DC/AC Circuit Reference

Resistor Networks

The purpose of this first laboratory assignment is to review fundamental concepts and to extend your DC circuit analysis skills. The assignment draws from the first four chapters of your text and guides you through the production of a your own reference on the analysis of circuits containing resistor networks, ideal voltage sources, and ideal current sources. Resistor networks are not very exciting to build, so much of our work will be theoretical. A secondary goal of this assignment is to introduce you to the Berkeley Spice circuit simulation program in a context in which you can easily verify results by hand.

Fundamental Quantities, Devices, and Concepts

Collect definitions for the following fundamental physical quantities, devices, and concepts.

Charge, Current, Voltage, Ground, Energy, Power, Resistance, Conductance, Resistor, Ideal Voltage Source, Ideal Current Source, Passive Sign Convention, Node, Essential Node, Branch, Loop, Mesh, and Planar Circuit.

Also consider what is meant by the phrases ``voltage across device X'' and ``voltage at point A.''

I don't expect you to come up with these on your own, but acknowledge ¹ the resources you use. We will discuss these in class.

Ohm's Law

``Ohmic'' devices show a linear relationship between applied voltage and current,

$\begin{displaymath} v = iR. \end{displaymath}$

(1)

This is known as Ohm's Law and applies to standard resistors and wires, among other things.

Experiment : Light bulbs are often thought of as resistors, but they are rumored to be nonlinear. That is, they are alleged to be non ohmic. Test this hypothesis experimentally with the light bulb supplied. (Lew will provide requested equipment, if we own it.) Choose a resistor (from one of the resistor boxes on the table at the west end of the lab) of similar resistance to the light bulb, and test it as well for comparison.

Give a written description of your experiment and results.

Kirchoff's Laws...

...Stated
Construct, in your own words, statements of Kirchoff's Laws. (Acknowledge the resources you use.)
...Applied to Voltage and Current Dividers
Apply Kirchoff's Laws to the circuits shown in Figure 1 to derive equations for the voltages and across the resistors in the ``voltage divider'' circuit (a) and for the currents and through the resistors in the ``current divider'' circuit (b).
...Applied to a Circuit with Two Loops
Design and solve a problem involving a circuit containing resistors, a single ideal voltage supply, and two loops. We will exchange problems as an exercise in class.

Equivalent Resistance

Resistors in Series and Parallel
Apply Kirchoff's Laws to circuits (a) and (b) in Figure 1 to derive the well known equations for the equivalent resistance of two resistors in series and in parallel. In each case, consider the special cases , , and .
Exercises
Find the equivalent resistance of the networks of resistors (c) and (d) in Figure 1, using $\Omega$ , $\Omega$ , and 50 $\Omega$ for unlabeled resistors.

Figure 1: (a) and (b) are simple (simplest) circuits with two resistors connected in series and parallel, respectively. (c) and (d) are resistor networks.

(a) $\scalebox{0.75}{ \includegraphics{lab1-ser.eps} }$ (b) $\scalebox{0.75}{ \includegraphics{lab1-par.eps} }$

(c) $\scalebox{0.75}{ \includegraphics{lab1-net1.eps} }$ (d) $\scalebox{0.75}{ \includegraphics{lab1-net2.eps} }$
$\Delta$ -Y Transformations
Practice working with Delta-Wye ( $\Delta$ -Y) transformations by finding the equivalent resistance of the resistor networks (a) and (b) in Figure 2.

Figure 2: Resistor networks requiring $\Delta$ -Y transformations. $\Omega$ , unlabeled resistors are 1 k $\Omega$ .

(a) $\scalebox{0.75}{ \includegraphics{lab1-net3.eps} }$ (b) $\scalebox{0.5}{ \includegraphics{lab1-net4.eps} }$
Experiment
Build the resistor network shown in Figure 2(a) and check your calculation of its equivalent resistance experimentally.

**Figure 2:** Resistor networks requiring $\Delta$ -Y transformations. $\Omega$ , unlabeled resistors are 1 k $\Omega$ .
(a) $\scalebox{0.75}{ \includegraphics{lab1-net3.eps} }$ (b) $\scalebox{0.5}{ \includegraphics{lab1-net4.eps} }$

Methods of Circuit Analysis

The Node Voltage Method
Construct, in your own words, a step by step description of the Node Voltage Method of circuit analysis. Include an example circuit diagram. (Acknowledge the resources you use.)
The Mesh Current Method
Construct, in your own words, a step by step description of the Mesh Current Method of circuit analysis. Include an example circuit diagram. (Acknowledge the resources you use.)
Exercises
Find the unknown currents and voltages for circuits (a) - (d) in Figure 3 using both the node voltage and mesh current methods.

Figure 3: DC circuits with resistor networks and power supplies.

(a) $\scalebox{0.75}{ \includegraphics{lab1-circ1.eps} }$ (b) $\scalebox{0.75}{ \includegraphics{lab1-circ2.eps} }$

(c) $\scalebox{0.75}{ \includegraphics{lab1-circ3.eps} }$ (d) $\scalebox{0.75}{ \includegraphics{lab1-circ4.eps} }$

The Spice Circuit Simulation Program

Use Spice to verify your calculations of Section 3. Hand in printouts of your circuit files and output. However, do not assume that they speak for themselves. Summarize important results.

Equivalent Circuits

Just as networks of resistors between two nodes may be replaced by a single equivalent resistance, entire circuits can be replaced by a combination of a single resistor and an ideal power source in two ways.

Thevenin Equivalent Circuits
Construct, in your own words, a step by step method of finding the Thevenin equivalent voltage and resistance. Include an example circuit diagram. (Acknowledge the resources you use.)
Norton Equivalent Circuits
Construct, in your own words, a step by step method of finding the Norton equivalent current and resistance. Include an example circuit diagram. (Acknowledge the resources you use.)
Exercises
Find the Thevenin and Norton equivalent circuits for terminals A and B of circuits (a) and (b) in Figure 4.

Figure 4: Circuits to be replaced with Thevenin and Norton equivalent circuits for terminals A and B.

(a) $\includegraphics{lab1-circ5.eps}$ (b) $\includegraphics{lab1-circ6.eps}$

(c) $\includegraphics{lab1-circ7.eps}$ (d) $\includegraphics{lab1-circ8.eps}$
Experiment
Build the circuit shown in Figure 4(a), and check your predictions for the voltage across and current through a 500 $\Omega$ load resistor.

Appendix : Resistor Color Codes

Most resistors you will encounter are marked with a set of bands, according to a standard color code, which you can use to determine their resistances. There are ten colors corresponding to numerical digits 0-9 (see the table below), and gold and silver bands indicating 5% and 10% accuracy in the coded resistance, respectively. Starting at the far end of the resistor from the gold/silver band, the first two bands are the first two digits in the resistance. The third band gives the power of ten by which you multiply the first two digits to obtain the resistance.

color	black	brown	red	orange	yellow	green	blue	violet	gray	white
digit	0	1	2	3	4	5	6	7	8	9
multiplier	1	10	100	1k	10k	100k	1M	10M	100M	1000M

$\displaystyle R = [band 1][band 2] \times 10^{[band 3]}$	$\textstyle \pm$	$\displaystyle 5\% (gold)$	(2)
	$\textstyle \pm$	$\displaystyle 10\% (silver)$

For example, Blue Yellow Red Gold gives $R = 64 \times 10^{2}~\Omega = 64 \times 100~\Omega = 6400~\Omega$ with a tolerance of 5%, or $320~\Omega$ .