IAN LANG ELECTRONICS

An AC waveform as the name suggests alternates in its intensity and often polarity. There are many kinds of AC waveform, and here are a few:

Working Through LCR Problems

Go Back sawtooth

The Sawtooth waveform, which would be output from an oscillator. Note how the amplitude goes up in a curve, and down again in a straight line. The curve indicates a capacitor is involved as it mirrors the charging time.

 

pulse train

The pulse waveform as above would be output from a digital device most commonly. The pulse is on for half the time in a window and off for the other half. The maximum amplitude is 5V and so as it is on for half and off for half if it does this approximately 500 times per second then any analogue device would see half the amplitude voltage and so 2.5V.

 

 

In the above, a direct current would be used and you will notice that polarity never changes, that is the voltage never goes below the zero line. DC cannot do this. However, if an alternator is used, then the polarity of the waveform does change with the rotation of the alternator. It gives the familiar sine wave:

sine wave degrees

and on this the amplitude (i.e maximum value) and polarity change. There's no real way of describing polarity but if we assume the hump above the line to be positive polarity and that below to be negative it's as good a way as any to think of it.

 

The time period of the waveform is given as one feature to the next adjacent feature. So it could be measured in any of these ways:

sine wave degrees
sine wave degrees
sine wave degrees
sine wave degrees

Whereever you measure from and to there is never more than one complete oscillation from peak to peak or trough to trough and never an incomplete oscillation. The time between the red lines as above is the period. If it oscillated once a second, this would give a frequency of one cycle per second, or 1 Hz.

If it was 0.5 seconds this would mean it did two cycles per second, or 2 Hz. If it does it ten times per second then it would do it every 0.1 seconds and have a frequency of 10 Hz.

 

The equation is:

Frequency =  1 / Period

Or in words, frequency is equal to one divided by the period. For example, let's say a cycle was completed in a time of 1/100th of a second, which in decimals is 0.01

 

F= 1/0.01  and doing that you get 100, which is your value in Hertz (Hz) which makes sense because if it's doing it every 1/100th of a second it must do 100 in a second. When you divide 1 by anything it's called a reciprocal. The reason is the answers reciprocate each other. So now put 1/100 in your calculator. Do you get 0.01 ?  Of course you do. That was the period. So:

Frequency =  1 / Period

 

and

 

Period  = 1 / Frequency

All AC waveforms have a frequency of alternation, which is not the case with DC. When we looked at DC on the previous page, we considered only resistance because no frequency was there to cause any other form of impedance to electron flow.

 

In AC circuits impedance is the equivalent of resistance in DC, and is made up of three components.

 

Like DC, there is resistance itself. In a resistor, the resistance is unchanging regardless of the frequency of the AC.  The symbol is R.

 

Capacitive reactance depends on the value (in Farads) of the capacitor and the frequency of the incoming AC. As frequency increases, capacitive reactance decreases.  The symbol is Xc.

 

Inductive reactance depends on the value (in Henries) of the inductor and the frequency of the incoming AC. As frequency increases, so does the inductive reactance. The symbol is XL.

 

The practical upshot is an inductor passes DC but blocks AC. A capacitor passes AC but blocks DC.

 

Now for some maths, as we work out the reactances of some circuits. We will start with capacitive, and a circuit borrowed from Humphris (Electronics Principles)

humphriscapreact

What we have here is a waveform at 1kHz (i.e it alternates 1000 times a second) and a 1 microFarad capacitor. Here comes the equation for capacitive reactance:

Xc = 1 / 2    fC

pibetter

There's a lot of ways to trip up here and the first is in mis-setting your units. Even now I do that. You must turn kHz into Hertz and uF into Farads before you plug them in.

 

1 uF = 1/1000000 (one one-millionth) of a Farad. So 1/1000000 = 0.000001

 

1 kHz = 1000 Hz

 

            is the greek character pi, pronounced pie. Pi is the ratio between the diameter of a circle and its arc. Pi is an irrational number and can be expressed to millions of decimals places without repeating, but for engineering purposes it is always equal to 3.142  leading to the joke on one of my favourite T-shirts:

 

            is approximately 3.142 times better than any other food.

pibetter pibetter

Let's solve that equation in little steps. The best thing is to start with the bottom line where we see

 

2 X pi X f X C

 

or in words two times pi times f times C.

 

Let's do 2 times pi. If pi is 3.142 then 2 times 3.142 is 6.284

 

Plug that in and we get:

 

Xc = 1 /  6.284 X f X c

 

and we worked out values for  f and C above and so let's plug those in:

 

Xc = 1 / 6.284 X  1000 X 0.000001

 

So now we multiply 6.284 by 1000 and get 6284 and so we plug that in:

 

Xc = 1 / 6284 X 0.000001

 

Next we multiply 6284 by 0.000001 and get the tiny number 0.006284 and so:

 

Xc = 1 / 0.006284

 

Spiffing, because now all we have to do is type in our calculators 1 divided by  0.006284 and we get 159.134 and plugging that in:

 

Xc =159.134

 

and don't forget to put your units after:

 

Xc =159.134 ohms.

 

Humphris gives the answer as 159.15R   (R being a typeface symbol to replace          ) .

omega

The discrepancy between the two answers is insignificant and I'm assuming Humphries used pi to a different number of places than we did up there.

 

Let's do another one from Duncan, (Success in Electronics) over the page.

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