Fully Example Driven Text
Prerequisite Modules
Voltage and Current concepts
Ohm’s Law
Circuit diagram symbols
Resistivity
A twoterminal 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 

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 

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 Tslide 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 Tslide to move that corner to the center of the wire. 

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 

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. 

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 


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 

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 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 

Some circuits can’t be reduced with series and parallel, such as this one In such cases, you can use the YΔ 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

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 
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 R1R2 ⇔ 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, 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 
R1+(R2(R3+R4)) R1+(R2R3)+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 productoversum 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, R1R2 means parallel 