Equivalent Capactiance and Expression for Uncertainty

Question

How do I determine an expression for the uncertainty of the equivalent capacitance of this circuit?

I know that circuits in series are 1/Ceq = 1/C1 + 1/C2 ect.

I know that circuits in parallel are Ceq = C1 + C2 ect.

 

SO how do I come up with a formula that has C1 and C2 parallel to each other but in series with C3?

 I also need help on how to find the actual value of the equivalent capacitance.

 

C1 = 47 x 10^-6 F ± 4.7µF
C2 = 220 x 10^-6 F ± 22µF

C3 = 4.7 x 10^-6 F ± .047µF

 

 

Answer 1

To calculate then uncertainty of the equivalent capacitor C_eq you have to differentiate the expression for the capacitor shown by Richard, and replace the symbol of differential by Δ symbol (means uncertainty or error of measurement). Let 

   

                                  C₀ = C₁ + C₂  (equivalent capacitor of parallel capacitors C₁ and C₂)

 

The uncertainty of this capacitor is 

 

                             ΔC₀ = ΔC₁ + ΔC₂

 

and the percentage of  it is ΔC₀/C₀ = ± 10%. Let's now differentiate the expression for the equivalent capacity Ceq.

 

 First of all, we have

 

                          1/Ceq = 1(/C₀₎ + (1/C₃₎

 

Its differential (after substituting by symbol Δ) is

 

                 ΔCeq/Ceq = Ceq [(1/C₀)(ΔC₀/C₀)    +  (1/C₃)( ΔC₃/C₃)]

 

By plugging all the numbers above into this expression we will obtain

 

                     ΔCeq/Ceq = ± 2.8%.

 

 

 

 

Answer 2

Since  C1 and C2 are in parallel they add to result in C1 + C2.  This combination ins in series with C3 so

 

  1/Ceq =  1/(C1 + C2) +  1/C3.      The values can be plugged in to get

  1/Ceq =  .2165 x 10^6   and  Ceq = 4.69 x 10^-6.   

Notice that this is very close to the value for C3.   This is not an accident.   C3 is a factor of 10 less than C1 and  a factor of ~47  less than C2.  The equivalent capacitance of a small capacitor in series with a large one is nearly that of the small one. 

 

By the same reasoning the uncertainty  on C3 (although only 1%) dominates the other errors.  This means that the uncertainty on Ceq is  of order 1% or 0.0469 x 10^-6 in absolute terms.