Resistive Circuit Analysis

Circuit analysis is the process of determining
how the current and voltage supplied to a particular system affects it and its individual components. The completion of this Instructable will teach you a few of the fundamental tools that are used during the analysis of a circuit. Specifically you will learn about Ohm’s Law and resistance equivalencies. You will then apply these tools to determine the current and voltages of components on a circuit board.

Although there are many different components that can be present in any given circuit, for the purposes of this Instructable, only two types of components will be used, batteries and resistors. The function of a battery is common knowledge and does not need explaining. However, the function of the resistor may not be as common as the battery’s. The purpose of the resistor is explained by its name. It resists the flow of current. Resistors are typically used to control the amount of current flowing into a specific component of a circuit that would otherwise be destroyed by a higher current.

There is no previous experience needed for this project. But basic algebra is required if you are not using a scientific calculator. The only tools needed for this project are a pencil and paper.

Due to their resistance (R), each resistor
maintains a relationship between its voltage (V) and the current (I) flowing through it. This relationship is described by Ohm’s Law which has the mathematical expression, V=IR. This shows that the voltage of a component is the product of its current and its resistance. Given that you are provided with at least two of the three variables you can determine the third. The units of measurement of voltage, current and resistance are volts, amps and ohms respectively. Ohms are typically depicted as the Greek letter Omega (Ω).

A very useful tool for circuit analysis is learning how to reduce a circuit to its equivalent resistance. The equivalent resistance of a circuit is the value of the algebraic summation of multiple resistors in an electric system. In some schematics of circuits that include multiple resistors, the schematic can be redrawn that only has a single resistor that is equivalent to the resistors that it replaced. This, in turn, makes the current flowing through the system easier to calculate. Before this is done, it is better that you have an understanding of the two basic component arrangements and how each of them affect the equivalent resistance (Req).

The first type of arrangement is similar to how the cars of a train are connected to each other. When the head of one resistor is connected to the tail of another, these two resistors are said to be in series. When two or more components share a connection, this connection is called a node. Components that are in series all share the same current.

The other type of component arrangement is formed when the head of one resistor is connected to the head of another resistor and the tail of both of those resistors are also connected to each other to form a loop. In other words, components in parallel share 2 nodes, as contrasted with sharing a single node in a series connection. Components that are in parallel have the same voltage.

In order for these arrangements to be true, the connection that each component shares has to be exclusive: The joint that the two components share cannot be shared by a third.

NOTE: Just because two components are not in series does not automatically mean that they are in parallel and vice versa. Some schematics contain components that are in neither parallel nor series and will use alternate methods that will not be discussed here.

When resistors are in series, the equivalent
resistance can be calculated by simply summing all of the values of the individual resistors together. Using this method, the equivalent resistance should always yield an equivalent resistance that is larger than the largest individual resistor.

Req = R1 + R2 + R3 + … (Series Equivalent Resistance)

When resistors are in parallel, the equivalent resistance is a bit more complex. The reciprocal of the equivalent resistor is equal to the summation of the individual reciprocal resistor values. Using this method, the equivalent resistance should always have a smaller value than the smallest individual resistor.

1/Req= 1/R1 + 1/R2 + 1/R3 + … (Parallel Equivalent Resistance)

Once the equivalent resistors have been calculated, additional equivalent resistors can also be calculated from those.

The following images will show you examples of resistors that are in series, parallel or neither. They will also show you how the components can be reduced.

The first image shows that the only resistors in
series are resistors R3 and R4. Because of this, those resistors can be combined into a single resistor.

Resistors R3 and R4­ have now been reduced to form the equivalent resistor Req1. Notice how this new resistor is now in parallel with resistor R2. These resistors can also be reduced to a single component.

Resistors R2 and Req1 have
now been combined to form the equivalent resistor Req2. Now all of the remaining resistors are in series and can be reduced to a single component.

After a circuit has been reduced it is often easier to find the current flowing through it. Once you have simplified the circuit, you can then go on to apply Ohm’s Law, V=IR, to determine the current.

The best way to retain what you have learned here is to practice it. Start by going over the guided example problem below. After that, try the next problem on your own. The answers will be provided at the end.

First, find the final equivalent resistance of the circuit (reduce the system to one resistor and the battery). Then, find the value of the current I.

The black values are the initial values of the resistors and battery.

The values that are boxed in red are the values
of the resistors that are to be reduced into a single component in the next step. In this is instance, the 15 ohm and the 30 ohm resistors are the ones to be combined using the Series Equivalent Resistance equation.

NOTE: It may appear that the 15 ohm and 45 ohm resistors are in parallel. Even though they do share a loop, the connection they share is also shared by the 30 ohm resistor, making the connection invalid. Also, don’t mistake the 30 ohm and the 60 ohm resistors to be in series. Those resistors do share a single node. However, the node is also shared by the 45 ohm resistor.

The red value is the new equivalent resistance of
the resistors in the precious step. Now this new resistor is in parallel with the 45 ohm resistor. Calculate the next equivalent resistance of these two resistors using the Parallel Resistance Equivalence equation.

TIP: When calculating the equivalent resistance of ONLY TWO resistors IN PARALLEL you can use this equation:

Req = R1R2/(R1+R2)

Now there are three resistors that are in series
and will be combined using Series Equivalent Resistance equation.

These 70 ohm and 162.5 ohm resistors will be reduced using the Parallel Resistance Equivalence equation.

For the final equivalent resistance, the 48.9 ohm and 55 ohm resistors will be reduced using the Series Equivalent Resistance equation.

Now that you know the final equivalent resistance of the circuit, you can use that to find the current of the system using Ohm’s Law, V=IR.

Solving for I you get, I=V/R. The current, I, flowing through the circuit is determined to be about .3849 amps or 384.9 milliamps .

Now try the next problem on your own. Remember: When reducing the circuit, it always helps to redraw the circuit with the new equivalent resistors in between each step that you take.

First, find the final equivalent resistance of the circuit (reduce the system to one resistor and the battery). Then, find the value of the current I.

Answers: Req = 75 ohms, I = 1/3 A