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AC Waveform and AC Circuit Theory

AC Waveform and AC Circuit Theory

AC Sinusoidal Waveforms are created by rotating a coil within a magnetic field and alternating voltages and currents form the basis of AC Theory

The AC waveform used the most in circuit theory is that of the sinusoidal waveform or sine wave. A periodic AC waveform in the form of a voltage source produces an EMF whose polarity reverses at regular intervals with the time required to complete one full reversal being known as the waveforms period.

Direct Current or D.C. as it is more commonly called, is a form of electrical current or voltage that flows around an electrical circuit in one direction only, making it a “Uni-directional” supply.

Generally, both DC currents and voltages are produced by power supplies, batteries, dynamos and solar cells to name a few. A DC voltage or current has a fixed magnitude (amplitude) and a definite direction associated with it. For example, +12V represents 12 volts in the positive direction, or -5V represents 5 volts in the negative direction.

We also know that DC power supplies do not change their value with regards to time, they are a constant value flowing in a continuous steady state direction. In other words, DC maintains the same value for all times and a constant uni-directional DC supply never changes or becomes negative unless its connections are physically reversed. An example of a simple DC or direct current circuit is shown below.

DC Circuit and Waveform

DC circuit and waveform

An alternating function or AC Waveform on the other hand is defined as one that varies in both magnitude and direction in more or less an even manner with respect to time making it a “Bi-directional” waveform. An AC function can represent either a power source or a signal source with the shape of an AC waveform generally following that of a mathematical sinusoid being defined as: A(t) = Amax*sin(2πƒt).

The term AC or to give it its full description of Alternating Current, generally refers to a time-varying waveform with the most common of all being called a Sinusoid better known as a Sinusoidal Waveform.

Sinusoidal waveforms are more generally called by their short description as Sine Waves. Sine waves are by far one of the most important types of AC waveform used in electrical engineering.

The shape obtained by plotting the instantaneous ordinate values of either voltage or current against time is called an AC Waveform. An AC waveform is constantly changing its polarity every half cycle alternating between a positive maximum value and a negative maximum value respectively with regards to time with a common example of this being the domestic mains voltage supply we use in our homes.

This means then that the AC Waveform is a “time-dependent signal” with the most common type of time-dependant signal being that of the Periodic Waveform. The periodic or AC waveform is the resulting product of a rotating electrical generator.

Generally, the shape of any periodic waveform can be generated using a fundamental frequency and superimposing it with harmonic signals of varying frequencies and amplitudes but that’s for another tutorial.

Alternating voltages and currents can not be stored in batteries or cells like direct current (DC) can, it is much easier and cheaper to generate these quantities using alternators or waveform generators when they are needed.

The type and shape of an AC waveform depends upon the generator or device producing them, but all AC waveforms consist of a zero voltage line that divides the waveform into two symmetrical halves. The main characteristics of an AC Waveform are defined as:

AC Waveform Characteristics

  • • The Period, (T) is the length of time in seconds that the waveform takes to repeat itself from start to finish. This can also be called the Periodic Time of the waveform for sine waves, or the Pulse Width for square waves.
  • • The Frequency, (ƒ) is the number of times the waveform repeats itself within a one second time period. Frequency is the reciprocal of the time period, ( ƒ = 1/T ) with the unit of frequency being the Hertz, (Hz).
  • • The Amplitude (A) is the magnitude or intensity of the signal waveform measured in volts or amps.

In our tutorial about Waveforms ,we looked at different types of waveforms and said that “Waveforms are basically a visual representation of the variation of a voltage or current plotted to a base of time”.

Generally, for AC waveforms this horizontal base line represents a zero condition of either voltage or current. Any part of an AC type waveform which lies above the horizontal zero axis represents a voltage or current flowing in one direction.

Likewise, any part of the waveform which lies below the horizontal zero axis represents a voltage or current flowing in the opposite direction to the first. Generally for sinusoidal AC waveforms the shape of the waveform above the zero axis is the same as the shape below it. However, for most non-power AC signals including audio waveforms this is not always the case.

The most common periodic signal waveforms that are used in Electrical and Electronic Engineering are the Sinusoidal Waveforms. However, an alternating AC waveform may not always take the shape of a smooth shape based around the trigonometric sine or cosine function. AC waveforms can also take the shape of either Complex Waves, Square Waves or Triangular Waves and these are shown below.

Types of Periodic Waveform

periodic AC waveform

The time taken for an AC Waveform to complete one full pattern from its positive half to its negative half and back to its zero baseline again is called a Cycle and one complete cycle contains both a positive half-cycle and a negative half-cycle. The time taken by the waveform to complete one full cycle is called the Periodic Time of the waveform, and is given the symbol “T”.

The number of complete cycles that are produced within one second (cycles/second) is called the Frequency, symbol ƒ of the alternating waveform. Frequency is measured in Hertz, ( Hz ) named after the German physicist Heinrich Hertz.

Then we can see that a relationship exists between cycles (oscillations), periodic time and frequency (cycles per second), so if there are ƒ number of cycles in one second, each individual cycle must take 1/ƒ seconds to complete.

Relationship Between Frequency and Periodic Time

frequency and periodic time relationship

AC Waveform Example No1

1. What is the periodic time, (T) of a 50Hz sinusoidal waveform. 2. what will be the oscillating frequency of a waveform that has a periodic time of 10mS.

1. Periodic Time

periodic time

2. Frequency

frequency

Frequency used to be expressed in “cycles per second” abbreviated to “cps”, but today it is more commonly specified in units called “Hertz”. For a domestic mains supply the frequency will be either 50Hz or 60Hz depending upon the country and is fixed by the speed of rotation of the generator. But one hertz is a very small unit so prefixes are used that denote the order of magnitude of the waveform at higher frequencies such as kHz, MHz and even GHz.

Definition of Frequency Prefixes

Prefix Definition Written as Periodic Time
Kilo Thousand kHz 1ms
Mega Million MHz 1us
Giga Billion GHz 1ns
Terra Trillion THz 1ps

Amplitude of an AC Waveform

As well as knowing either the periodic time or the frequency of the alternating quantity, another important parameter of the AC waveform is Amplitude, better known as its Maximum or Peak value represented by the terms, Vmax for voltage or Imax for current.

The peak value is the greatest value of either voltage or current that the waveform reaches during each half cycle measured from the zero baseline. Unlike a DC voltage or current which has a steady state that can be measured or calculated using Ohm’s Law, an alternating quantity is constantly changing its value over time.

For pure sinusoidal waveforms this peak value will always be the same for both half cycles ( +Vm = -Vm ) but for non-sinusoidal or complex waveforms the maximum peak value can be very different for each half cycle.

Sometimes, alternating waveforms are given a peak-to-peak, Vp-p value and this is simply the distance or the sum in voltage between the maximum peak value, +Vmax and the minimum peak value, -Vmax during one complete cycle.

The Average Value of an AC Waveform

The average or mean value of a continuous DC voltage will always be equal to its maximum peak value as a DC voltage is constant. This average value will only change if the duty cycle of the DC voltage changes. In a pure sine wave if the average value is calculated over the full cycle, the average value would be equal to zero as the positive and negative halves will cancel each other out. So the average or mean value of an AC waveform is calculated or measured over a half cycle only and this is shown below.

Average Value of a Non-sinusoidal Waveform

AC waveform average value

To find the average value of the waveform we need to calculate the area underneath the waveform using the mid-ordinate rule, trapezoidal rule or the Simpson’s rule found commonly in mathematics. The approximate area under any irregular waveform can easily be found by simply using the mid-ordinate rule.

The zero axis base line is divided up into any number of equal parts and in our simple example above this value was nine, ( V1 to V9 ). The more ordinate lines that are drawn the more accurate will be the final average or mean value. The average value will be the addition of all the instantaneous values added together and then divided by the total number. This is given as.

Average Value of an AC Waveform

coordinate rule

Where: n equals the actual number of mid-ordinates used.

For a pure sinusoidal waveform this average or mean value will always be equal to 0.637*Vmax and this relationship also holds true for average values of current.

The RMS Value of an AC Waveform

The average value of an AC waveform that we calculated above as being: 0.637*Vmax is NOT the same value we would use for a DC supply. This is because unlike a DC supply which is constant and and of a fixed value, an AC waveform is constantly changing over time and has no fixed value. Thus the equivalent value for an alternating current system that provides the same amount of electrical power to a load as a DC equivalent circuit is called the “effective value”.

The effective value of a sine wave produces the same I2*R heating effect in a load as we would expect to see if the same load was fed by a constant DC supply. The effective value of a sine wave is more commonly known as the Root Mean Squared or simply RMS value as it is calculated as the square root of the mean (average) of the square of the voltage or current.

That is Vrms or Irms is given as the square root of the average of the sum of all the squared mid-ordinate values of the sine wave. The RMS value for any AC waveform can be found from the following modified average value formula as shown.

RMS Value of an AC Waveform

AC waveform rms

Where: n equals the number of mid-ordinates.

For a pure sinusoidal waveform this effective or R.M.S. value will always be equal too: 1/2*Vmax which is equal to 0.707*Vmax and this relationship holds true for RMS values of current. The RMS value for a sinusoidal waveform is always greater than the average value except for a rectangular waveform. In this case the heating effect remains constant so the average and the RMS values will be the same.

One final comment about R.M.S. values. Most multimeters, either digital or analogue unless otherwise stated only measure the R.M.S. values of voltage and current and not the average. Therefore when using a multimeter on a direct current system the reading will be equal to I = V/R and for an alternating current system the reading will be equal to Irms = Vrms/R.

Also, except for average power calculations, when calculating RMS or peak voltages, only use VRMS to find IRMS values, or peak voltage, Vp to find peak current, Ip values. Do not mix them together as Average, RMS or Peak values of a sine wave are completely different and your results will definitely be incorrect.

Form Factor and Crest Factor

Although little used these days, both Form Factor and Crest Factor can be used to give information about the actual shape of the AC waveform. Form Factor is the ratio between the average value and the RMS value and is given as.

AC waveform form factor

For a pure sinusoidal waveform the Form Factor will always be equal to 1.11. Crest Factor is the ratio between the R.M.S. value and the Peak value of the waveform and is given as.

AC waveform crest factor

For a pure sinusoidal waveform the Crest Factor will always be equal to 1.414.

AC Waveform Example No2

A sinusoidal alternating current of 6 amps is flowing through a resistance of 40Ω. Calculate the average voltage and the peak voltage of the supply.

The R.M.S. Voltage value is calculated as:

rms voltage

The Average Voltage value is calculated as:

average voltage

The Peak Voltage value is calculated as:

peak voltage

The use and calculation of Average, R.M.S, Form factor and Crest Factor can also be use with any type of periodic waveform including Triangular, Square, Sawtoothed or any other irregular or complex voltage/current waveform shape. Conversion between the various sinusoidal values can sometimes be confusing so the following table gives a convenient way of converting one sine wave value to another.

Sinusoidal Waveform Conversion Table

Convert From Multiply By Or By To Get Value
Peak 2 (√2)2 Peak-to-Peak
Peak-to-Peak 0.5 1/2 Peak
Peak 0.707 1/(√2) RMS
Peak 0.637 2/π Average
Average 1.570 π/2 Peak
Average 1.111 π/(2√2) RMS
RMS 1.414 2 Peak
RMS 0.901 (2√2)/π Average

In the next tutorial about Sinusoidal Waveforms we will look at the principal of generating a sinusoidal AC waveform (a sinusoid) along with its angular velocity representation.

328 Comments

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  • Very Good

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  • @xi@g@me

    Very good article, but I regret a bit some values are given with no explanations. I will try to cover a bit here.

    First equation, is for the sine wave, sin(2.Pi.f.t).
    f is the frequency of the waveform. if frequency is one, you will need to make a complete turn around the trigonometric circle to have a full period of the sine wave in 1 second (t = 1). A full turn equals to 2 Pi radians, thus the 2 Pi factor here. The formula can also be written sin(omega.t). Omega is equal to 2.Pi.f here, and is called angular velocity.

    To compute average voltage of a sinusoidal waveform, instead of using the smae method as before, we could try to compute the area under the curve, and divide by length.
    The area can be found by calculating the integral of sin(x) between 0 and Pi.
    Let’s say f(x) = sin(x). We need to compute int(0, Pi) f(x) dx. The best way is to find F(x), which is the primitive of f(x), and then compute F(Pi) – F(0). For f(x) = sin(x), F(x) is -cos(x).
    F(Pi )- F(0) = -cos(Pi) – -cos(0), or cos(0) – cos(Pi). 1 – (-1) = 2, so int(0, Pi) sin(x) dx = 2.
    Once we get the area, simply divide by the length (distance on X) to find the average. 2 / Pi = 0.637. Here it is.

    For the R.M.S. voltage, do the same by integrating sin²(x). The primitive of sin²(x) is a bit more complex, and is x/2 + sin(2x) / 4. In that case (F(Pi) – F(0)), we find Pi/2, and dividing by Pi gives 1/2. Square root of (1/2) has a value of 0.707 (which also is 2 divided by the square root of 2).

    Also, here is another way to solve the second exercise. From V_RMS, you can find V_max by simply dividing by 0.707. Then multiply by 0.637 and you will get V_avg. No need to use form factor or crest factor.

    Hope this helps!

    • Wayne Storr

      To clarify your misunderstanding.

      The expression: A(t) = AMAXsin(2πƒt) to which you refer is the generalised equation we can use to find the instantaneous voltage, or current, at any instant in time, or at any angle. Since a sinusoidal waveform is the graph of the mathematical function of: A = sin(x) rotating at a constant speed, we can relate maximum and instantaneous values by the sine of the rotational angle corresponding to that instant.

      If “A” makes ƒ revolutions or complete cycles per second, then ƒ is called the frequency of the waveform given in Hertz. Since one full revolution of “A” around the circumference of a circle generates 360 degrees, it therefore follows that: A = 360ƒt giving the total angle in degrees, generated in a time of “t” seconds. Thus, from the previous equation having a frequency of ƒ hertz, we can also write it as: A(t) = AMAX360ƒt. We can also say that as ƒ is the number of revolutions per second, and since each revolution occurs during 360 degrees, it follows that 360ƒ can be called the angular velocity of the sinusoid in degrees.

      One full revolution of a circle (the circumference) is equal to the mathematical expression of: 2π*radius. If the length of the radius is placed around the circumference of the circle, the central angle of the arc created is given in radians. Since we can put a radius length around the circumference 2π times, it therefore follows that as each arc forms one radian, there must be 2π radians around the whole circle. Then ω (omega) can be defined as: 2πƒ. That is angular velocity in radians per second.

      The average voltage of a sinusoid is covered in the Average Voltage Tutorial

      The RMS, or Root Mean Squared voltage of a sinusoid is covered in the RMS Voltage Tutorial

      The objective of the tutorial and website is to educate the reader about all aspects of electrical and electronic engineering. The Crest Factor and Form Factor are used to describe the shape and quality of a sinusoidal waveform. For a pure sine-wave, the form factor is equal to 1.11, since it is the ratio between the average value and the RMS value. The crest factor is 1.414 (√2) since it is the ratio of the maximum value to the RMS value.

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