ALTERNATING CURRENT
Alternating current | Mention of expression for instantaneous current |
Peak and rms values of alternating current and voltage. | |
AC voltage applied to a resistor | Expression for current |
Phase relation between voltage and current | |
Phasor representation | |
AC voltage applied to an inductor | Expression for current |
Phase relation between voltage and current | |
Phasor representation | |
AC voltage applied to a capacitor | Current amplitude |
Average power | |
Phasor diagram | |
AC voltage applied to series LCR circuit | Impedance |
Phasor diagram solution | |
Analytical solution | |
Electrical resonance : resonant frequency, sharpness of resonance | |
Power in AC circuit | Power factor |
Inductive and capacitive circuit | |
Meaning of wattles current | |
LC oscillations | Qualitative explanation |
Frequency of LC oscillations and total energy of LC circuit | |
LC oscillators | |
Transformer | Principle, construction and working |
Step up and step down transformers | |
Sources of energy losses | |
Numericals | Concept based problems |
An alternating voltage v = vm sinωt applied to a resistor R drives a current i = im sinωt in the resistor.
where,
The current is in phase with the applied voltage.
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/1a.png?resize=70%2C35)
For an alternating current i = im sinωt passing through a resistor R, the average power loss P (averaged over a cycle) due to joule heating is
(1/2) im2R .To express it in the same form as the dc power (P = I2R), a special value of current is used. It is called root mean square (rms) current and is denoted by I:![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/2a-2.png?resize=109%2C35)
Similarly, the rms voltage is defined by,
We have, P = IV = I2R
(1/2) im2R .To express it in the same form as the dc power (P = I2R), a special value of current is used. It is called root mean square (rms) current and is denoted by I:
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/2a-2.png?resize=109%2C35)
Similarly, the rms voltage is defined by,
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/2b-1.png?resize=110%2C35)
Inductive reactance:
An ac voltage v = vm sinωt applied to a pure inductor L, drives a current in the inductor i = im sin (ωt – π/2),
where im = vm /XL.
where im = vm /XL.
- XL = ωL is called inductive reactance. The current in the inductor lags the voltage by π/2.
- The average power supplied to an inductor over one complete cycle is zero.
Coulomb’s Law:
An ac voltage v = vm sinωt applied to a capacitor drives a current in the capacitor:
i = im sin (ωt + π/2).
Here,
it is called capacitive reactance.
The average power supplied to a capacitor over one complete cycle is zero.
Here,
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/4a-2.png?resize=131%2C33)
The average power supplied to a capacitor over one complete cycle is zero.
For a series RLC circuit driven by voltage v = vm sinωt, the current is given by I = im sin (ωt + φ)
where
and
where is the phase difference between voltage across the source and current in the circuit is,
is called the impedance of the circuit.
The average power loss over a complete cycle is given by,
P = V I cosφ.
The term cosφ is called the power factor
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/5a-1.png?resize=130%2C30)
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/5b-1.png?resize=112%2C30)
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/5c-1.png?resize=154%2C25)
The term cosφ is called the power factor
An interesting characteristic of a series RLC circuit is the phenomenon of resonance. The circuit exhibits resonance, i.e., the amplitude of the current is maximum at the resonant frequency,
The quality factor Q defined by
is an indicator of the sharpness of the resonance, the higher value of Q indicates sharper peak in the current.
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/6a-4.png?resize=86%2C40)
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/6b-2.png?resize=119%2C40)
Simple harmonic motion:
A circuit containing an inductor L and a capacitor C (initially charged) with no ac source and no resistors exhibits free oscillations. The charge q of the capacitor satisfies the equation of simple harmonic motion:
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/7a-4.png?resize=102%2C40)
and therefore, the frequency ω of free oscillation is
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/7a-4.png?resize=102%2C40)
and therefore, the frequency ω of free oscillation is
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/7b-3.png?resize=86%2C40)
Transformer:
A transformer consists of an iron core on which are bound a primary coil of Np turns and a secondary coil of Ns turns. If the primary coil is connected to an ac source, the primary and secondary voltages are related by
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/8a-1.png?resize=99%2C50)
and the currents are related by
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/8b-1.png?resize=97%2C49)
If the secondary coil has a greater number of turns than the primary, the voltage is stepped-up (Vs > Vp). This type of arrangement is called a step–up transformer.
If the secondary coil has turns less than the primary, we have a step-down transformer.
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/8a-1.png?resize=99%2C50)
and the currents are related by
![](https://i0.wp.com/drrajkumars.com/drla/wp-content/uploads/2020/12/8b-1.png?resize=97%2C49)
If the secondary coil has a greater number of turns than the primary, the voltage is stepped-up (Vs > Vp). This type of arrangement is called a step–up transformer.
If the secondary coil has turns less than the primary, we have a step-down transformer.