In this post we discuss AC waveform circuit theory and various formulas to calculate AC waveform.
Difference Between AC Waveform and DC Waveform
The sinusoidal, or sine, wave, is the most often used AC waveform in circuit theory. The term “waveform period” refers to the regular intervals at which an EMF with reversed polarity is produced by a periodic AC waveform from a voltage source.
Direct current, sometimes known as DC, is a unidirectional source as it only flows in one direction. Usually power supplies, batteries, dynamos, and solar cells produce DC currents and voltages.
A DC voltage or current has a specific direction, and a defined magnitude. For example +9V denotes 9 volts in the positive direction, while -12V denotes 12 volts in the negative direction, and so on.
DC power supplies stay in a consistent steady state direction by holding their value fixed throughout time. Until its connections are explicitly reversed, a unidirectional DC supply remains unchanged and never becomes negative.
In contrast to a direct current, an Alternating Current (AC) waveform is “bi-directional” because it exhibits balanced variations in both amplitude and direction over time.
The formula A(t) = Amax x sin(2π̒t), where Amax is the maximum amplitude, ƒ is the frequency, and t is the duration, is commonly used to depict it as a sinusoidal waveform.
What is Sinusoidal Waveform
In electrical engineering, sine waves, sometimes referred to as sinusoidal waveforms are an essential class of AC waveform.
Every half cycle, the polarity of these waves alternates between peaks that are positive and negative. The mains voltage source in a home serves as a common example.
Periodic waveforms, often known as AC waveforms or “time-dependent signals,” are produced by spinning electrical generators.
Alternating voltages and currents, in contrast to direct current, cannot be stored in batteries; instead they are more effectively produced as needed by power generators or alternators.
The generator determines the exact features and structure of an AC waveform, although all AC waveforms include a zero-voltage line that symmetrically divides them in half.
AC Waveform Characteristics:
- Period (T): Time for one cycle, called Periodic Time for sine waves or Pulse Width for square waves.
- Frequency (ƒ): Cycles per second, measured in Hertz (Hz), with ƒ = 1/T.
- Amplitude (A): Magnitude in volts or amps.
- Waveforms visually represent voltage or current variations over time.
- The horizontal baseline in AC waveforms signifies zero voltage or current.
- Above the zero axis indicates flow in one direction; below signifies the opposite.
- Sinusoidal AC waveforms exhibit symmetry above and below the zero axis.
- Non-power AC signals, like audio, may deviate from this pattern.
- Sinusoidal waveforms are common in Electrical and Electronic Engineering, but not all AC waveforms follow a smooth sine or cosine shape.
Relationship Between Periodic Time, Cycle and Frequency
A cycle (symbolized as “T”) in an AC waveform is the length of time it takes to generate one entire sequence, covering both positive and negative halves.
The frequency denoted by the symbol “ƒ,” is the number of cycles per second expressed in Heinrich Hertz units (Hz). There is a clear link between cycles, periodic time, and frequency: if ƒ cycles per second, then 1/ƒ seconds is needed for each cycle to finish.
The following formula explains the connection between frequency and periodic time:
Frequency (f) = 1 / Periodic Time = 1 / T Hertz
Or, Periodic Time (T) = 1 / Frequency = 1 / f seconds
Calculating AC Waveform (Example#1)
Problem: What is the 50 Hz sinusoidal waveform’s periodic time (T), and what would the oscillation frequency of a waveform with a 10 mS periodic time be?
Solution:
Periodic Time (T) = 1 / f = 1 / 50 = 0.02 seconds or 20 ms.
Frequency (f) = 1 / T = 1 / 10 x 10-3 = 100 Hz.
Originally defined by “cycles per second” (cps), frequency is now more frequently expressed in Hertz (Hz).
Depending on the country in question, a home mains supply’s frequency is usually 50Hz or 60Hz, which is determined by the utility generator’s rotation speed.
Since the Hertz is a tiny quantity, higher frequencies are denoted by prefixes like kHz, MHz, and even GHz.
AC Waveform and its Amplitude
Two essential elements of the AC waveform are frequency, amplitude (represented as Vmax for voltage and Imax for current), and periodic time.
The waveform’s amplitude is the highest or peak value it reaches throughout each half cycle. AC is constantly changing over time, as contrast to DC, which keeps a consistent state.
Peak values in complex or non-sinusoidal waveforms might change, but, in sinusoidal waveforms, the peak value is constant for both half cycles (+Vm = -Vm).
The voltage difference between the maximum and minimum peak values inside a single cycle can alternatively be represented by an alternate waveform, as a peak-to-peak (Vp-p) value.
AC Waveform Average Value
Since DC voltage is constant, it’s highest peak value and average value are always equal.
When calculating the average value of a pure sine wave across its whole cycle, the cancellation of its positive and negative parts results in a value of zero, and therefore an AC waveform’s average value is calculated across one half cycle.
Mathematical techniques like Simpson’s rule, the mid-ordinate rule, and the trapezoidal rule can be used to get the waveform’s average value.
The region below the curve may be approximated for nonlinear waveforms with ease using the mid-ordinate rule.
Increasing the number of ordinate lines improves the reliability of the computed average or mean value.
The zero-axis baseline is divided into equal sections (e.g., V1 to V9). By adding up each instantaneous value and dividing the result by the total number, the average value can be found.
Calculating Average Value of an AC Waveform
Vaverage = V1 + V2 + V3 + V4 + …. + Vn
Where “n” is the number of mid-ordinates, the average or mean value for a pure sinusoidal waveform is consistently 0.637 times Vmax, and this holds true for current averages as well.
AC Waveform and its RMS Value
The average value 0.637 x Vmax, that we computed for the AC waveform is not the same as the value for a DC supply.
An AC waveform has no set value, and is constantly changing, in contrast to a steady and fixed-value DC supply.
“Effective value” refers to the equivalent value in an alternating current system that provides the same electrical power as a DC equivalent circuit.
The I2 x R heating effect which a continuous DC supply would produce in a load is equal to the effective value of a sine wave, sometimes referred to as Root Mean Squared (RMS).
Vrms or Irms are the square roots of the mean of the squared voltage or current values, respectively.
In order to calculate this root mean square (RMS) value, we take the square root of the average of the sum of the squared sine wave mid-ordinate values.
The RMS value of an AC Waveform can be calculated using the following formula:
VRMS = √(V21 + V22 + V23 + V24 + …. V2n) / n
Where “n” signifies the number of mid-ordinates.
The RMS value for a pure sinusoidal waveform is always equal to 1/√2 times the peak value, or 0.707 times Vmax, for both voltage and current.
With the exception of a rectangular waveform, where they are identical because of a continuous heating effect, the RMS value is always higher than the average.
Multimeters are usually used to measure RMS values rather than averages.
Since the average, RMS, and peak values in a sine wave are different from one other, you should use VRMS for IRMS and Vp for Ip when doing the calculations.
Mixing the two might lead to inaccurate results.
Calculating Form Factor and Crest Factor
Although they are rarely used these days, Form Factor and Crest Factor reveal the true form of the AC waveform.
This is accomplished using the form factor, which is defined as the ratio of the average to the RMS values.
Form Factor = R.M.S. Value / Average Value = 0.707 x Vmax / 0.637 x Vmax
The Form Factor, which shows the ratio between the average and RMS values, is always 1.11 in a pure sinusoidal waveform.
Crest Factor = Peak Value / R.M.S. Value = Vmax / 0.707 x Vmax
Conversely, in these waveforms, the Crest Factor—which represents the ratio of RMS to peak values—is consistently 1.414.
Calculating AC Waveform (Example#2)
Calculate the average and peak voltage of a sinusoidal AC with 5 amps flowing through a 50Ω resistance.
The RMS voltage is determined by the formula:
VRMS = I x R = 5 x 50 = 250 V
The average voltage can be calculated using the following formula:
Form Factor = VRMS / VAverage = 250 / 1.11 = 225 V
To calculate Peak Voltage, we can use the following formula:
Peak Voltage = R.M.S. x 1.414 = 250 x 1.414 = 353 V
References: AC Waveform