Sensor - Calculate the Capacitance

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SUMMARY

The discussion revolves around calculating the additional capacitance required for an airbag sensor that utilizes accelerometers. The key parameters include a voltage of 0.15 V, acceleration of 10 m/s², plate area of 250 cm², and a spring constant of 500 N/m. The capacitance is derived from the formula C = εA/d, where ε is the permittivity, A is the area, and d is the distance between plates. The participants emphasize using conservation of energy and the relationship between displacement and capacitance to solve the problem.

PREREQUISITES
  • Understanding of basic capacitor formulas, specifically C = εA/d
  • Knowledge of damped harmonic motion and differential equations
  • Familiarity with energy conservation principles in mechanical systems
  • Basic circuit analysis, including voltage division in capacitors
NEXT STEPS
  • Research the application of conservation of energy in mechanical systems to find maximum displacement.
  • Learn about the superposition principle in circuit analysis for capacitors.
  • Study the relationship between capacitance and voltage in capacitor circuits.
  • Explore the derivation of capacitance formulas for multiple capacitors in series and parallel.
USEFUL FOR

Students studying electrical engineering, particularly those focusing on sensor technology, circuit analysis, and mechanical systems involving capacitors.

  • #61
1. Vc = Vin * C1/(C1+Cc)

2. Vc = Vin * C2/(C2+Cc)

Are this to equations are correct? What should I do then?
 
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  • #62
2013 said:
1. Vc = Vin * C1/(C1+Cc)

2. Vc = Vin * C2/(C2+Cc)

Are this to equations are correct? What should I do then?

No, they are incomplete. Both expressions should contain all three capacitances, and you must put a value to Vin for each. Look at the circuit and identify all the capacitances that correspond to the variables in the voltage divider equation.
 
  • #63
1. Vc = C1/(C1+C2+Cc) (I don´t know how I put a value Vin for each. How should I do that?)

2. Vc = C2/(C1+C2+Cc) (I don´t know how I put a value Vin for each. How should I do that?)Can you show me, how I should do that?
 
  • #64
2013 said:
1. Vc = C1/(C1+C2+Cc) (I don´t know how I put a value Vin for each. How should I do that?)

2. Vc = C2/(C1+C2+Cc) (I don´t know how I put a value Vin for each. How should I do that?)


Can you show me, how I should do that?

Can't you just look at the circuits and tell?
 
  • #65
I really don`t know. Maybe:
1. Vc = C1 * Vin/(C1+C2+Cc)

?
 
  • #66
2013 said:
I really don`t know. Maybe:
1. Vc = C1 * Vin/(C1+C2+Cc)

?

You've just rearranged the formula; you haven't replaced Vin with its numerical value.
 
  • #67
Ahh, okay sorry I don´t understand what you mean.

1. Vc = 12V * C1/(C1+C2+Cc)

2. Vc = 12V * C2/(C1+C2+Cc)

Is this right?
 
  • #68
2013 said:
Ahh, okay sorry I don´t understand what you mean.

1. Vc = 12V * C1/(C1+C2+Cc)

2. Vc = 12V * C2/(C1+C2+Cc)

Is this right?

Not quite. Did you pay careful attention to the polarities of the sources in the circuit?
 
  • #69
Can you show me what you mean?

Do you mean, that:
1. Vc = -12V * C1/(C1+C2+Cc)

2. Vc = -12V * C2/(C1+C2+Cc)

?
 
  • #70
2013 said:
Can you show me what you mean?

Do you mean, that:
1. Vc = -12V * C1/(C1+C2+Cc)

2. Vc = -12V * C2/(C1+C2+Cc)

?

Surely you can read a circuit diagram? Do both sources have the same polarity with respect to the node where Vc is?
 
  • #71
1. Vc = 12V * C1/(C1+C2+Cc)

2. Vc = -12V * C2/(C1+C2+Cc)?
 
  • #72
2013 said:
1. Vc = 12V * C1/(C1+C2+Cc)

2. Vc = -12V * C2/(C1+C2+Cc)


?

Yes. Now each of those expressions for Vc contribute to the actual Vc (you need to sum them; that's how superposition works). Solve for Cc.
 
  • #73
You mean:
1. + 2.

2Vc = (C1+C2) / (2C1+2C2+2Cc)

Is this right? Have I understand you right?
 
  • #74
2013 said:
You mean:
1. + 2.

2Vc = (C1+C2) / (2C1+2C2+2Cc)

Is this right? Have I understand you right?

No. First, each of the expressions contribute to the total of Vc. Think of them as Vca and Vcb, and so the total is Vc = Vca + Vcb. That's how superposition works. You sum up the contributions due to each source to yield the result of both sources acting together. (Surely you must have covered superposition in your circuit theory classes?)

Second, you have to add the fractions properly. You don't add the common denominators; Is 1/3 + 1/3 equal to 2/6?
 
  • #75
We have done superposition, but only in a short launch. We have set different priorities in the first semester.

I`m so sorry that I have no idea what I exactly should do.

Can you show me the beginning?
 
  • #76
2013 said:
We have done superposition, but only in a short launch. We have set different priorities in the first semester.

I`m so sorry that I have no idea what I exactly should do.

Can you show me the beginning?

Sum the two expressions (their right hand sides). That will yield the final expression for Vc. This is just algebra; you should be able to sum fractions, especially when they have the same denominators.
 
  • #77
12V * C1/(C1+C2+Cc) + (-12V * C2/(C1+C2+Cc)) = Vc
Vc = (C1+C2) / (C1+C2+Cc)

?
 
  • #78
2013 said:
12V * C1/(C1+C2+Cc) + (-12V * C2/(C1+C2+Cc)) = Vc
Vc = (C1+C2) / (C1+C2+Cc)

?

What happened to the +12V and -12V terms :frown:
 
  • #79
12V * C1/(C1+C2+Cc) - 12V * C2/(C1+C2+Cc)) = Vc
12V * (C1/(C1+C2+Cc) - (C2/(C1+C2+Cc)) = Vc

Vc = 12V * ((C1-C2)/(C1+C2+Cc))

?
 
  • #80
2013 said:
12V * C1/(C1+C2+Cc) - 12V * C2/(C1+C2+Cc)) = Vc
12V * (C1/(C1+C2+Cc) - (C2/(C1+C2+Cc)) = Vc

Vc = 12V * ((C1-C2)/(C1+C2+Cc))

?

Yes.
 
  • #81
Vc = 12V * ((C1-C2)/(C1+C2+Cc))
Vc*Cc=12V * ((C1-C2)/(C1+C2)
Vc*(Qc/Vc)=12V * ((C1-C2)/(C1+C2)
Qc=12V * ((C1-C2)/(C1+C2)=12V*((2,46*10^-11As/V - 2,01*10^-11As/V) / (2,46*10^-11As/V+2,01*10^11As/V)
= 1,21As

Cc=(Qc/Vc)=1,21As/0,15V=8,05As/V

Is this right? Why is it not the same capacity as in the other method?
 
  • #82
2013 said:
Vc = 12V * ((C1-C2)/(C1+C2+Cc))
Vc*Cc=12V * ((C1-C2)/(C1+C2)
Vc*(Qc/Vc)=12V * ((C1-C2)/(C1+C2)
Qc=12V * ((C1-C2)/(C1+C2)=12V*((2,46*10^-11As/V - 2,01*10^-11As/V) / (2,46*10^-11As/V+2,01*10^11As/V)
= 1,21As

Cc=(Qc/Vc)=1,21As/0,15V=8,05As/V

Is this right? Why is it not the same capacity as in the other method?
The answers should be identical. Your algebra has errors.
 
  • #83
Can you say me at which step, I have done a mistake.

?
 
  • #84
2013 said:
Can you say me at which step, I have done a mistake.

?
Your second line is incorrect. You cannot eliminate a single additive term from the denominator by multiplying the fraction by that term. Also, why are you introducing Qc at this point? You have Cc alone as a variable in the expression, and all other quantities have known values. Don't make things harder! Just do the algebra to isolate Cc.
 
  • #85
Vc=12V*(C1-C2)/(C1+C2+Cc)
Vc*(C1+C2+Cc)=12V*(C1-C2)
how can I get C1 and C2 to the other site?
 
  • #86
2013 said:
Vc=12V*(C1-C2)/(C1+C2+Cc)
Vc*(C1+C2+Cc)=12V*(C1-C2)
how can I get C1 and C2 to the other site?
Elementary operations: Divide both sides by Vc. This will leave the LHS (Left Hand Side) as a simple sum of terms. Move everything that isn't Cc to the RHS.
 
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  • #87
I have to say thank you to you, I have already the solution and it is the same as in the other method.
Thank you, Thank you for helping me :) :)
 
  • #88
2013 said:
I have to say thank you to you, I have already the solution and it is the same as in the other method.
Thank you, Thank you for helping me :) :)

*Whew*! I'm glad we got there in the end :smile:

You're welcome. Good luck in your studies.
 

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