$\dfrac{V_{1}}{V_{2}} = \dfrac{N_{1}}{N_{2}} = \dfrac{I_{2}}{I_{1}}$. Voltage is proportional to turns; current is inversely proportional. For ideal transformer, $V_{1}I_{1} = V_{2}I_{2}$ (power in = power out).

Name the two main types of power loss in a transformer and how each is reduced.

Copper (I²R) losses in the windings — reduced by using thicker, lower-resistance conductors; and iron (core) losses comprising hysteresis (reduced with silicon-steel) and eddy-current losses (reduced by laminating the core). Transformer efficiency $\eta = \dfrac{P_{out}}{P_{out} + \text{losses}}$.

Explain the operating principle of a three-phase induction motor.

Three-phase currents in the stator windings create a rotating magnetic field. This field cuts the rotor conductors, inducing currents (transformer/induction action). The induced rotor currents interact with the field to produce torque, dragging the rotor round in the direction of field rotation.

Define synchronous speed and slip for an induction motor, with formulas.

Synchronous speed $N_{s} = \dfrac{120 f}{p}$ (rev/min), where $p$ is the number of poles. Slip $s = \dfrac{N_{s} - N_{r}}{N_{s}} \times 100\%$, where $N_{r}$ is the rotor speed. An induction motor must have slip to develop torque.

Why won't a single-phase induction motor self-start, and how is starting achieved?

A single-phase winding produces a pulsating (not rotating) field, giving no starting torque. A starting method is added — e.g. a start (auxiliary) winding with a capacitor or a shaded pole — to create a phase displacement and a rotating field until the motor runs up.

Describe the construction and principle of a DC generator.

Armature coils rotate in a magnetic field produced by field windings/poles; an EMF is induced ($e = Blv$). A commutator and brushes rectify the alternating coil EMF into a unidirectional (DC) output at the terminals.

What is the function of the commutator in a DC machine?

The commutator is a segmented rotating switch that reverses the connection to the external circuit twice per revolution. In a generator it converts the AC induced in the armature into DC output; in a motor it keeps torque acting in one direction by reversing armature current as coils pass the brushes.

Compare series, shunt and compound DC machine field connections.

Series: field winding in series with the armature — high starting torque, speed varies greatly with load. Shunt: field in parallel with the armature — nearly constant speed. Compound: has both series and shunt windings, combining good starting torque with reasonable speed regulation.

Write the back-EMF equation for a DC motor and explain its role.

Back-EMF $E_{b} = V - I_{a}R_{a}$, where $V$ is supply voltage, $I_{a}$ armature current and $R_{a}$ armature resistance. The rotating armature generates $E_{b}$ opposing the supply; it limits armature current and rises with speed, so at start (zero $E_{b}$) current is high and a starter is needed.

Give the formula for energy stored in a charged capacitor and in an inductor.

Capacitor: $W = \tfrac{1}{2}CV^{2}$ (joules). Inductor: $W = \tfrac{1}{2}LI^{2}$ (joules). A capacitor stores energy in an electric field; an inductor stores it in a magnetic field.

How do you combine capacitors in series and in parallel?

Series: $\dfrac{1}{C_{T}} = \dfrac{1}{C_{1}} + \dfrac{1}{C_{2}} + \dots$ (total is smaller than the smallest). Parallel: $C_{T} = C_{1} + C_{2} + \dots$. Note this is opposite to the rules for resistors.

Define the time constant for an RC and an RL circuit and its significance.

RC: $\tau = RC$; RL: $\tau = \dfrac{L}{R}$ (seconds). The time constant is the time to reach about $63.2\%$ of the final change; a transient is essentially complete after about $5\tau$.

State the capacitance formula for a parallel-plate capacitor and name typical in-service functions of capacitors.

$C = \dfrac{\varepsilon_{0}\varepsilon_{r}A}{d}$, where $A$ is plate area, $d$ the separation and $\varepsilon_{r}$ the relative permittivity. In service capacitors are used for power-factor correction, motor starting/running, smoothing and filtering, and energy storage.