# Astable Multivibrator Calculations.

Discussion in 'General Electronics Discussion' started by eptheta, Dec 23, 2010.

1. ### eptheta

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Dec 20, 2009
I understood the concept of operation of the multivibrator (astable) from the Wikipedia article, but I am having trouble framing equations for the math.

From what i can see, the capacitor needs to get charged to 0.6V before it discharges and is charged from the other side.
But for the time in which it gets charged to 0.6V, the other capacitor has already charged a whole lot(a value which I can't figure out because the setup does not work until i reach one of the two states-Q1 is on or Q2 is on)

So how do i frame equations for the time period and frequency, which has conveniently become T=ln(2)*2RC (if R1=R2 and C1=C2)

Thank you.

2. ### eptheta

188
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Dec 20, 2009
I understood it in terms of the 555 chip where op-amps fix the maximum and minimum voltage across the capacitor at 2/3Vcc and 1/3Vcc respectively.

For anyone who cares,
Then it is a simple matter of charging-discharging the capacitor through the resistor using Vo=Ve^(-t/RC), where V is 1/3Vcc and Vo is 2/3Vcc.
which gets t=ln(2)RC

However I do not understand how this works in the astable multivibrator mode where there are no op-amps to actually 'fix' the maximum and minimum voltages.
Only the transistor's base prevents voltage build ups by switching on.

Which is why I still can't figure out the equations needed for an astable multivibrator.

Thanks.

3. ### janagyjr

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Dec 17, 2010
Here are some equations I picked up from some text books I still have (they are for a 555 Timer's astable mode of operation)

Th = 0.693*C*(R1+R2) (Time High)
Tl = 0.693*C*R2 (Time Low)
T = Th + Tl = 0.693*C*(R1+ (2*R2)) (time, or period)

PRR (or PRF) = 1/T = 1.44/C*(R1+(2*R2)) (Pulse Rate Repetition or Pulse Repetition Frequency)

D = (R1 + R2)/(R1 + (2*R2)) (duty cycle)

I hope this helps.

4. ### eptheta

188
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Dec 20, 2009
Yeah thanks, i got that too, but only with respect to the 555 timer.
What I don't seem to understand is how this applies to a standard multivibrator circuit with just transistors resistors and capacitors like this:

It's not that there are op-amps which control the max and min voltage....
The only reason the equation Tl=0.693C(R2) exists is because we assume that the capacitor starts at 1/3Vcc and goes up to 2/3Vcc in time Tl using the capacitor discharge equation.

But in the above astable multivibrator, there is no indication(that I can see at least) of voltage limiting. So in the equation V(t)=Voe^(-t/RC), what is Vo and what is V(t)?

Thank you.

5. ### janagyjr

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Dec 17, 2010
Seems the forums block wiki links?

My education went right into 555 timers and skipped the basics. Sorry that I misunderstood.

6. ### eptheta

188
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Dec 20, 2009
Hey! The link was fine a while ago. Well you got the idea... Thanks anyway.
Steve, are you there ?

Skipped the basics? How did that happen ?

7. ### janagyjr

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Dec 17, 2010
Let me rephrase that:

I was a student at ITT Tech nearly 12 years ago. I had the basics back then. I went over some of the basics at TTC (my current school) but not into the basic-basics. I understand everything, but don't remember all the math (I have my text books). Give me about an hour and I should be able to find those formulas (I'm cooking atm).

8. ### eptheta

188
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Dec 20, 2009
That will be nice! What textbooks did you study from anyway, maybe I can get them myself ?
Most of what I know I've scavenged from the internet... No formal electronics edu, just cheap 12th std physics.... which is why I have no idea about the math (and very little about everything else for that matter)

Thanks

9. ### janagyjr

67
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Dec 17, 2010
Grob's Basic Electronics (they are up to 11th edition, I have 8th), Operational Amplifiers and Linear Integrated Circuits: Theory and Applications, Digital Systems: Principles and Applications (7th edition, don't know what they are up to now) edit: DS: PaA is an experiment book fwiw.

Edit: Grob's 8th edition does not have the astable multivibrator broken down into individual components. They use two amplifiers and go from there.

Looking again at the circuit diagram you provided...all I can assume is that as a whole they provide the voltage limiting. The base of Q1 is connected to the collector of Q2 with C2 in series, and Q2's base is in series with the collector of Q1 with C1 in series. Perhaps that set up (seems like it provides feedback??) is what limits the voltage?

Though as I understand it, the three equal resistors in the 555 are what provide the voltage limiting for the comparators (sorry to go back to it again) in either astable or monostable.

Last edited: Dec 23, 2010
10. ### eptheta

188
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Dec 20, 2009
Yeah, the 3 resistors in the 555 do go to the comparators... I saw an internal diagram of the 555 which then made a lot of sense...

I'm guessing those 5k resistors can be any value as long as there are three of them dividing Vcc into 1/3 and 2/3 for the op amps...
Turns out the 555 was easy to understand... now i have to go and understand how op amps work...

I can kind of understand why the upper limit voltage is 0.6V (I checked it on a simulator circuit sim!-- it does go to 600mV and then discharges), but that's because the base of the transistors are triggered at that voltage.....

Now when C1 is charged to 600mV, I found(on the simulator) that C2 has a voltage of 3.7V(approx) across it. At this time, C1 discharges and C2 begins to be charged the other direction so it goes 3.7V-->0V-->-600mV....

I'm not sure where this maximum 3.7V voltage came from.. I can easily see that the cap will never charge up to Vcc because it will inevitably discharge earlier(because the other cap has to charge only to 600mV), that part is fine.... But I can't seem to find an equation or a definite relation for the maximum voltage on C.

Another thing i can't get my mind on is it's initial state.
All the references start the explanation of the circuit from a state where Q1 is either just turned off or on. But by this time, 3.7V has already been generated across C2.
No one accounts for the time taken to charge the capacitor up to 3.7V, and so there is a missing time period too....

I really don't know how they ended up with the same time period equation: ln(2)RC

If you find any concrete equations it would be really helpful.
Thank you !

11. ### janagyjr

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Dec 17, 2010
Wish I could be of more help, but that's some really theoretical stuff you're dealing with. Perhaps I could ask my instructor at the beginning of next trimester (Jan 3)

edit: I also don't understand the voltage maximums (because the 555 timers come in packages where the operational voltage ranges from 4.8V to 18V). Though you can see this if you take the output on an oscilloscope.

Last edited: Dec 23, 2010
12. ### eptheta

188
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Dec 20, 2009
This is what I understood, please correct me if I am wrong:
The maximum is got by taking out a line from in-between the 3-resistor-voltage-divider in such a way that 2/3Vcc is selected. It doesn't have a fixed value like the astable multivibrator(that's fixed at 0.6V), over here its just 2/3Vcc, so I think their operating voltages actually have nothing to do with that.....The time period still stays constant(charging faster, but to a higher value)
So when the comparator sees anything higher than 2/3Vcc, it switches on the transistor's base, forcing the capacitor to discharge. And since the op-Amp has its own Vc(which can be regulated), again the operational voltage range doesn't matter.

Is the output of the 555 entirely dependent on Vcc (i.e if Vcc is 12V, then the output is a 12V square-wave),
OR is it that after a Vcc of say 5V, all outputs are regulated to 5V...?

If I only had an oscilloscope, I'd find out myself.(Right now I have a use my laptop's sound-card as a cheap oscilloscope--After much disappointment)

Last edited: Dec 24, 2010