Hi guys,
I have a homework question that has me a little stumped.
It asks the following,
If an astable waveform (rectangular waveform) has a mark/space ratio of 0.42, what would be the Duty Cycle for the same waveform expressed in %?

Now I understand how to calculate the T high and T low and then therefore determine the mark/space ratio by T high/T low and even how to calculate the Duty Cycle for a waveform as T high/ T X 100.
The thing that is confusing is that I have no other information other than the ratio. I know a ratio of 1 gives a T high and T low of 50% as they have the same timing.
Just not sure how to approach this. I'm not looking for the answer, however I would benefit from a few clues.

Regards
Dan

 

How would you write the algebraic equation for T(total) in terms of T(high) and T(low)? Given that ratio, can you write the equation for T(high or low) in terms of that ratio and T(low or high)?

 

Mathematics is your language of choice.
Write down what you know about T(high), T(low), T(total).
Write down the definition of mark/space ratio and duty cycle.

 

Thanks for the feedback, this is what I have so far.
T total = T high + T low
MSR = T high/T low
DC = T high/ T x 100
Solve for T high
T high = MSR x T low
Put this into DC formula gives
DC = MSR x T low/ T high + T low x 100
replacing T high in the denominator
DC = MSR x T low/(MSR x T low) + T low x 100
Now solve for t low in MSR = T high/T low
T low = T high /MSR
substitute T low in derived DC formula
DC = MSR x (T high/MSR)/MSR x (T high/MSR) + T high/MSR x 100
Top and bottom cancel leaving
DC = T high/MSR x 100
So I think I have made an error as I believe T high should have cancelled somewhere, not sure where though. Or the other alternative is that this is completely wrong.
Would appreciate feedback/guidance.

Regards
Dan

 

DC = T high/ T x 100

Typo? Have another look at the definition of duty cycle.

 

Leave out the x100 for now. Apply x100 afterwards, i.e. 0.25 = 25%

DC = TH / (TH + TL)
MSR =TH / TL

Substitute for TH or TL.
Either will do.

 

Hi Guys,
I'm still missing something here as I keep getting the same answer. I must be doing something wrong but cannot pin point the error.
This is what I did.
DC = TH / (TH + TL)
I substituted for TL
DC = TH / (TH + TH/MSR)
I tried to implement the ear rule
DC = TH / 1/ (TH / 1 + TH / MSR)
DC = TH MSR / 2TH
DC = MSR / TH

This is opposite to what I had earlier. Maybe I should sleep on it as I am pretty tired.

 

Hi Guys,
I'm still missing something here as I keep getting the same answer. I must be doing something wrong but cannot pin point the error.
This is what I did.
DC = TH / (TH + TL)
I substituted for TL
DC = TH / (TH + TH/MSR)
I tried to implement the ear rule
DC = TH / 1/ (TH / 1 + TH / MSR)
DC = TH MSR / 2TH
DC = MSR / TH

This is opposite to what I had earlier. Maybe I should sleep on it as I am pretty tired.

What's the "ear rule"? I assume some mnemonic to help you remember how to do some basic algebraic manipulation?

Look at your next to last line. What are the units? Then look at your last line. What are the units?

You need to be much more careful with your order of operations. When you write

X = Y / 2Z

what you are really writing is

X = YZ/2

This is because mutliplication and division have the same precedence and are left associative, so the division is done first and then the multiplication

X = (Y/2)·Z

When you write

DC = TH / 1/ (TH / 1 + TH / MSR)

I have no idea what YOU meant for the order of operations. What you wrote is

DC = (TH / 1) / ( (TH / 1) + (TH / MSR) )

While this is correct, is it how YOU are interpreting it?

How do you get from this to your next line?

Be very careful and explicit in your steps and make liberal use of parenthesis to ensure that your expressions actually match what you intended them to be.

 

Hi Guys,
I'm still missing something here as I keep getting the same answer. I must be doing something wrong but cannot pin point the error.
This is what I did.
DC = TH / (TH + TL)
I substituted for TL
DC = TH / (TH + TH/MSR)

You are correct to this point.
Now multiply numerator and denominator by MSR and show me what you have.

Edit: I have never heard of the "EAR" rule.

 

Hi Guys,
I appologise for the confusion WBhan, in reflection of my post last night it does look messy. I was very tired.
Anyway as for the Ear rule, it's probably not it's real name, however I remember it as that due to my maths teacher coining the term. it is as follows.
(a) / (b) / (c) / (d) = (ad) / (bc)
Looking again at my work I see where I have made my error. Please stay with me.
DC = (TH / 1) / ((TH / 1) + (TH / MSR))
Using the the "Ear" rule
DC = (TH x MSR) / (TH + (TH x MSR))
I can't explain how I got the denominator to be 2 TH last night, as I said tired and weiry. Anyway If I now substitute the value for MSR back in I get the following.
DC = 0.33TH / (1 + 0.33TH)
One of my errors last night that I made was with the denominator (TH + (TH x MSR)) I for what ever reason had this to be 2TH x MSR which is incorrect.
I thought about that part differently today by using this simple equation x + 2x which is obviously equal to 3x.
So when I apply this to the TH coefficient the denominator is equal to (1 + 0.33TH).
So therefore
DC = 0.33TH / 1.33TH
Obviously TH can now cancel.
DC = 0.33 / 1.33
DC = 0.2481 x 100
DC = 24.81%

Thank you all for trying to bump me in the right direction, it really must have been like watching a car crash.
Thanks for not giving up on me.

Regards
Dan

 

Your math is sloppy.
Where did you state MSR = 0.33?
In post #1, MSR = 0.42

Do not substitute values until the very end.

(1 + 0.33TH) is not equal to 1.33TH.
You will be able to spot and avoid errors such as this if you pay attention to units.
What are the units of 1?
What are the units of 0.33TH?
You cannot add or subtract quantities of different units.

This is correct so far:
DC = (TH x MSR) / (TH + (TH x MSR))

Can you factorize this?
(TH + (TH x MSR))

 

Hi MrChips,
I didn't realise I had entered 0.33. This was from another problem that is similar to the original.

Can you factorize this?
(TH + (TH x MSR))

In factored form TH(1 + MSR)
Ok now please follow.
DC = (TH x MSR) / TH(1 + MSR)
TH can now be cancelled from top and bottom, leaving
DC = MSR / (1 + MSR)
Now substituting in values
DC = 0.42 / (1 + 0.42)
DC = 0.2957 x 100
DC = 29.57%

Your right about the units, it definitely helped looking at it from that perspective. This has identified to me I need to re-acquaint myself with algebraic manipulation and especially factoring. To be honest I was completely oblivious to this before you had pointed it out.
Thanks again for staying with it.
Regards
Dan