Fully Example Driven Text

Prerequisite Modules

Voltage and Current concepts

Ohm’s Law

Circuit diagram symbols

Resistivity

A two-terminal resistor net can be replaced by a single resistor with the “equivalent resistance” without changing the voltages and currents in the rest of
the circuit.

Networks with more than two terminals do not have an equivalent resistance.

Example: Equivalent Resistance

Goal: find the equivalent resistance.

In other words, this group of resistors will act like one equivalent resistor. What is its resistance?

Phase 1: redrawing

Circuit diagrams are an abstraction of a real circuit, there are many ways to draw and redraw them.

This example shows you how to redraw circuit diagrams to suit our purposes.

We want stuff that looks like these.


Just trust me.

All the resistors oriented vertically.

Terminals exiting from the middle out the top and bottom.

Ultimately we want only vertical resistors. Our first task is try to resolve this diagonal part on the left side. We don’t like
diagonals.


The <–> symbol means you can swap one for the other freely, whichever makes your diagram easier to understand and more
convenient to analyze.

Stretch and Bend: Length, orientation, and shape of the ideal wires in diagrams don’t matter.

The length and shape of these connections doesn’t affect the voltages and currents across the elements (For a real circuits, wire
lengths do make a difference).

As our first move, we’ll stretch some wire to make that T junction a little nicer to look at.

Rotate: The orientation of a circuit element doesn’t matter.

Let’s rotate them to get them vertical. We’ll make them immediately recognizable later, after we work on other parts of the circuit.

Slide You can slide an ideal resistor along a piece of ideal conductor.

Now let’s clean up the right side a bit. we can slide the 4 ohm resistor around the corner, YAY!

We can also T-slide points A and B to coincide with the junction of the 10 ohm resistor

T slides can go around corners too!

Now we want to work on getting that 8 ohm resistor in line. First, let’s do a T-slide to move that corner to the center of the wire.
This gives us terminals poking out the top and bottom.l.

Now we slide resistor 8 along the wire around the corner onto the vertical segment.

Now our circuit is nicely transformed. After doing this a few times, you will be able to do all this in your head. It just takes a
little practice.

These “legal moves” are analogous to operations on a breadboard, too.

Phase 2: Reduction

Use parallel and series formulas to decompose circuit one piece at a time.

This circuit is nice. The circled segments look like those special patterns mentioned earlier in the example.

These configurations are common and important enough to have names.

The one on the right is called “Series”

Two resistors are in series when they share a single node that no other element is connected to.


Two resistors in series

We can replace a series pattern of resistors with a single resistor with the “Equivalent resistance” denoted with the symbol Req

Req=R1+R2

How to understand and remember this formula?: Remember computing resistivity? Putting two identical resistors in series is like
doubling the length of a wire and longer wires have proportionally more resistance.

If you want to see a proof of how this was obtained, click here. You have all the tools to follow it.

Req1=R1+R2=4+2=6

Apply the series equivalent resistance formula

Our answer is reasonable: a series combination increases resistance.

Redraw the diagram with the equivalent resistor.

The circled configuration is also a special configuration. it is called “Parallel”

Resistors are in parallel if they are all connected at both ends by ideal wires.

It has an equivalent resistance of

Req=1/(1/R1+1/R2)

“product over sum rule”

Req=R1*R2/(R1+R2)

Want to see a proof? Check it out here

How to understand and remember this formula? Remember the formula with resistivity? Putting two identical resistors is like doubling
the area of a piece of wire. This lowers resistance by providing a wider path for the current.

Req2=1/(1/R1+1/R2)=1/(1/20+1/30)=12

Apply the parallel resistance formula!

Note that our answer is reasonable: a parallel configuration reduces resistance.

Redraw the diagram with the parallel resistors replaced with their equivalent resistance.

Search for easy reductions. The 8 ohm resistor and Req2 (12 ohms together) are in series.

Req3=8+12=20

Apply the series equivalent resistance formula.

Let’s redraw the diagram with the third reduction inserted.

We’re in the home stretch,

After doing a few of these, you will be able to do more than one step at a time.


Req= 1/( 1/R1+1/R2+1/ R3)

Req=1/( 1/R1+1/R2+1/ R3+1/R4)

The parallel formula works for more than two resistors as well. The three or four resistor version of the “product over sum” rule is
ugly. don’t use it

In general..

For parallel combinations of resistors

Req= 1/( 1/R1 + 1/R2 + 1/ R3 +1/R4 + … )

Req=1/(1/20+1/10+1/6)=3.157Ω

Apply the parallel resistance formula to three resistors. We have succeeded! The whole mess we started with behaves as though it were a
single resistor with a resistance of about 3.1 ohms.

Some circuits can’t be reduced with series and parallel, such as this one

In such cases, you can use the Y-Δ Transform to reduce it

We could have done this all symbolically (as a general formula that doesn’t use the numbers until the end).There are many benefits to
doing it that way, but this way is easier to think about at first. Don’t worry about it for now.

Many circuit reduction problems can be solved with this simple strategy. Reducing circuits is often just a small part of a larger problem.

  • Redraw the circuit to make everything immediately recognizable.

  • look for elements that are in series or parallel

  • apply the formula

  • redraw

  • Repeat

Try it yourself: Find the equivalent resistance for each of the networks shown in Figure 2.4. [Hint for (b): R3 and R4 are in parallel.]

Answer a. 3Ώ ; b. 5Ώ ; c. 52.1Ώ ; d. 1.5 kΏ

Example: One resistance is unknown

Given the resistances of three elements, what must the resistance of the fourth element be to make the equivalent resistance equal to
the desired value?

Phase one: Stretch Circuit

Most problems with equivalent resistances have this as the first step.

Luckily, we just need to bend the corners a little to get it to line up.

Phase 2: Solving

Since one of the resistances is an unknown, we can’t just reduce numerically, we have to find the symbolic solution.

R1+R2 ⇔ Resistor one is in series with resistor two

R1||R2 ⇔ Resistor one in parallel with resistor two

When doing problems symbolically, this notation is helpful.

We can see two easy series resistances, resistors 1 and 2 are in series, and resistor 3 is in series with the unknown resistor 4.

We make our reductions, but we will do things symbolically. Let’s use our new notation, and write it this way

R1+R2 should be read “Resistor 1 is in series with resistor two”. When written symbolically in this context, the R’s represent objects,
not numbers.

This notation makes solving problems fast once you recognize combinations easily.

(R1+R2)||(R3+R4)

These combinations are immediately recognizable as parallel. To write this, we use the || symbol. This statement should be read “The
series combination of resistor 1 and resistor two is in parallel with the series combination of resistor 3 and resistor 4.


R1+(R2||(R3+R4)) R1+(R2||R3)+R4

Parentheses matter!

Req=1/(1/ (R1+R2)+1/(R3+R4))

Let’s turn this statement into an equation we can solve. We’ve got one equation with only one unknown (R4), so we are in the clear.

Req=(R1+R2)*(R3+R4)/(R1+R2+R3+R4)

Since we have exactly two elements in parallel, we can use the product-over-sum rule instead. it is often easier to manipulate

10=(4+6)*(2+R4)/(4+6+2+R4)

We can work it totally symbolically, but let’s plug in our knowns this time.

10=10*(2+R4)/(12+R4)

(2+R4)=(12+R4)

Cancel the 10’s, multiply both sides by denominator

IMPOSSIBLE! NO SOLUTION!

We arrive at the impossible answer 2=12. Since this is impossible, that means our original task was impossible.

There is no possible resistor R4 that can make the equivalent resistance equal 10

This problem solving strategy still works on problems that have solutions, though.

Let’s Review!

Legal Moves List

Rotate

T Slide

T slide around corner


T Fusion

Rotate and Slide

Important Equations and notation

Resistance of resistors in parallel Req= 1/( 1/R1 + 1/R2 + 1/ R3 +1/R4 + … )

Resistance of resistors in series Req=R1+R2+R3+R4+…

R1+R2 means series, R1||R2 means parallel

Standard