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Wednesday, January 31, 2024

Alternating Current Circuits : Class 12 Physics Free Classnotes

Introduction to Alternating Current, Direct Current, EMF, Current Equations and Basic AC Circuits

Alternating Current

An alternating current or voltage is one whose magnitude changes continuously with time between zero and a maximum value and whose sense of flow reverses periodically. An alternating current varies sinusoidally. Electric generator and power plants produce alternating current or AC.

Alternating current
Alternating current
  • Safe to transfer over longer distances and can provide more power.
  • The frequency of alternating current is 50 Hz or 60 Hz depending upon the country.
  •  It reverses its direction while flowing in a circuit.
  • It is the current of magnitude varying with time.
  • Obtained from AC generator through mains.

Direct Current

When a battery is connected to a circuit, the current moves moves steadily in only one direction, from positive terminal to negative terminal. This is called a direct current or DC.

Direct Current
Direct Current
  • Voltage of DC cannot travel very far until it begins to lose energy.
  • The frequency of Direct current is zero.
  • It flows in one direction in the circuit.
  • It is the current of constant magnitude.
  • Obtained from cell or battery.

EMF and Current Equation

Alternating current or voltage varying as sine function can be represented as 

AC and DC Current

  • Time taken to complete one cycle of variations is called Time Period(T)     T=2π/ω
  • T = Time Period, ω = Angular velocity in rad/s , t = time at any instant
  • i or V reaches to the maximum value at t=T/4 sec.

Peak Value

The maximum value of alternating quantity(i or V) is called its peak value or amplitude.

Mean Square Value (`V^2`) or <`i^2`>

 The average value of instantaneous values in one cycle is called mean square value. It is always positive for one complete cycle.
<`V^2`> = `V_0^2`/2
<`i^2`> = `i_0^2`/2

Root Mean Square (rms) Value of AC/Effective/Virtual

For an alternating current (AC) or voltage that follows a sinusoidal waveform, the root mean square (RMS) value is commonly used to represent the effective or equivalent value. The RMS value is calculated by taking the square root of the mean of the squares of the instantaneous values over a complete cycle.

For a sinusoidal AC current (I) or voltage (V), the RMS value can be calculated using the following formulas:

1. RMS Current `(\(I_{\text{rms}})\))`:
   `\[ I_{\text{rms}} = \frac{I_{\text{peak}}}{\sqrt{2}} \]`

   `\( I_{\text{rms}} \)` is the root mean square current.
   `\( I_{\text{peak}} \)` is the peak value of the AC current.

2. RMS Voltage `(\(V_{\text{rms}})\))`:
  `\[ V_{\text{rms}} = \frac{V_{\text{peak}}}{\sqrt{2}} \]`

   `\( V_{\text{rms}} \)` is the root mean square voltage.
   `\( V_{\text{peak}} \)` is the peak value of the AC voltage.

These formulas are applicable when dealing with sinusoidal waveforms, which are commonly encountered in AC circuits. If the waveform is not sinusoidal, more complex calculations involving integration may be necessary to determine the RMS values.

Mean or Average Value (`i_a` or `V_a`)

The mean or average current `(\(I_{\text{avg}}\))` and voltage `(\(V_{\text{avg}}\))` can be calculated based on the time-dependent current and voltage waveforms. For periodic waveforms like sinusoidal AC, the average values are zero. However, for time-varying but non-periodic waveforms, the average values can be calculated using integration.

For a general time-varying current `(\(i(t)\))` or voltage `(\(v(t)\))` over a time period `\(T\)`, the average can be found using the following formulas:

1. Mean or Average Current `(\(I_{\text{avg}})\)`:
`\[ I_{\text{avg}} = \frac{1}{T} \int_{0}^{T} i(t) \, dt \]`

2. Mean or Average Voltage `(\(V_{\text{avg}})\)`:
`\[ V_{\text{avg}} = \frac{1}{T} \int_{0}^{T} v(t) \, dt \]`

These formulas involve integrating the instantaneous values of current or voltage over a given time period and dividing by the total time. For periodic waveforms, the time period `\(T\)` would be the period of one complete cycle.

For sinusoidal AC waveforms, where the average over one complete cycle is considered, the average current `(\(I_{\text{avg}}\))` and average voltage `(\(V_{\text{avg}}\))` are both zero. The average power `(\(P_{\text{avg}}\))` in a sinusoidal AC circuit is given by:

`\[ P_{\text{avg}} = I_{\text{rms}} \cdot V_{\text{rms}} \cdot \cos(\phi) \]`

`\(I_{\text{rms}}\)` is the root mean square (RMS) current.
`\(V_{\text{rms}}\)` is the root mean square (RMS) voltage.
`\(\phi\)` is the phase angle between the current and voltage waveforms.

This formula provides the average power for sinusoidal AC circuits, taking into account the phase relationship between current and voltage.

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