In this post we are going to discuss about how AC waveform works in circuit theory and we also see some main formulas which we can use to calculate different AC waveform values.
Difference Between AC Waveform and DC Waveform
So first we must understand that AC and DC are totally different. In electronics we mostly deal with both AC and DC types, but their behavior is not same at all.
Now the sinusoidal waveform or what we simply call sine wave, that is the most common and widely used AC waveform in all circuit theories. We can see that this type of waveform keeps changing direction, and this change happens in fixed time intervals. So this fixed time after which the polarity of voltage changes or reverses, that we call waveform period.
But if we talk about direct current or DC, then that is totally unidirectional. Means it always flows in only one direction and never changes. This is why we call it “direct”.
Mostly DC comes from power supplies, or batteries, or dynamos or solar panels. In all these, current flows in one direction only.
So in DC, voltage or current will have one fixed direction and one fixed level or magnitude. For example when we say +9V, it means 9 volts flowing in positive direction. And if we say -12V, it means 12 volts in reverse or negative direction. That is how we show DC.
Now DC power supplies always stay stable and fixed unless we manually reverse the connections. So it keeps its value same all the time, never going up or down like AC.
But in Alternating Current or AC, the case is totally different. This one is bi-directional, means it keeps swinging in both directions, positive and negative and also goes up and down in amplitude or strength.
We can represent AC mathematically like this:
A(t) = Amax × sin(2πft)
Here,
- A(t) is the value of the AC at any time t,
- Amax is the highest value it can go (peak),
- f is frequency,
- t is time.
This is the usual sinusoidal formula for AC signals.
What is Sinusoidal Waveform
In electrical engineering when we say sine wave or sinusoidal waveform, we are simply talking about one very standard AC type waveform.
This waveform keeps changing its polarity, one moment it goes up positive, then comes down zero, then goes down negative. Then again back up. So this goes on continuously. This full up and down we call one cycle.
Like in our homes, the AC mains voltage is exactly of this sinusoidal type. Thats why it works so well for appliances.
Such AC waveforms are normally produced by rotating electric generators or alternators. These are called time-varying signals, because they keep changing with time.
And one important thing: alternating current or voltage cannot be stored like DC. We cannot put it in batteries. It has to be generated fresh using generators, whenever we need it.
The shape and frequency of AC will depend on the generator design. But all AC signals will always have one zero axis line in the middle. The waveform goes equally above and below this line.
AC Waveform Characteristics
When we analyze AC waveform, we need to look at these main things:
- Period (T) – This is time taken for one full cycle. In sine wave it’s called Periodic Time but in square waves it’s also called Pulse Width.
- Frequency (f) – This is how many cycles are happening every second. Unit is Hertz (Hz). Formula is: f = 1 / T
- Amplitude (A) – This is how strong the wave is. Like how many volts or amps it is reaching at peak.
Now waveforms basically show us how voltage or current is changing over time.
The middle horizontal line in the waveform is zero level. When signal goes above it, current flows in one direction. When below it, current goes in opposite direction.
Sine waves are always symmetric, they go same height above and below the zero axis. But other waveforms like audio signals, they can be uneven.
Even though sinusoidal is most popular in power systems, not all AC signals are sine-shaped. Some can be square, triangle, sawtooth, etc.
Relationship Between Cycle, Periodic Time and Frequency
So now let is understand how cycle, time, and frequency are linked.
One cycle (we also call it T) is the time it takes to complete one full up-and-down wave, that is positive half plus negative half together.
Now frequency (ƒ) which we write in Hertz (Hz), tells us how many such cycles happen in one second.
These are connected through a basic formula:
- f = 1 / T
- T = 1 / f
So if we know how many cycles per second, then we can know time per cycle. And if we know time per cycle, then we can find frequency.
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.
Long time ago, people were using the word “cycles per second” (cps) to define how many times the waveform is repeating in one second. But now, in modern systems we just use the word Hertz (Hz) to show the same thing. So 1 Hz means 1 cycle per second, simple.
Now depending on the country where we live, the AC mains frequency can be either 50 Hz or 60 Hz. Like in India and UK it’s 50 Hz, in USA it’s 60 Hz. This value is fixed based on how fast the generator or turbine is spinning in the power plant.
Because Hertz is a very small unit when frequency becomes high we use shortcut prefixes like:
kHz = kilohertz = 1000 Hz,
MHz = megahertz = 1 million Hz,
GHz = gigahertz = 1 billion Hz.
So for high frequency stuff like radio, Wi-Fi, etc., we use MHz and GHz.
AC Waveform and its Amplitude
So in AC waveform, we always talk about three main things:
Frequency – how many cycles in 1 second,
Amplitude – how high the voltage or current reaches,
Periodic Time – time taken for one full cycle.
Now the Amplitude is normally written as Vmax for voltage and Imax for current. This is the peak value of the waveform – means the highest point it can reach in any half cycle.
AC is not steady like DC. In DC, the value stays same all the time. But in AC, the value keeps changing, from zero to peak, then down to zero, then to negative peak, then back to zero again.
In complex or non-sinusoidal waveforms, this peak value can change from one cycle to next. But in pure sinusoidal waveform, the peak value is always same, so we say +Vm = -Vm.
Sometimes instead of showing just one peak, we want to show full swing from top to bottom. That time we use peak-to-peak voltage which we write as Vp-p. This means the voltage from maximum positive point to maximum negative point.
Average Value of AC Waveform
In DC, there is no up-down, so the average value is same as peak value.
But in AC, if we try to calculate the average value over one full cycle, it becomes zero. Because positive half cancels negative half.
So to get a meaningful average value for AC, we must calculate it only for one half cycle, not full cycle. That gives us the proper average.
How to Find Average Value of AC Using Math Rules
Now, if we want to calculate the average value of an AC waveform properly then we can use some smart math tricks. These are called Simpson’s rule, mid-ordinate rule and trapezoidal rule.
For nonlinear waveforms, means where the curve is not smooth or not regular, we can use the mid-ordinate rule which is actually very simple and works quite nicely. It helps us to remove the area under the curve which is same as average value, right?
But one thing here, if we want more accurate result then we must increase the number of ordinate lines, means the more vertical slices we make in the waveform, the better and correct our average value becomes.
Here is how we do it:
We divide the zero-axis baseline into equal parts like V1, V2, V3, up to V9 or more. So now we have several points on the waveform.
Then we take each of these instantaneous values, add them all together and then divide the total by the number of parts. That gives us the mean or average value of the waveform for that half cycle.
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
So now, earlier we saw that the average value of a sine wave is around 0.637 × Vmax, right? But we must understand one thing, this average value is not same as DC supply value.
In DC, the voltage or current stays fixed and never changes but in AC the waveform is always changing, up and down, right? So, AC has no one fixed value.
That is why, we have something called “Effective Value” or sometimes also called RMS value which tells us how much power the AC will give us if we compare it with DC.
Means, if we apply an AC waveform and a DC supply to same load then the RMS value of the AC will produce the same heat or power in the load like the DC does using I² × R formula.
So that is why, effective value or RMS (Root Mean Square) value is very important. We show it like Vrms or Irms, and it is found by taking the square root of the mean of all squared instantaneous values of the waveform.
To do that, we take all mid-ordinate values of the sine wave then square them, and then add all of them, divide by total count, and then finally take the square root of that result. That gives us the RMS.
Here’s the formula to find it:
VRMS = √(V21 + V22 + V23 + V24 + …. V2n) / n
Where “n” indicates the number of mid-ordinates.
RMS Value Compared with Average and Peak in AC
So now in AC sine wave, the RMS value is always equal to 1 divided by √2 times the maximum value or we can say 0.707 × Vmax, same for voltage and for current also. This rule always works for pure sinusoidal waveform.
But now we must know one more thing that the RMS value is usually more than average value, unless we are using a rectangular waveform where all values are same always, due to constant heating effect.
So in most waveforms, RMS is higher than average and that is why we use multimeters to measure RMS values, not the average ones.
In calculations we must be very careful. We must use VRMS only with IRMS and Vpeak only with Ipeak. We cannot mix them or else we will get wrong and confusing answers.
What is Form Factor and Crest Factor
Now let us understand two more things, Form Factor and Crest Factor. These are not very common today, but they help to know how the actual waveform shape looks.
So Form Factor means the ratio between RMS and Average value. We can write it like this:
Form Factor = RMS Value / Average Value
= (0.707 × Vmax) / (0.637 × Vmax)
When we solve this, we get:
Form Factor = 1.11 (for sine wave always)
Then Crest Factor tells us the ratio between peak value and RMS. So it looks like this:
Crest Factor = Peak Value / RMS Value
= Vmax / (0.707 × Vmax)
When we calculate this:
Crest Factor = 1.414 (for pure sine wave always)
So now we know, for sine waveform, form factor is 1.11 and crest factor is 1.414.
AC Waveform Calculation (Example #2)
Now let us take one example to understand better.
Suppose we have 5 amps current flowing through a 50-ohm resistor in a sinusoidal AC circuit.
First, we calculate VRMS like this:
VRMS = I × R = 5 × 50 = 250 V
Now to get the average voltage we use Form Factor:
Average = RMS / Form Factor = 250 / 1.11 = 225 V
Then we calculate the peak voltage like this:
Vpeak = RMS × 1.414 = 250 × 1.414 = 353 V
So, this is how we understand and calculate all important values in AC, like average, RMS, and peak using simple formulas and sinewave basics.
References: AC Waveform
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