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###### HNC MATHS A2 Task 3 (Radio Transmitters)

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7 months ago

Any pointers for the question below? struggling as to where to start.

Your company produces radio transmitters and part of that circuit uses a ring modulator to multiply a high frequency sinusoidal carrier signal with a much lower frequency audio signal to transmit over long distances using amplitude modulation, (AM).

The ring modulator circuit works by multiplying together the 125 kHz carrier signal, Vc, and the 10 kHz audio signal, Va, to produce the transmitted modulated signal output, Vo. The two signals and the output from the ring modulator are described as follows:

𝑉𝑐=4cos(2𝜋125∗103𝑡)

𝑉𝑎=2cos(2𝜋10∗103𝑡)

𝑉𝑜=[4+2cos(2𝜋10∗103𝑡)]cos(2𝜋125∗103𝑡)=[4+𝑉𝑎][𝑉𝑐4]=𝑉𝑐+[𝑉𝑎𝑉𝑐4]

Use one of the graphical packages listed at the beginning of this assignment to plot 𝑉𝑐, 𝑉𝑎, 𝑎𝑛𝑑 𝑉𝑜, and include a screenshot of your plot as part of your submission. Ensure the scale of both axes is carefully chosen to clearly show the detail of all 3 waveforms.

If you drew a smooth curve linking the peaks of the positive half cycles of the output waveform, Vo, which is called the envelope of the waveform, what can you observe about that curve?

Your company produces radio transmitters and part of that circuit uses a ring modulator to multiply a high frequency sinusoidal carrier signal with a much lower frequency audio signal to transmit over long distances using amplitude modulation, (AM).

The ring modulator circuit works by multiplying together the 125 kHz carrier signal, Vc, and the 10 kHz audio signal, Va, to produce the transmitted modulated signal output, Vo. The two signals and the output from the ring modulator are described as follows:

𝑉𝑐=4cos(2𝜋125∗103𝑡)

𝑉𝑎=2cos(2𝜋10∗103𝑡)

𝑉𝑜=[4+2cos(2𝜋10∗103𝑡)]cos(2𝜋125∗103𝑡)=[4+𝑉𝑎][𝑉𝑐4]=𝑉𝑐+[𝑉𝑎𝑉𝑐4]

Use one of the graphical packages listed at the beginning of this assignment to plot 𝑉𝑐, 𝑉𝑎, 𝑎𝑛𝑑 𝑉𝑜, and include a screenshot of your plot as part of your submission. Ensure the scale of both axes is carefully chosen to clearly show the detail of all 3 waveforms.

If you drew a smooth curve linking the peaks of the positive half cycles of the output waveform, Vo, which is called the envelope of the waveform, what can you observe about that curve?

Reply 1

6 months ago

Have you got anywhere with this as of yet? Im currently trying to work out the exact same question.

Ive got a final answer but it looks kind of weird and even more strange when plotted in the desmos graphing sim.

Ive got a final answer but it looks kind of weird and even more strange when plotted in the desmos graphing sim.

Original post by constantine20

Have you got anywhere with this as of yet? Im currently trying to work out the exact same question.

Ive got a final answer but it looks kind of weird and even more strange when plotted in the desmos graphing sim.

Ive got a final answer but it looks kind of weird and even more strange when plotted in the desmos graphing sim.

I did manage to get all 3 graphs done and they seemed correct i didn’t compare my findings though that was my final wave form

Reply 3

6 months ago

Original post by csutton18

I did manage to get all 3 graphs done and they seemed correct i didn’t compare my findings though that was my final wave form

do you have an image of what your graph looks like with all 3 plotted? ive got something plotted but i can not seem to adjust the axis correctly to get all 3 clearly visible. was your image above just the Vo waveform?

Original post by constantine20

do you have an image of what your graph looks like with all 3 plotted? ive got something plotted but i can not seem to adjust the axis correctly to get all 3 clearly visible. was your image above just the Vo waveform?

That graph above is with the 2 put together as 1 equation 1 graph should be as the 4 cos equation the 2nd as the 2 cos equation then the 3rd as above with both together which you can see from the equation above it

Reply 5

6 months ago

Original post by csutton18

That graph above is with the 2 put together as 1 equation 1 graph should be as the 4 cos equation the 2nd as the 2 cos equation then the 3rd as above with both together which you can see from the equation above it

okay so it appears mine is widly different, I can assume ive gone terribly wrong somewhere. ive got an incredibly high frequency for the Vo waveform with 6 cos (3*pi*312625000) i did the 𝑉𝑐 + [𝑉𝑎𝑉𝑐/4] and thats what it came out at. Any hints on where it could have gone wrong?

Im also a little stuck on this part, Ive input the equations into desmos and put them all onto the same graph to compare, but i must be missing something, it seems way too simple for a question.

"If you drew a smooth curve linking the peaks of the positive half cycles of the output waveform, Vo, which is called the envelope of the waveform, what can you observe about that curve?" All i can say is the peaks follow the curve of Va graph... doesn't seem like the right answer lol

My two plots zoomed in and out a little

"If you drew a smooth curve linking the peaks of the positive half cycles of the output waveform, Vo, which is called the envelope of the waveform, what can you observe about that curve?" All i can say is the peaks follow the curve of Va graph... doesn't seem like the right answer lol

My two plots zoomed in and out a little

Reply 8

6 months ago

How did you scale the axis on this? ive plotted the graphs but cant for the life of me find the correct scale to get the curve looking like yours? cheers

(Original post by OsirisMB)Im also a little stuck on this part, Ive input the equations into desmos and put them all onto the same graph to compare, but i must be missing something, it seems way too simple for a question.

"If you drew a smooth curve linking the peaks of the positive half cycles of the output waveform, Vo, which is called the envelope of the waveform, what can you observe about that curve?" All i can say is the peaks follow the curve of Va graph... doesn't seem like the right answer lol

My two plots zoomed in and out a little

how did you scale your axis? ive plotted my graphs but now cant seem to get the axis right to show the curves correctly?

(Original post by OsirisMB)Im also a little stuck on this part, Ive input the equations into desmos and put them all onto the same graph to compare, but i must be missing something, it seems way too simple for a question.

"If you drew a smooth curve linking the peaks of the positive half cycles of the output waveform, Vo, which is called the envelope of the waveform, what can you observe about that curve?" All i can say is the peaks follow the curve of Va graph... doesn't seem like the right answer lol

My two plots zoomed in and out a little

how did you scale your axis? ive plotted my graphs but now cant seem to get the axis right to show the curves correctly?

Original post by constantine20

How did you scale the axis on this? ive plotted the graphs but cant for the life of me find the correct scale to get the curve looking like yours? cheers

(Original post by OsirisMB)Im also a little stuck on this part, Ive input the equations into desmos and put them all onto the same graph to compare, but i must be missing something, it seems way too simple for a question.

"If you drew a smooth curve linking the peaks of the positive half cycles of the output waveform, Vo, which is called the envelope of the waveform, what can you observe about that curve?" All i can say is the peaks follow the curve of Va graph... doesn't seem like the right answer lol

My two plots zoomed in and out a little

how did you scale your axis? ive plotted my graphs but now cant seem to get the axis right to show the curves correctly?

(Original post by OsirisMB)Im also a little stuck on this part, Ive input the equations into desmos and put them all onto the same graph to compare, but i must be missing something, it seems way too simple for a question.

"If you drew a smooth curve linking the peaks of the positive half cycles of the output waveform, Vo, which is called the envelope of the waveform, what can you observe about that curve?" All i can say is the peaks follow the curve of Va graph... doesn't seem like the right answer lol

My two plots zoomed in and out a little

how did you scale your axis? ive plotted my graphs but now cant seem to get the axis right to show the curves correctly?

I think it’s the cog on the left hand side for the scale. If I remember I think it’s -0.00001 to 0.00001

Reply 10

6 months ago

Original post by csutton18

I think it’s the cog on the left hand side for the scale. If I remember I think it’s -0.00001 to 0.00001

amazing thank you for that, that has sorted me out for the image of your second one where its very similar to one of the other plotted graphs. now to play around and find out how to get the graph looking like your top image.

Original post by constantine20

amazing thank you for that, that has sorted me out for the image of your second one where its very similar to one of the other plotted graphs. now to play around and find out how to get the graph looking like your top image.

Your black graph isn’t far off my the looks of it. It did take some playing around to get that graph but not 100% sure if that was correct or not he never mentioned it when he marked my paper so yours could be good enough

Reply 13

5 months ago

was any body able to tell what to label the axes for these sine waves im abit stuck with it dont want to miss anything as im aiming for distinction

Reply 15

5 months ago

Original post by OsirisMB

I didnt actually put any labels on the axes, must have forgotten that but i got a distinction and there were no comments on it from the market. Only thing i did was label the graph with what was plotted, Vo=4cosetcetc

no worries i just ended up labelling the x axis time and the y axis amplitude.

can you give me any tips for part 2

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 115 kHz, 125 kHz, and 135 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

ive been stuck on this for a while at its all i need to complete the assignment as im aiming for distinction level as well am currently doing my study with unicourse but just cant quite get my head around this section

Reply 16

4 months ago

did you figure it out im stuck on this also ?

How did you guys get on? I just submitted mine, not sure if its correct or not but here it is.

First, we must rewrite the signal in a form that shows the components.

Vo = VC + Va Vc/4

Plug the expressions for the two signals into this equation to give.

Vo =4cos(2π125*10^3t) + [2cos(2π10*10^3t) * 4cos (2π125*10^3t)/4]

Simplify to

Vo =4cos(2π125*10^3t) + 2cos(2π10*10^3t) * cos (2π125*10^3t)

Now apply the conversion cos(A)cos(B)=1/2[cos(A−B) +cos(A+B)] to give

Vo = 4cos (2 π125*10^3t) + ½ [cos (2 π125-10) *10^3t) + cos (2 π (125+10) *10^3t)]

This shows that the multiplier circuit comprises frequency components are 115kHz, 125KHz and 135kHz.

Vo = 4cos (2 π125*10^3t) + ½ [cos (2 π 115) *10^3t) + cos (2 π (135) *10^3t)]

First, we must rewrite the signal in a form that shows the components.

Vo = VC + Va Vc/4

Plug the expressions for the two signals into this equation to give.

Vo =4cos(2π125*10^3t) + [2cos(2π10*10^3t) * 4cos (2π125*10^3t)/4]

Simplify to

Vo =4cos(2π125*10^3t) + 2cos(2π10*10^3t) * cos (2π125*10^3t)

Now apply the conversion cos(A)cos(B)=1/2[cos(A−B) +cos(A+B)] to give

Vo = 4cos (2 π125*10^3t) + ½ [cos (2 π125-10) *10^3t) + cos (2 π (125+10) *10^3t)]

This shows that the multiplier circuit comprises frequency components are 115kHz, 125KHz and 135kHz.

Vo = 4cos (2 π125*10^3t) + ½ [cos (2 π 115) *10^3t) + cos (2 π (135) *10^3t)]

Original post by OsirisMB

Im also a little stuck on this part, Ive input the equations into desmos and put them all onto the same graph to compare, but i must be missing something, it seems way too simple for a question.

"If you drew a smooth curve linking the peaks of the positive half cycles of the output waveform, Vo, which is called the envelope of the waveform, what can you observe about that curve?" All i can say is the peaks follow the curve of Va graph... doesn't seem like the right answer lol

My two plots zoomed in and out a little

"If you drew a smooth curve linking the peaks of the positive half cycles of the output waveform, Vo, which is called the envelope of the waveform, what can you observe about that curve?" All i can say is the peaks follow the curve of Va graph... doesn't seem like the right answer lol

My two plots zoomed in and out a little

Hi mate, i'm on this part and got the same answer as you, and yes it does seem too simple, did you go with this answer or have I missed something? cheers...

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