**Understanding**

➔ Alternating current (ac) generators

➔ Average power and root mean square (rms) values of current and voltage

➔ Transformers

➔ Diode bridges

➔ Half-wave and full-wave rectification

**Applications ****and** **skills**

➔ Explaining the operation of a basic ac generator, including the effect of changing the generator frequency

➔ Solving problems involving the average power in an ac circuit

➔ Solving problems involving step-up and step- down transformers

➔ Describing the use of transformers in ac electrical power distribution

➔ Investigating a diode bridge rectification circuit experimentally

➔ Qualitatively describing the effect of adding a capacitor to a diode bridge rectification circuit

### Equations

➔ rms and peak values: I_{rms }=I_{0}/√2

➔ potential difference: V_{rms }=V_{0}/√2

➔ resistance: *R *= V_{0}/I_{0}_{ }= V_{rms }/I_{rms}

➔ maximum power: *P*max = *I*_{0} *V*_{0}

➔ average power: *P _{average }*= 1/2

*I*

_{0}

*V*

_{0}

➔ transformer equation: ∈_{p}/O_{s} =N_{p}/N_{s}=I_{s}/I_{p}

### 11.2 Transmission of Power

The alternating current (ac) generator

Rotating coil in a region of magnetic field.

A magnetic field that cuts the rotating coil.

Relative movement between the coil and the magnetic field, causing emf to be induced and current to flow.

Rotation of coil: caused by a turbine in a power plant.

Two slip rings attached to the ends of the coil, rotating along with it and touching carbon brushes that transfer current to the outside world.

Current: When the left-hand wire is moving upwards and the right-hand wire is moving downwards, current is counter-clockwise. Half a period later, the current will be opposite.

Induced emf, by Faraday’s law, is the minus rate of change of the flux linkage.

If the angle speed (w) increases, frequency and emf amplitude increase.

Increasing N, B or A causes the emf to increase, without changing the frequency

Power in ac circuits always positive, with a period of half the frequency.

Average power dissipated: half the peak’s value.

Same phase as current and emf.

π/2 out of phase with the graph of change of magnetic flux linkage (it is its derivative)

**Root mean square (rms)**

In order to fund the average value of the current and of the emf (given that they are both negative and positive), we the root of the peak value divided by 2.

Current rms = Irms = sqrt(Io²/2)

Voltage rms = Vrms = sqrt(Vo²/2)

Mean power = (Irms)(Vrms)/2 = Irms²R/2 = Vrms²/(2R)

**The transformer**

Changes the potential difference from one alternating current into another potential difference.

Process:

Alternating current produces magnetic field in primary coil.

Flux in the core is created.

Changing flux is linked to the secondary coil.

If coil is part of a circuit, current flows.

Formulas: εp/εs = Np/ Ns = Vp/Vs = Is/Ip. (primary = p; secondary = s).

N = number of coils.

Frequency remains unchanged.

Step-up transformer: Output voltage > Input voltage: Ns > Np.

Step-down transformer: Output voltage < Input voltage: Ns < Np.

Graph of secondary coil = gradient of graph of primary coil.

Core material: soft magnetic material (avoids magnetic hysteresis).

Can be rapidly magnetized and demagnetized.

Core design/shape: ensures flux does not leak out of the core (less power loss).

Laminators: prevent the formation of currents inside the core itself, known as eddy currents, which lead to heating and power loss.

### Transformers and power transmission

Power loss in cables proportional to (current)².

Reducing power loss: If pd is increased by transformer then I is decreased, so power falls, as R is constant.

Other benefits: Smaller I – Smaller temperature – Smaller R – Lower genetic damage.

Rectification of alternating current

Process of converting an alternating current supply into direct current.

Rectifier: a diode, which only allows current to pass in one direction.

When current passes through it, the diode is said to be forward biased.

When no current passes through it, it is reverse biased.

Half-wave rectification: half of the power is lost.

Current and voltage is not constant: It is zero during half a period.

Full-wave rectification: usage of the whole power.

Rectification with capacitors: using a circuit with a capacitor (in parallel with resistor), which charges when alternate current is forward and discharges with reverse bias.

Useful to overcome the problem of zero current, by creating small ripples.

### ALTERNATING CURRENT

### ALTERNATING AND DIRECT CURRENT

#### ADVANTAGES OF A.C. OVER D.C.

- The generation of A.C. is cheaper than that of D.C.
- Alternating voltage can be easily stepped up or stepped down by using a transformer.
- A.C. can be easily converted into D.C. by rectifier. D.C. is converted to A.C. by an inverter.
- A.C. can be transmitted to a long distance without appreciable loss.

### AVERAGE AND RMS VALUE OF ALTERNATING CURRENT

- Time period : The time taken by A.C. to go through one cycle of changes is called its period. It is given as
- Phase : It is that property of wave motion which tells us the position of the particle at any instant as well as its direction of motion. It is measured either by the angle which the particle makes with the mean position or by fraction of time period.
- Phase angle : Angle associated with the wave motion (sine or cosine) is called phase angle.
- Lead : Out of the current and emf the one having greater phase angle will lead the other e.g., in equation

- Lag : Out of current and emf the one having smaller phase angle will lag the other. In the above equations, the emf lags the current by .

### RESISTANCE OFFERED BY VARIOUS ELEMENTS (INDUCTOR, RESISTOR AND CAPACITOR) TO A.C.

### IMPEDANCES AND PHASES OF AC CIRCUIT CONTAINING DIFFERENT ELEMENTS

#### CIRCUIT CONTAINING ONLY RESISTOR (R)

#### CIRCUIT CONTAINING ONLY INDUCTOR (L)

#### CIRCUIT CONTAINING ONLY CAPACITOR (C)

E = E0 sin ωt.

#### CIRCUIT CONTAINING RESISTANCE AND INDUCTANCE IN SERIES (L-R SERIES CIRCUIT)

#### CIRCUIT CONTAINING RESISTANCE AND CAPACITANCE IN SERIES (C–R SERIES CIRCUIT)

#### CIRCUIT CONTAINING INDUCTANCE AND CAPACITANCE IN SERIES (SERIES L-C CIRCUIT)

#### CIRCUIT CONTAINING RESISTANCE, INDUCTANCE AND CAPACITANCE IN SERIES (SERIES LCR CIRCUIT)

- If XC > XL, the value of φ is positive, i.e., current leads the applied emf.
- If XC < XL, the value of φ is negative, i.e., current lags behind the applied emf.
- If XC = XL, the value of φ is zero, i.e., current and emf are in same phase. This is called the case of resonance and resonant frequency for condition XC = XL, is given by :

### PARALLEL RESONANT CIRCUIT

Q-factor or quality factor of the circuit and is given by

### POWER IN AN A.C. CIRCUIT

L-C circuit, the power factor is zero ( φ = 90º); for R-circuit

cos φ = 1 ( φ = 0) and for all other circuit cos φ = R/Z, where

Z = impedance.

- Unless mentioned otherwise, all a.c. currents and voltages are r.m.s. values.
- For resonance to occur, the presence of both L and C elements in the circuit is a must.
- In series resonant circuit, current is maximum at resonance. In a parallel resonant circuit, current is minimum (or zero) at resonance but p.d across the combination is maximum.
- To depict oscillatory motion mathematically we may use sines, cosines or their linear combination. This is because changing the zero position transforms one into another.
- While adding voltage across different elements in an a.c. circuit we should take care of their phases.
- The average current over a complete cycle in an a.c circuit is zero but the average power is not zero.
- An inductor offers negligibly low resistance path to d.c. and a resistive path for a.c.
- A capacitor acts as a block for d.c and a low resistance path to a.c.
- The principle of electric meter is heating effect of current. These meters give the reading of Irms. It is important to note that these meters can measure D.C. as well as A.C.
- D.C. flows through the cross-section of the conductor whereas A.C. flows mainly along the surface of the conductor. This is also known as Skin Effect. The skin effect is directly proportional to the frequency.

### VARYING CURRENT

#### GROWTH OF CURRENT

#### DECAY OF CURRENT

### LC OSCILLATIONS

So q = q0cosωt, then

### TRANSFORMER

- Step up transformer : It converts low voltage into high voltage.
- Step down transformer : It converts high voltage into low voltage.

#### COMPARATIVE STUDY OF STEP-UP TRANSFORMER AND STEP-DOWN TRANSFORMER

#### POWER LOSSES IN A TRANSFORMER

- Copper loss. This is due to resistance of the winding of primary and secondary coil (I2 R)
- Iron loss or Eddy current loss.
- Loss due to leakage of magnetic flux.
- Hysteresis : Due to repeated magnetisation and demagnetisation of iron core.
- Humming loss : Due to vibration.