GTU BE01000051 Basic Electrical Engineering Winter 2025 Solved Paper & Solutions

Kirchhoff's Voltage Law (KVL) states that the algebraic sum of all the voltages around a closed loop is zero. The supply voltage equals the sum of the voltage drops.

In the series circuit, the same current $I$ flows through both resistors.

Thevenin's theorem: a network seen from two terminals can be replaced by a voltage source $V_{th}$ in series with a resistance $R_{th}$.

From the figure: $V = 24$ V, $R_1 = 4\ \Omega$, $R_2 = 8\ \Omega$, and the load $R_L = 6\ \Omega$ is connected across $R_2$.

The three star resistances are $R_1 = 10\ \Omega$, $R_2 = 20\ \Omega$, $R_3 = 30\ \Omega$, joined at terminals $1$, $2$ and $3$.

Inductive load: an inductor opposes any change in the current. The voltage across it is $v = L\dfrac{di}{dt}$. So the voltage is largest when the current is changing fastest, and the current reaches its peak a quarter cycle later. Hence the current lags the voltage. For an R-L load, $\phi = \tan^{-1}\left(\dfrac{X_L}{R}\right)$ is positive, and the current lags by $\phi$. The power factor $\cos\phi$ is therefore called lagging.

Capacitive load: the current into a capacitor is $i = C\dfrac{dv}{dt}$. The current is largest when the voltage changes fastest, which is when the voltage is zero. So the current reaches its peak a quarter cycle before the voltage. Hence the current leads the voltage, and the power factor is leading.

The sign of the angle depends on which of $X_L$ and $X_C$ is larger. An inductor never makes the current lead, and a capacitor never makes it lag.

In a star connection, one end of each of the three windings is joined at a neutral point. In a delta connection, the three windings are joined end to end in a closed loop.

The total power in both is $P = \sqrt{3}\,V_L I_L\cos\phi$. The two connections differ in how the voltage and current are shared between the lines and the windings, and this decides where each one is used.

Line and phase voltages

In star, the line voltage is $\sqrt{3}$ times the phase voltage and leads it by $30^\circ$. In delta, the line voltage is equal to the phase voltage.

Line and phase currents

The same rule applies to the currents.

In star, the line current is the same as the phase current. In delta, the line current is $\sqrt{3}$ times the phase current.

Typical applications

• Star: power distribution (the 415 V line and 230 V phase supply to houses), where a neutral is needed; for starting motors at reduced voltage; for the secondary of step-up transformers at the power station.
• Delta: industrial motors and heavy loads that run at full voltage; the primary of distribution transformers; where no neutral is needed and a high current is required.

In the example of the magnetic circuit, the current in the coil gives the mmf, and the flux flows in the iron ring.

A single-phase transformer works on the principle of mutual induction. When an alternating voltage is applied to the primary winding, an alternating flux is set up in the core. This flux links with the secondary winding and induces an emf in it, as given by Faraday's law. The emf is proportional to the number of turns, so $\dfrac{V_2}{V_1} = \dfrac{N_2}{N_1}$. The frequency stays the same.

Why it is called static: a transformer has no moving or rotating parts. Energy passes from the primary to the secondary through the magnetic field only, and not through any mechanical motion. A motor or a generator has a rotating part, and so it is a rotating machine. For the same reason, a transformer is quiet, and it has a high efficiency, since there are no friction and windage losses.

A single-phase induction motor runs on a single-phase supply. It is used where a three-phase supply is not available.

Construction

• Stator: a laminated steel core with slots. It carries a main winding, and an auxiliary (starting) winding placed $90^\circ$ apart in space. A capacitor is often put in series with the auxiliary winding.
• Rotor: a squirrel cage rotor, made of copper or aluminium bars short-circuited by end rings. It has no brushes or slip rings.
• Centrifugal switch: it cuts out the starting winding when the motor reaches about 75 percent of the full speed.

Working

1. The single-phase current in the main winding gives a pulsating flux, which cannot start the rotor alone. It can be seen as two equal fluxes rotating in opposite directions, so the starting torque is zero.
2. The auxiliary winding has a different phase of current, because of the capacitor or its resistance. The two windings together produce a rotating magnetic field, which gives a starting torque.
3. The rotating field cuts the rotor bars and induces emf and current in them. This current gives a torque, and the rotor starts to turn in the direction of the field.
4. At about 75 percent of the synchronous speed, the centrifugal switch disconnects the auxiliary winding. The motor then runs on the main winding, and the rotor is kept in motion.

Household applications

• Ceiling fans and table fans.
• Washing machines and mixer grinders.
• Refrigerators, water pumps and air conditioners.

Importance of earthing in domestic wiring

• It protects persons from shock. If a fault makes the metal body of an appliance live, the fault current flows to the earth and not through the person.
• It makes the fuse or MCB operate quickly, since a large fault current flows.
• It protects the appliances from damage by leakage current.
• It keeps the metal parts at the earth potential, so they are safe to touch.

Two common types of earthing

1. Pipe earthing: a perforated GI pipe is buried vertically in the ground with charcoal and salt around it.
2. Plate earthing: a copper or GI plate is buried in a pit with charcoal and salt, and the earth wire is joined to it.

An electricity bill is made from the energy used in a billing period, which is measured by the energy meter in units (kWh).

General procedure

1. Meter reading: the reader notes the present reading of the energy meter.
2. Units used: subtract the previous reading from the present reading.
3. Energy charge: multiply the units used by the tariff. Many boards use slabs, in which the rate per unit rises with the use.
4. Add the other charges: the fixed (meter and connected load) charge, a fuel adjustment charge, the electricity duty and the taxes.
5. Total bill: the sum of all the charges, with the due date. The consumer pays it by the due date, or a late fee is added.
6. Check: the consumer can compare the meter reading on the bill with the actual reading on the meter.

Energy in kilowatt-hours

One kWh (one unit) is the energy used by a load of 1 kW in 1 hour.

Example: a fan of 75 W runs for 8 hours, three bulbs of 10 W run for 5 hours, and a TV of 100 W runs for 4 hours each day.

For 30 days it is $1.15 \times 30 = 34.5$ units. At an assumed rate of Rs 5 per unit, the energy charge is $34.5 \times 5 =$ Rs 172.50.

A voltmeter measures the potential difference between two points. It is connected in parallel with the part of the circuit, and it has a very high resistance, so it takes very little current.

In a moving coil voltmeter, a coil carrying a small current is placed in the field of a permanent magnet. The current gives a torque, which turns the pointer against a spring. A high resistance (multiplier) is in series with the coil. The current is $I = V/R_{\text{total}}$, so it is proportional to the voltage, and so is the deflection.

Characteristics of a battery are:

1. Voltage (emf): the voltage between its terminals, for example 1.5 V or 12 V.
2. Capacity: the charge it can give, in ampere-hour (Ah).
3. Internal resistance: the resistance inside the battery, which causes a voltage drop and heat when it supplies current.
4. Energy density: the energy stored per unit mass or volume.
5. Efficiency: the energy given out divided by the energy put in during charging.
6. Cycle life: the number of charge and discharge cycles it can give before its capacity falls much.
7. Self-discharge: the loss of charge when the battery is not used.

A Switch Fuse Unit (SFU) is a single unit that combines a switch and fuses in one metal enclosure. It is used to switch a circuit on and off by hand and to protect it from overload and short circuit.

Construction

• A metal-clad (iron) enclosure that protects the parts and is earthed.
• A switch with a handle, which makes and breaks the circuit on all the lines together.
• Fuses (HRC fuses) in fuse carriers, one in each phase line. HRC means high rupturing capacity, so the fuse can break a very large fault current safely.
• A neutral link in a four-pole unit, which is a solid link and does not have a fuse.

Working

1. When the handle is turned to ON, the switch connects the supply to the load through the fuses.
2. In a normal run, the current flows through the fuses without harm.
3. If an overload or a short circuit occurs, the fuse element melts and the circuit opens.
4. The supply can be isolated by hand with the switch. The fuse is replaced after a fault, with one of the same rating, and the switch is closed again.

Uses

• At the main incoming point of a house, a shop or a factory, to control and protect the whole installation.
• In the control of motors and other heavy loads, and in distribution boards.