BETTER UNDERSTANDING OF OHM'S AND KIRCHHOFF'S LAWS ~ The Young Engineerzz

THIS IS A MINI PROJECT CARRIED OUT BY A FRIEND

Introduction: George Simon Ohm and Gustav Kirchhoff loom large in panoply of 19^th Century scientists who developed the theories behind electrical circuits. Before any sophisticated understanding of electrical systems can occur, the student must master the laws put down by these two men.

Theory: Circuit construction. A circuit, in electronics, plumbing or rally motoring, is a closed loop. The general theory is that an item (electron, drop of water, or sports car) begins at some starting point and proceeds around the loop, perhaps with some choice of paths but never doubling back, and ends at the beginning. There are two primary kinds of loops, series and parallel. These two can be mixed to produce many variations on the theme, but these complexities can usually be reduced to series and parallel elements. The series loop is a path with no choices. The electron flow starts at the source (battery, generator) and proceeds through each circuit element. A circuit element is a resistor, capacitor, inductor, or any other electronic part or device. See figure 1:

Figure 1

Notice how one line, which represents a wire, connects each circuit element to the next. (Hint for circuit building: whenever you build a circuit, trace out the route on paper and make sure it follows the same route you have connected). Each element has a wire which enters the device at one end and exits out the other end.

A parallel loop consists of choices. Does the flow (of water or electrons) go this way or that? Each intersection is called a junction, and each path is called a branch. See figure 2:

Figure 2

In theory, a wire is completely transparent, meaning it doesn't affect the circuit. A connection in parallel is like having two separate wires carrying the flow. Another way of looking at the parallel circuit is to think of the left sides of the devices being the same "place'', that is, consisting of one point instead of three. This is true as long as the wire doesn't interfere with the circuit. Of course, the right sides of the devices are at the same place as well.

You put devices in series and parallel all the time, and probably don't even think about it. When you put batteries one after the other in a long flashlight you are putting the batteries in series. When might you put batteries in parallel? One can combine series and parallel loops in endless complex circuits. One simple combination is shown in figure 3:

Figure 3

The generic name for this is a series parallel combination. Notice how two components are in parallel, and that they are in series with the other device. Passive components such as resistors and active devices such as oscilloscopes can all be connected in series or parallel, or any combination thereof. The same rules of connection apply for each. In this and future labs you will be asked to assemble circuits in series or parallel, and you must know what that means. We will wire some circuits with open wires, and we will breadboard other circuits. A breadboard is a small table with many small holes. These holes are connected together underneath, eliminating the need for open wires. One inserts the lead (wire) of a component into a hole and the lead of another device into a hole in the same line. These two components are now connected together. A breadboard is shown in figure 4:

Figure 4

The 25 holes along each of the 8 bus strips are all connected, as are the 5 holes in each of the terminal strips. The yellow dividers show that there is no connection across them.

Sometimes the components are so big that breadboarding is impractical. In these cases terminals are provided on each component. Use spade or alligator clips on these. Avoid twisting wires together. Other times the components are too small to wire conveniently, so we use a breadboard. Many times a combination of the two methods is used, so the breadboard has three binding posts (places to connect big wires) mounted on it. These posts are called banana jacks, for reasons which have been lost in antiquity. Anyway, they are NOT connected under the breadboard, so they must be jumped (plugged) into the board itself.

We will use resistors in this lab. Resistors are measured in ohms (

) and are marked in 3 color bands to denote their resistance. The color code follows:

Resistor Color Codes
Black = 0	Green = 5
Brown = 1	Blue = 6
Red = 2	Violet = 7
Orange = 3	Grey = 8
Yellow = 4	White = 9

Table 1

The fourth band relates to uncertainty, called tolerance, in these components: None = 20% Silver = 10% Gold = 5%.

One reads a resistor by the colors. For instance, if the first two bands (opposite from the tolerance bands) are green and red, the first two digits are 5 and 2. If the third band is yellow, follow the 52 with 4 zeros. Hence the resistance has a value of 520000

, or 520 k

. If the fourth band is gold, this value may be 2.5% high or 2.5% low, a manufacturing problem. It also is the uncertainty you need in your analyses.

Theory: Ohm's Law. Not a fundamental law of Physics but an empirical relationship, Ohm's Law describes the relationship between current and voltage in the presence of resistance, as long as currents aren't too high, the components don't get too hot, and other constraints. It is a simple relationship:

Equation 1

In practical terms, for a given voltage V, current I increases as resistance R decreases. Voltage sources are generally assumed to be constant, as the 120V from a wall outlet should be, so if driving a very low resistance load the load consumes a lot of current and producing a lot of heat as a byproduct.

Another application of Ohm's law is to discover the voltage drop across a resistance. In a series circuit the current is the same in every component; therefore the relative drop in voltage, or potential difference, for each component can be easily measured or calculated. On the other hand, in a parallel circuit, the voltage across each component is the same, and the current through each component depends on its resistance.

These rules imply different ways to add resistances, depending on the configuration. For series:

Equation 2

For parallel:

Equation 3

With these and Ohm's Law, given either the current or voltage the other variable is easily calculated.

To correctly measure voltage and current for a component one must put a meter (VOM) in parallel and series, respectively.

Figure 5

The not-so-obvious implication is that the VOM must have very high resistance when used as a voltmeter and also must have a very low resistance when used as an ammeter. Otherwise, the VOM will adversely affect the very component you are trying to read.

Theory: Kirchhoff's Loop Laws. Some circuits are simplly resistors in series or parallel or some non-ambiguous combination of the two. Most circuits are, however, a bit more complicated, involving two or more overlapping loops.

Inserting a voltage source (battery) into our diagram and substituting the symbol for a resistor for the black rectangle used in previous figures, a basic two loop circuit looks like this:

Figure 6

One measures the current through and voltage across each resistor in the manner outlined above, but to calculate the theoretical values one applies the following two rules:

Kirchhoff's Junction and Loop Laws

Equation 4	Equation 5

Equation 4 states that the sum of the currents entering a junction equals the sum of the currents exiting a junction (red arrows below). Equation 5 states that the sum of the voltage gains and drops around a closed loop equal zero (green loops below).

Figure 7

In this case let's assume that the voltage sources and resistances are known and you are seeking the values for the currents through each resistor. The current through R₁ (I₁) plus the current through R₂ (I₂) equals the current through R₃ (I₃). Additionally, V₁ - R₁I₁ - R₃I₃ = 0, and V₂ - R₂I₂ - R₃I₃ = 0. Note that there are three equations for the three unknown currents; solve. It is entirely conceivable that you are given two currents and one resistance, or one current and two resistances, or any other such combination. As long as there are three unambiguous loops/junctions, three unknows can be calculated. (Choosing the junction below R₃ in addition to the junction above R₃ would be redundant and therefore ambiguous. Can you see why?)

An important point: you may choose your current directions in any manner, but once you make that choice you must stick with it. For example, suppose you decided that both loops are clockwise in figure 7; that means that for the right-hand loop there is a voltage gain across R₃ since you are going from a lower to a higher potential, as determined by the left-hand loop direction. The worst thing that can happen is that, after calculations are complete, one or more of the current solutions may have negative sign. This indicates that initial choice for direction is wrong, but the magnitude is still correct.

These circuits are not limited to 1 junction and 2 loops. The culmination of your lab today will be something more interesting.

Task:

To identify 5 resistors using the color code in Table 1
To verify Ohm's Law in three circuits
To verify Kirchhoff's Laws in two circuits

Equipment:

resistors
voltage sources
VOM
breadboard

Procedure:

Use Circuit Maker to duplicate each circuit and snip the images for your report. You may also use Circuit Maker to test your calculations.

Select 5 resistors from the boxes. Identify the resistance of each and record these values for your report in a table similar to this one:

Resistor	Value in k	Tolerance %
R₁
R₂
R₃
R₄
R₅

Order them so that R₁ will have the least resistance and R₅ will have the most. You may wish to use some tape to label them for the circuits to come.
Let V₁ = 4.5V, V₂ = 6V, and V₃ = 3V.

Examine the three circuits below:


Figure 8	Figure 9

Figure 10

For each, calculate the current through and the voltage drop across each resistor. Build each circuit on your breadboard, then measure these values for each resistor, and present the results in a table citing the theoretical and experimental values including uncertainty.