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Hello

I'm looking for some help with the following question:

Using the information contained in Section 2.3 of Workbook 2 on Trigonometric Identities, develop an alternative expression for Vo, to show that the output from the multiplier circuit comprises frequency
components at 230 kHz, 250 kHz, and 270 kHz.

Subsequently, plot your alternative mathematical expansion to demonstrate that the two versions of the output signal, Vo, from Part 1 and Part 2, are identical.

Vc = 8cos (2pi250*10^3t)
Va = 5cos (2pi20*10^3t)
Vo = Vc + VcVa/8

Using cos(A)cos(B) = 1/2 [cos(A+B)+cos(A-B)]
Alternative expression:
Vo = 8cos(2pi*250*10^3t)+cos(2pi*270*10^3t)+cos(2pi*230*10^3t)

I have put the original Vo and alternative Vo into desmos and they are not the same, which they should be.

Any help is appreciated.

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Reply 1
Original post by thomas0611
Hello

I'm looking for some help with the following question:

Using the information contained in Section 2.3 of Workbook 2 on Trigonometric Identities, develop an alternative expression for Vo, to show that the output from the multiplier circuit comprises frequency
components at 230 kHz, 250 kHz, and 270 kHz.

Subsequently, plot your alternative mathematical expansion to demonstrate that the two versions of the output signal, Vo, from Part 1 and Part 2, are identical.

Vc = 8cos (2pi250*10^3t)
Va = 5cos (2pi20*10^3t)
Vo = Vc + VcVa/8

Using cos(A)cos(B) = 1/2 [cos(A+B)+cos(A-B)]
Alternative expression:
Vo = 8cos(2pi*250*10^3t)+cos(2pi*270*10^3t)+cos(2pi*230*10^3t)

I have put the original Vo and alternative Vo into desmos and they are not the same, which they should be.

Any help is appreciated.

I have shared the desmos graph for viewing

https://www.desmos.com/calculator/ug1fslgd8n
Reply 2
Original post by thomas0611
Hello

I'm looking for some help with the following question:

Using the information contained in Section 2.3 of Workbook 2 on Trigonometric Identities, develop an alternative expression for Vo, to show that the output from the multiplier circuit comprises frequency
components at 230 kHz, 250 kHz, and 270 kHz.

Subsequently, plot your alternative mathematical expansion to demonstrate that the two versions of the output signal, Vo, from Part 1 and Part 2, are identical.

Vc = 8cos (2pi250*10^3t)
Va = 5cos (2pi20*10^3t)
Vo = Vc + VcVa/8

Using cos(A)cos(B) = 1/2 [cos(A+B)+cos(A-B)]
Alternative expression:
Vo = 8cos(2pi*250*10^3t)+cos(2pi*270*10^3t)+cos(2pi*230*10^3t)

I have put the original Vo and alternative Vo into desmos and they are not the same, which they should be.

Any help is appreciated.

Just using the given product to sum identity on the second, alternative vo expression, then va must be 2cos not 5cos for the two expressions to match. However if the first expression for vo is correct then there must be a multiplier in front of the second and third cos terms in the alternative expression, so it really depends on which you think is correct.
Reply 3
Original post by mqb2766
Just using the given product to sum identity on the second, alternative vo expression, then va must be 2cos not 5cos for the two expressions to match. However if the first expression for vo is correct then there must be a multiplier in front of the second and third cos terms in the alternative expression, so it really depends on which you think is correct.

Vc and Va are the expressions given in the question. To get the first expression I inputted the Vc and Va into the expression given for Vc and that gave me the answer so I don't see how that would not be correct.

I've looked at another example and that works correctly but this example does not seem to match so I must be missing something
Reply 4
Original post by thomas0611
Vc and Va are the expressions given in the question. To get the first expression I inputted the Vc and Va into the expression given for Vc and that gave me the answer so I don't see how that would not be correct.

I've looked at another example and that works correctly but this example does not seem to match so I must be missing something

If va=5... then when you derive the alternative expression for vo you seem to have a multiplier of 2 rather than 5. However, youd have to post your working to highlight where the error is.
Reply 5
Original post by mqb2766
If va=5... then when you derive the alternative expression for vo you seem to have a multiplier of 2 rather than 5. However, youd have to post your working to highlight where the error is.

My workings as follows:
Vo = Vc + VaVc/8

Vo = 8cos(2pi250*10^3t)+[5cos(2pi20*10^3t)8cos(2pi250*10^3t)/8]

Vo = 8cos(2pi250*10^3t)+5cos(2pi20*10^3t)cos(2pi250*10^3t)

Apply cos(A) cos(B) = 1/2(cos(A+B)+cos(A-B)

Vo = 8cos(2pi25010^3t) + 1/2[cos(2pi250+20)10^3t) + cos(2pi(250-20)*10^3t)]

Vo = 8cos(2pi25010^3t) +5/2cos(2pi27010^3t) + cos(2pi230*10^3t)]

I have put 5/2 as I am not fully sure on this part which may well be where the error occurrs

Thanks
Reply 6
Original post by thomas0611
My workings as follows:
Vo = Vc + VaVc/8

Vo = 8cos(2pi250*10^3t)+[5cos(2pi20*10^3t)8cos(2pi250*10^3t)/8]

Vo = 8cos(2pi250*10^3t)+5cos(2pi20*10^3t)cos(2pi250*10^3t)

Apply cos(A) cos(B) = 1/2(cos(A+B)+cos(A-B)

Vo = 8cos(2pi25010^3t) + 1/2[cos(2pi250+20)10^3t) + cos(2pi(250-20)*10^3t)]

Vo = 8cos(2pi25010^3t) +5/2cos(2pi27010^3t) + cos(2pi230*10^3t)]

I have put 5/2 as I am not fully sure on this part which may well be where the error occurrs

Thanks

After applying the product to sum identity you have a 1/2 multiplier but no 5, why? For the identity you have 5cos(A)cos(B) = ...

Your final line is half right, but it seems to be luck, so maybe work though how the appropriate multiplier applies to both cos terms
Reply 7
Original post by mqb2766
After applying the product to sum identity you have a 1/2 multiplier but no 5, why? For the identity you have 5cos(A)cos(B) = ...

Your final line is half right, but it seems to be luck, so maybe work though how the appropriate multiplier applies to both cos terms

That's where I am not fully confident. So the 1/2 should be a 5?

And that 5 applies to both terms?
Reply 8
Original post by thomas0611
That's where I am not fully confident. So the 1/2 should be a 5?

And that 5 applies to both terms?

Youve got 5 cos(A)cos(B) terms, not just 1. So you must be able to work out how that affects the right hand side in the identity
cos(A)cos(B) = 1/2 (cos(A+B)+cos(A-B)
Whatever you do to one side of an equation, you must do to the other to keep balance.
(edited 2 months ago)
Reply 9
Original post by mqb2766
Youve got 5 cos(A)cos(B) terms, not just 1. So you must be able to work out how that affects the right hand side in the identity
cos(A)cos(B) = 1/2 (cos(A+B)+cos(A-B)
Whatever you do to one side of an equation, you must do to the other to keep balance.

Should this be the next step
Vo = 8cos(2pi25010^3t) +5cos(2pi250+20)10^3t) +5 cos(2pi(250-20)*10^3t)]
Reply 10
Original post by thomas0611
Should this be the next step
Vo = 8cos(2pi25010^3t) +5cos(2pi250+20)10^3t) +5 cos(2pi(250-20)*10^3t)]

You could plot it in desmos and its half right, but youve not worked through numbers properly. Start with the previous post and be clear about what youre doing to each side.
Reply 11
Original post by mqb2766
You could plot it in desmos and its half right, but youve not worked through numbers properly. Start with the previous post and be clear about what youre doing to each side.

Is it the multiplier infront of the cos that I am getting wrong?
Reply 12
Original post by thomas0611
Is it the multiplier infront of the cos that I am getting wrong?

yes, just work through the steps clearly and check in desmos if necessary
Reply 13
Original post by mqb2766
yes, just work through the steps clearly and check in desmos if necessary

Is this first bit correct?

Vo = 8cos(2pi250*10^3t)+[5cos(2pi20*10^3t)8cos(2pi25010^3t)/8]
Vo = 8cos(2pi25010^3t)+5cos(2pi20*10^3t)cos(2pi250*10^3t)
Reply 14
Original post by thomas0611
Is this first bit correct?

Vo = 8cos(2pi250*10^3t)+[5cos(2pi20*10^3t)8cos(2pi25010^3t)/8]
Vo = 8cos(2pi25010^3t)+5cos(2pi20*10^3t)cos(2pi250*10^3t)

Yes, you can check in desmos.
Reply 15
Original post by mqb2766
Yes, you can check in desmos.

Does the 8 have to be multiplied by 5? I am completely lost with this now, don't know the steps to take with this
Reply 16
Original post by thomas0611
Does the 8 have to be multiplied by 5? I am completely lost with this now, don't know the steps to take with this

You have roughly
8*cos + 5*cos*cos
The first term is correct, you use the identity for the second term and its multplied by 5. You should be able to do this basic algebra and if unsure, check each derived expression with the original one in desmos.
Reply 17
So is it just the cos that gets multiplied? Sorry I haven't had any examples to go off in the workbook and getting myself confused
Reply 18
Original post by thomas0611
So is it just the cos that gets multiplied? Sorry I haven't had any examples to go off in the workbook and getting myself confused

What do you mean? It should be just basic expressions.
Reply 19
Original post by mqb2766
What do you mean? It should be just basic expressions.

Should be but I'm somehow lost how to solve it

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