Combinations

The questions below are due on Tuesday October 14, 2025; 10:00:00 PM.
 
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Just like we saw with resistors, we can simplify circuits that contain series or parallel combinations of capacitors or inductors by replacing those combinations with a single equivalent capacitor/inductor.

1) Capacitor Combinations

1.1) Capacitors in Series

Firstly, let's consider two capacitors connected in series:

By virtue of being in series, the capacitors share the same current: i_1 = i_2 = i.

We also have v = v_1 + v_2. Taking the derivative of both sides of this equation, we have v' = v_1' + v_2'. Using our constitutive equation and assuming that the box can be replaced with an equivalent capacitance C_{\rm eq}, we find:

{i\over C_{\rm eq}} = {i\over C_1} + {i\over C_2}

We can then divide each term by i to remove that from the equation, leaving us with an equation for C_{\rm eq}:

C_{\rm eq} = {C_1C_2\over C_1 + C_2}

Note that this is a special case for two capacitors in series. The more-general equation is that {\displaystyle {1\over C_{\rm eq}} = \sum_n{1\over C_n}}.

1.2) Capacitors in Parallel

By virtue of being in parallel, the capacitors share the same voltage drop: v = v_1 = v_2. And via KCL, we can say that i = i_1 + i_2. Using the constitutive equation and assuming the box can be replaced with an equivalent capacitance C_{\rm eq}, we have:

C_{\rm eq}v' = C_1v' + C_2v'

We can then divide each term by v' to arrive at an equivalent capacitance:

C_{\rm eq} = C_1 + C_2

2) Inductor Combinations

We'll leave it up to you to derive the relationships for inductors in series and parallel (note, though, that you can follow similar steps to the derivations above).

3) Exercises

Find the equivalent capacitance of each figure below. Enter your answer as an exact fraction or decimal.


Equivalent capacitance (in \mu{\rm F}):


Equivalent capacitance (in \mu{\rm F}):

Now, find the equivalent inductance of each figure below. Enter your answer as an exact fraction or decimal.


Equivalent inductance (in {\rm mH}):


Equivalent inductance (in {\rm mH}):