Resistor Combinations Tutorial

The questions below are due on Monday February 10, 2025; 10:00:00 PM.
 
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"Looking Ahead" Exercises

Many problem sets will include an exercise that serves as an introduction to ideas that will be formalized and practiced in the following week. This exercise is the first such exercise this semester. It is intended to be approachable using the material already covered in class just like the other exercises in this week, but it is also intended to provide as a first introduction to next week's topics, with which we'll get a lot more practice in next week's materials.

This exercise is all about resistors in series and parallel, and finding their equivalent resistances. For each kind of combination, we'll start by finding an exact expression for the equivalent resistance.

But then we'll also talk about some "tricks of the trade" that we can use to find exact equivalent resistances for parallel combinations efficiently (and without the need for a calculator!). We won't give the same treatment to seris combinations since those are generally easier to think about.

1) Series Combinations

Let's start by considering two resistors (with arbitrary resistances R_1 and R_2, respectively), connected in series:

This combination of resistors exerts a constraint on the variables i and v defined in the drawing above, and that constraint could be replicated by a single resistor with resistance R_s:

Solve for R_s in terms of R_1 and R_2. Enter your answer in the box below as a Python expression involving variables R_1 and/or R_2:
 
R_s =~

2) Parallel Combinations

Now let's instead consider two resistors (still with arbitrary resistances R_1 and R_2, respectively), connected in parallel:

This combination of resistors exerts a constraint on the variables i and v defined in the drawing above, and that constraint could be replicated by a single resistor with resistance R_p:

Solve for R_p in terms of R_1 and R_2. Enter your answer in the box below as a Python expression involving variables R_1 and/or R_2:
 
R_p =~

2.1) Easy Cases of Parallel Combinations

In general, the easiest case of an equivalent parallel resistance to solve for is when all of the resistances are the same. For example, here we have two resistors with resistance R connected in parallel, and we've labeled the current through each individual resistor with its own name:

How does i relate to i_1 and i_2? Enter your answer as a Python expression in terms of i_1 and i_2.
 
i =~

If the voltage drop across the whole combination is v, what is i_1? Enter your answer in terms of R and/or v.
 
i_1 =~

If the voltage drop across the whole combination is v, what is i_2? Enter your answer in terms of R and/or v.
 
i_2 =~

Given this, what is i, not in terms of i_1 and i_2, but in terms of v and R?
 
i =~

Given the result above, if we wanted to express that same constraint as v = iR_{eq} for some value R_{eq}, what value of R_{eq} should we use?

R_{eq} =~

Now, imagine we had not two but three of these resistors in parallel:

What is the equivalent resistance of this combination?

What if we had some arbitrary number N of resistors in parallel instead of 3? What would be the equivalent resistance then?

2.2) Reducing Toward the Easy Case

As we hopefully saw in the last section, finding the equivalent resistance of a parallel combination is relatively easy when the resistors all have the same value. In this section, we'll try to explore a little bit of how we can reduce other kinds of combinations down to that case. For example, let's consider the following combination of resistors:

Simply plugging these resistances into our formula is likely to be tedious, and it's not likely to provide a lot of intuition. But there is a neat little trick we can use to make this combination much easier to think about.

What we would like to do is to express 11\Omega, 22\Omega, and 33\Omega each as parallel combinations of some number of resistors with the same resistance so that we can construct a parallel combination using only that one resistance, but whose equivalent resistance is the same as that of the combination above. We can do this by first finding the least common multiple of our resistances.

What is the least common multiple of 11, 22, and 33?

Now, each of the resistances in our original combination can be expressed in terms of that new resistance (let's call it R_{\rm LCM}).

How many R_{\rm LCM} resistors in parallel would we need to replicate the 11\Omega resistance?

How many R_{\rm LCM} resistors in parallel would we need to replicate the 22\Omega resistance?

How many R_{\rm LCM} resistors in parallel would we need to replicate the 33\Omega resistance?

Now let's think about putting all of those combinations in parallel with each other. This should give us a new combination consiting only of resistors with value R_{\rm LCM} that has the same equivalent resistance as the combination we started with.

How many total R_{\rm LCM} resistors will be in this combination?

Using this idea, what is the equivalent resistance of the combination of 11\Omega, 22\Omega, and 33\Omega in parallel with each other? Enter your answer (in Ohms) below: