AC Fundamentals

The ac fundamentals are necessary for analyzing the electrical networks.

Alternating waveform:

• The alternating quantity such as voltage, current, power etc can take any shape such as triangular, square, sine etc.
• But the most popular shape is sine wave as shown in figure below:

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Some Definitions Related to Alternating Quantities:

Instantaneous Value:

• The instantaneous value of an AC quantity is defined as the value of that quantity at a particular instant of time.
• For example in Figure A, V(t1) is the instantaneous value of the ac voltage at t1 or V(t2) is its instantaneous value at t2.

Cycle:

• In an ac waveform, a particular portion consisting of one positive and negative part repeats many times. Each repetition consisting of one positive and one identical negative part is called as one cycle of the waveform. Refer Figure A.
• If the waveform is plotted by plotting angle on the X-axis in place of time, then cycle is that portion of the waveform corresponding to an angle span of 2ϖ radians.

1 Cycle = 2ϖ radians

Time Period or Periodic Time (T):

• Time period (T) is defined as the time taken in seconds by the waveform of an ac quantity to complete one cycle. After every T seconds, the cycle repeats itself.
• Time period T = Time corresponding to one cycle

Frequency:

• Frequency is defined as the number of cycles completed by an alternating quantity in one second. It is denoted by "f" and its units are cycles/second or Hertz (Hz).
• As the time period (T) is the time in seconds per cycle denoted in seconds/cycle, hence relation between frequency and time period is as follows:
• Frequency (f) = cycles/second = 1/(second/cycle) = 1/T
• Therefore, f = 1/T Hz

Amplitude:

• The maximum value or peak value of an ac quantity is called as its amplitude.
• The amplitude is denoted by V_m for voltage, I_m for current waveform etc.

Angular Velocity (ω):

• The angular velocity (ω) is the rate of change of angle ωt with respect to time.
• Therefore, ω = dθ/dt, where dθ is the change in angle in time dt.
• If dt = T i.e. time period, (one cycle) then the corresponding change in θ is 2ϖ radians.
• Therefore, dθ = 2ϖ
• Therefore, ω = 2ϖ/T
• But I/T = f.
• Therefore, ω = 2ϖf

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Mathematical Expression  for an Alternating Quantity:

The alternating voltage is mathematically expressed as follows:

v(t) = V_msin(2ϖfot)

• v(t) = instantaneous voltage
• V_m  = Peak Value (or maximum value)
• fo = frequency in Hz
• and "sin" represents the shape of the waveform

It can also be represented as 

v(t) = V_msin(ωot)  or  v(t) = V_msin(θ)

• where θ = ωot = 2ϖfot

Similarly an alternating current is mathematically represented as,

i(t) = I_msin(2ϖfot)

• i(t) = instantaneous voltage
• I_m = Peak value
• fo = frequency in Hz

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Effective or RMS Value:

Definition: The effective or RMS value of an AC current is equal to the steady state or DC current that is required to produce the same amount of heat as produced by the AC current provided that the resistance and time for which these currents flow are identical.

RMS value of a sinusoidal waveform (sine or cosine) is equal to 0.707 times its peak value.

I_rms = 0.707 I_m  and  V_rms = 0.707 V_m

• All the ac currents or voltages are expressed as RMS values unless clearly specified other.
• RMS value of ac current is denoted by I_rms and RMS voltage is denoted by V_rms.
• RMS value is called as the heat producing component of ac current.

Average Value:

Definition: The average value of an alternating quantity is equal to the average of all, the instantaneous values over a period of half cycle.

The average value of ac current denoted by I_av or I_dc. The average value of a sinusoidal waveform is equal to 0.637 times its peak value. And is expressed as:

I_av = 0.637 I_m  and  V_av = 0.637 V_m

• The dc ammeters or voltmeters indicate the average value.
• Average value of a full cycle of a symmetrical ac waveform is zero.

Form Factor:

The form factor of an alternating quantity is defined as the ratio of its RMS value to its average.

Therfore, Form factor, K_f = RMS value / Average value

• Form factor is dimensionless quantity and its value is always higher than one.
• Form factor of a sinusoidal alternating current is given by:
• K_f = I_rms / I_av = 0.707I_m / 0.637I_m
• Therefore, K = 1.11