Counting Capacitor Circuits

An electric circuit uses exclusively identical capacitors of the same value $C$.

The capacitors can be connected in series or in parallel to form sub-units, which can then be connected in series or in parallel with other capacitors or other sub-units to form larger sub-units, and so on up to a final circuit.
Using this simple procedure and up to $n$ identical capacitors, we can make circuits having a range of different total capacitances. For example, using up to $n=3$ capacitors of $\pu{60 \mu F}$ each, we can obtain the following $7$ distinct total capacitance values:

If we denote by $D(n)$ the number of distinct total capacitance values we can obtain when using up to $n$ equal-valued capacitors and the simple procedure described above, we have: $D(1)=1$, $D(2)=3$, $D(3)=7$, $\dots$
Find $D(18)$.
Reminder: When connecting capacitors $C_1, C_2$ etc in parallel, the total capacitance is $C_T = C_1 + C_2 + \cdots$,

whereas when connecting them in series, the overall capacitance is given by: $\dfrac{1}{C_T} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \cdots$

A quick review of connectivity in terms of Capacitors:

Capacitors in Parallel: Total Capacitance C_Tot = C1 + C2 + . . . + Cn

This implies that connecting capacitors in parallel is a more straightforward procedure, when two capacitors are connected in parallel, their total capacitance is simply the sum of individual capacitances.

Capacitors in Series: 1 / C_Tot = 1 / C1 + 1 / C2 + . . . + 1 / Cn

For capacitors connected in series, the total capacitance is found by adding the reciprocals of the individual capacitances, and then taking the reciprocal of the sum.

In terms of forming a strategy to solve this problem, we will essentially form a plan to connect capacitors in all possible combinations of series and parallel arrangements up to n capacitors.

According to the problem, D(1) = 1, D(2) = 3, D(3) = 7.

If we take a systematic approach and analyze each step:

For D(1), there’s only one capacitor, so only one unique capacitance value can be formed (let’s call this capacitance C).

For D(2), there are 2 capacitors. They can be arranged in the following unique ways: 2 in parallel (gives 2C), 2 in series (gives C/2), and 1 alone (C). This makes 3 distinct capacitances: C, 2C, C/2.

For D(3), it’s getting a little more complicated. Here are the unique ways we can arrange them: 3 in parallel (3C), 3 in series (C/3), 2 in parallel + 1 alone (2C, C), 2 in series + 1 alone (C/2, C), 1 alone + 1 alone + 1 alone (C, C, C). After eliminating duplicates, we will get 7 unique capacitances: C, 2C, 3C, C/2, C/3.

Hence, the number of distinct capacitances that can be obtained increases as we include more capacitors. Each time, we are considering all possible arrangements without repetition.

To find D(18), we would need to make a comprehensive list of all possible combinations of series and parallel arrangements using up to 18 capacitors, and compute their total capacitance, removing duplicates.

However, calculating this manually would be an extremely daunting task, especially given that the function D(n) grows rapidly with n because the number of unique combinations of series and parallel arrangements grows exponentially.

So for higher values of n like D(18), this becomes a computational problem best solved by a computer program iterating over all possible unique arrangements up to 18 capacitors. It’s a complex recursive problem that requires a quite sophisticated program involving number theory and recursion techniques.

More Answers:
Sums of Square Reciprocals
Investigating Gaussian Integers
Exploring Pascal’s Pyramid

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