## Chapter : 7

## What are Alternating Current ?

## Alternating Current

- Current which changes continuously in magnitude and in direction periodically, is called alternating current.
- It can be represented by a sine curve or a cosine curve.

I = I_{0} sin *ω*t or I = I_{0} cos *ω*t

where I_{0} is amplitude or peak value, I is the instantaneous value of alternating current.

*ω*= 2π/T = 2πv, where*ω*is the angular frequency (rad s^{-1}) T is period of ac and is the frequency of ac.- Alternating emf can similarly be represented as alternating current i.e., E = E
_{0}sin*ω*t and E = E_{0}cos*ω*t.

## Peak And Rms Value Of Alternating Current/voltage

- Peak value is defined as the maximum value reached by an alternating quantity (current/voltage) in one cycle.
- Root mean square (RMS) value of alternating current is defined as that value of steady current which would generate the same amount of heat in a given resistance in a given time, as is done by the alternating current when passed through the same resistance for the same time.

I_{rms}=I_{0}/*√2=*0.707I_{0, }

where I_{0} is the peak value or amplitude of alternating current.

- Similarly, for alternating voltage, E
_{rms}=(E_{0}/√2)=0.707E_{0}

All ac instruments measures rms value of ac.

## Reactance And Impedance

**Inductive Reactance :**It is the opposition offered by the inductor to the flow of alternating current through it. X_{L}= ωL=2πvL**Capacitive Reactance :**It is the opposition offered by the capacitor to the flow of alternating current through it.

X_{C} = (1/ωC)=(1/2πvC)

**Impedance:**The total effective resistance offered by the RLC circuit is called impedance.

$$\text{Z}=\sqrt{\text{R}^2+(\text{X}_\text{L}-\text{X}_\text{C})^2}$$

- Inductive reactance is zero for dc (v= 0) and has a finite value for ac.
- Capacitive reactance in infinite for ac.
- Capacitive reactance in infinite for dc (v= 0) and has a finite value for ac.

## LCR Series Circuit

For V=V_{0} sinωt, I=I_{0}sin(ωt-Φ),

where, I_{0} =(V_{0}/Z)

Here Z is the impedance of the series LCR circuit and is given as,

$$\text{Z}=\sqrt{\text{R}^2+(\text{X}_\text{L}-\text{X}_\text{C})^2} \space \text{or}\space \text{Z}=\sqrt{\text{R}^2+(\omega_\text{L}-\frac{1}{\omega_\text{c}})^2}$$

- The alternating current lags behind the voltage by a phase angle Φ, tan Φ =(X
_{L}-X_{C}/R) - If X
_{L}> X_{C}, tan Φ is positive and the circuit is inductive. - If X
_{L}< X_{C}, tan Φ is negative and the circuit is capacitive.

## Power In AC Circuits

**Instantaneous power:**The power in the ac circuit at any instant of time is called instantaneous power. It is equal to the product of values of alternating voltage and alternating current at that time.**Average Power (P**The power averaged over one full cycle of ac is known as average power or true power._{av}):

P_{av} = V_{rms} I_{rms} cos Φ =(V_{0}I_{0}/2)cosΦ

- Apparent Power (P
_{v}) : The product of virtual voltage (V_{rms}) and virtual current (I_{rms}) in the circuit is known as virtual power. - P
_{v}= V_{rms}I_{rms}=(V_{0}I_{0}/2)

## Power Factor

- Power factor is defined as the ratio of true power to apparent power. It can also be defined as the ratio of the resistance to the impedance of an ac circuit.
- cos Φ =(R/Z)
- Power factor is a unitless and dimension less quantity.

## Wattless Current

The average power associated over a complete cycle with a pure inductor or pure capacitor is zero, even though a current is flowing through them. This current is known as wattless current.

## Ac Generator

- An ac generator produces electrical energy from mechanical energy of rotation of a coil.
- It is based on the phenomenon of electromagnetic induction.
- If a coil of N turns and area A is rotated at revolutions per second in a uniform magnetic field B, then the motional emf produced is ε = NBA
*ω*sin ωt or ε=NBA(2πv) sin*ωt*

## Transformer

- Transformers are used for converting a low alternating voltage to a high alternating voltage and vice-versa.
- It is based on the phenomenon of mutual induction.
- For an ideal transformer, (E
_{s}/E_{p})=(I_{p}/I_{s})=(n_{s}/n_{p})=K, where n_{p}is the number of turns in primary coil and n_{s}is the number of turns in secondary coil. - For a step up transformer, K > 1
- For a step down transformer, K < 1.