Megamox

A friend asked me to help him answer a question for his past paper revision notes, and apparantly i got it wrong. Now i look silly!. The question was:

What value of capacitor must be added in series with a 1k ohm resistor, and signal source (1v peak @ 1khz) to complete the potential divider and reduce the peak voltage to half.

It should be 92nf right???

His Teacher marked it wrong and said the answer was 159uf - with a little note on the side "Reactance of 90nf does NOT equal 1k ohm at 1khz".

Who's right??

Megamox

Roff

You're both wrong. The answer is 276nF.

Vo/Vin=(1/RC)/(jw+1/RC), where w=radian frequency=2*pi*F

|Vo/Vin|=(1/RC)/sqrt(w^2+(1/RC)^2)=0.5, by problem definition

F=1000, R=1000

solving for C,

C=sqrt(3)/(w*R)

C=275.66nF

The teacher's solution (probably 159nF, not uF) does indeed give Xc=1k@1kHz, but due to the associated 45 degree phase shift, the resulting amplitude is 0.707V.

Tell your friend to take this back to the teacher.

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Ron

Megamox

I know why i got 92nf now, i was using the input as 1v RMS. When you feed in 1V RMS you get 0.5v RMS out, with a 92nf cap and 1kohm, @ 1khz

Roff

RMS, peak, whatever - it doesn't matter, so long as you are consistent. The answer is still 276nF. How did you come up with 92nF?

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Ron

Megamox

I made the magnitude of the impedance = 2k and solved for the capacitive reactance. Then rearranged to get C.

My simulator gives out 0.5V RMS, with an input 1v RMS?

Roff

2k :?: :?: :?: Why 2k :?: :?: :?:
Anyhow, the impedance of 92nF at 1kHz is 1.73k.

And your simulator is either wrong, or you are using it wrong, or misinterpreting it. Can you post the schematic and the waveform (or Bode plot)?

Below is a sim from SwitcherCAD.

Attached Images

	rc_divider_sch.png (5.0 KB, 315 views)
	rc_divider_waves.png (17.8 KB, 315 views)

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Ron

ljcox

I agree with Megamox. If you assume the output voltage is across the resistor then.

Vo/Vi = R/(R + 1/SC) = 0.5 Let SC = jwC

Vo/Vi = R/(R - j/wC) = 0.5

Thus wRC/(wRC -j) = 0.5

Take the modulus wRC/sqrt{(wRC)^2 + 1} = 0.5

Square both sides and rearrange 4(wRC)^2 = (wRC)^2 + 1

3(wRC)^2 = 1 gives C = 1/{sqrt(3) wR}

C = 92 nF

Edit: After posting this, I noticed Ron's latest post. He has the output across the capacitor. Whereas, I assumed it was across the resistor. So I'll re-do the maths to see if it agrees with Ron's simulation.
If R = 1000 ohm, w = 2 pi * 1000

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Len

Megamox

Someone tell me im not going crazy :shock:

Attached Images

reactance1k.jpg (38.2 KB, 309 views)

ljcox

See my edit.

I would appear that the difference is whther you take the output across the resistor or the cap. I'll do the maths and see if I get the same answer as Ron if the output is across the cap.

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Len

ljcox

I transposed the R and C and re-did the maths. This way, the answer is 275 nF as Ron said.

So we are all right! (except your teacher)

I had a feeling that the Xc of the cap would not = 1 k for the output to be half.

The relationship between the supply voltage and the voltages across the R and the C is not linear. They are related by Pythagorus (I'm not sure about the spelling)

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Len

Megamox

Ahhhhhh.... yeah i got the same. Thanks guys!
Id hate to be a teacher and have to mark all this

Megamox

Roff

Len, you're right. Megamox, please accept my apologies. As Len said, I assumed the lowpass configuration. I think the original wording implied the highpass config, which I didn't even consider. I guess I was only half wrong. ops:
Anyhow, the teacher needs some re-education.

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Ron

Roff

Pretty amazing, when you consider Pythagoras had never heard of a capacitor (or even a condenser, or a Leyden jar, for that matter).

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Ron

ljcox

Its all Greek to me! You were not actually wrong Ron, I just assumed the output was across the resistor as it seemed to be easier in the maths.

Actually it is slightly easier the other way.

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Len

Roff

Yeah, I know. I just felt bad for chiding Megamox while not considering that there were two solutions.
God, I hope Megamox has a way to get this back to the teacher. We have enough guys giving bad advice in this forum. We don't need teachers creating more of them! :roll:

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Ron