So far we kept limited our discussion to Direct Current (DC) circuits. In contrast to the DC circuit, Alternating Current (AC) circuit have such elements that vary its magnitude and direction with time. In the case of the DC circuit, the direction and magnitude of current and voltage in the circuit don’t change with time expect transients in inductor and capacitors. The following figure shows the comparison of DC and AC, it is shown that the direction and magnitude of DC are constant while they are changing for AC around the zero point with respect to time.

The above graph shows that DC can be represented by only magnitude and direction while these terms are insufficient for AC. To introduce the concept of AC completely, we have to define some other terms.

The following animation shows the behavior of AC current through the resistor, assuming that the magnitude is changed according to the waveform.

## alternating current definitions:

### Waveform:

The trace of the quantity i.e. voltage or current against time, degrees, radian etc. is called a waveform.

### Instantaneous value:

The magnitude of any quantity like voltage or current against the specific value of time, degrees, radian etc. is called instantaneous value.

### Peak Amplitude:

The maximum value of quantity like voltage and current it reaches the waveform.

### Peak-Peak value:

The difference between the maximum and minimum value of the waveform is called peak to peak value.

### Periodic waveform:

If a portion of a waveform repeats itself after a specific time, degree, radian etc. The waveform is called a periodic waveform. And that portion of the waveform is called cycle.

### Time Period:

The time required for the waveform to complete itself is called Time period. It is denoted by T in seconds.

### Cycle:

The waveform of one Time Period is called cycle.

### Frequency:

The number of cycles in one Time Period is called frequency. Frequency is represented in cycles per seconds or Hertz (name after scientist).

$f=\frac{1}{T} $ (in Hertz OR $\frac{cycle}{second}$)

### Sinewave:

The waveform of AC voltage is most of the time sinusoidal (sine wave or cosine wave) unless it is stated. It is mainly that resistance, capacitance or inductance responses have no effect on the waveform. The sinusoidal waves can be plotted against degrees or radians. Where degrees and radian has the following relationship

$360 \quad deg=2\pi \quad radians$

Where $\pi $ is the ratio between the circumference of circle and diameter of the circle and is equal to 22/7.

### Angular Velocity:

If a radius vector of a circle rotates around its center at a constant speed, the vertical projection of the vector can be plotted as a sine wave. The speed at which the vector rotates is called angular velocity.

$Angular Velocity=\frac{Distance (time or radians)}{Time (seconds)}$

$\omega =\frac{\alpha }{t}=\frac{2\pi }{T}=2\pi f$

The sine wave for voltage can be represented by the following formula.

$e=E_{m} Sin \omega t$

Where $E_{m}$ shows the peak voltage of the voltage waveform and $\omega $ is the angular velocity of the waveform.

### Average Value of a waveform:

The average value of sine wave is the ratio of the area under the sine wave to the length of the sinusoidal. Looking at the waveform, it is clear that for the first half it is positive and for the second half its is negative and both are same in magnitude for each complete cycle. So the average of AC sinusoidal is zero. The formula for average calculation is

$V_{avg}=\frac{\int_{0}^{T}{V_{m} sin\alpha d\alpha }}{T-T_{0}}$

Where T-T$_{0}$ is the time period of the AC sine wave and $V_{m}\sin \alpha $ is the AC signal.

### Root-Mean-Square (RMS) Value:

Th RMS value of the AC signal provides a relationship between the AC and DC. The RMS value of the AC sinusoidal gives the information of delivering the same power by the corresponding DC source. The RMS value of the AC signal is used generally unless it is stated. The formula for the RMS value of an AC signal is

$V_{rms}=\frac{V_{m}}{\sqrt{2}}$

Where $V_{m}$ is the peak voltage of the AC sinusoidal. The relationship holds for current too.

### Sinusoidal voltage and current format:

The basic mathematical format for sinusoidal voltage is

$v=V_{m}\sin \alpha $

Where

$\alpha =\omega t$

We get

$v=V_{m}\sin \omega t$

Where V$_{m}$ is the peak voltage of the waveform, ω is the angular velocity of the waveform and t is the time duration from the rotation beginning. The ω, angular velocity represents the position of the instant use value at the waveform where the time t, determine the numbers of rotations.

Remember that the lowercase letter indicates the instantaneous value where the uppercase letter with subscripts of ‘m’ means the peak value i.e. v and V_{m}.

### Phase relation:

In simple words, the phase shows the relationship among multiple waveforms, how soon and late each one of them started, achieved the peak and/or reach it’s zero and final value. Suppose we have a first waveform of $I_{m}\sin (\alpha +0)$ and a second waveform of $V_{m}\sin (\alpha +30^{o })$, their waveforms could be shown as follow.

The first waveform will start from the zero point at axes where the second will start 30^{o} later.

The general form of the phase relation equation is

$v=V_{m}\sin \alpha \pm \theta $

The phase relation of two or more waveforms can be found through an analog and digital oscilloscope. Where the magnitude of AC quantiles is measured in RMS value through digital multi-meter and voltmeter and ammeter. A special instrument for measurement of frequency meters is used for measurement of frequency.

## Conclusion :

Alternating current changes its magnitude and direction every moment where direct current remains same in direction and magnitude. Our home appliances are using AC where electronic devices like handheld devices are using DC power.