The circuit photographed below includes two identical light bulbs and three identical batteries, wired as seen in the photograph. In this photograph the switch is OFF, that is, the circuit is open at that point. Notice that the intensity level of the two light bulbs is identical.
The question is: What will happen when the switch is turned ON, closing that part of the circuit?
The upper light bulb will:
 (a) become brighter.
 (b) become dimmer.
 (c) stay the same.
The lower light bulb will:



(a) become brighter.
(b) become dimmer.
(c) stay the same.
The answer is (c): the bulbs remain at the same brightness, as can be seen by comparing the two photographs below.

Two important points are relevant to the explanation of this result. First, note that the potential at the point where the third battery joins the circuit of the other two remains the same when the switch is closed. This is so because all of the batteries are the identical, and the potential along the light bulb wire is divided equally between the bulbs because they are identical. Therefore, closing the switch does not do anything to the circuit.

Resistance and Ohm’s Law
Materials in general have a characteristic behavior of resisting the flow of electric charge. This physical property, or ability to resist current, is known as resistance and is represented by the symbol R. The resistance of any material with a uniform cross-sectional area A depends on A and its length L. In mathematical form,
R


A
L where ρ is known as the resistivity of the material in ohmmeters. Good conductors, such as copper and aluminum, have low resistivities, while insulators, such as mica and paper, have high resistivities.
The circuit element used to model the current-resisting behavior of a material is the resistor. For the purpose of constructing circuits, resistors are usually made from metallic alloys and carbon compounds. The resistor is the simplest passive element. Georg Simon
Ohm (1787–1854), a German physicist, is credited with finding the relationship between current and voltage for a resistor.
Ohm’s law states that the voltage v across a resistor is directly proportional to the current i flowing through the resistor.
That is, v ~ i
Ohm defined the constant of proportionality for a resistor to be the resistance, R. (The resistance is a material property which can change if the internal or external conditions of the element are altered, e.g., if there are changes in the temperature.) Thus, v = iR which is the mathematical form of Ohm’s law. R is measured in the unit of ohms, designated

.
The resistance R of an element denotes its ability to resist the flow of electric current; it is measured in ohms (

).
We may deduce that
R = v / i so that
1

= 1 V/A
To apply Ohm’s law we must pay careful attention to the current direction and voltage polarity. The direction of current i and the polarity of voltage v must conform with the passive sign convention. This implies that current flows from a higher potential to a lower

potential in order for v = iR. If current flows from a lower potential to a higher potential, v = −iR.
Since the value of R can range from zero to infinity, it is important that we consider the two extreme possible values of
R. An element with R = 0 is called a short circuit. For a short circuit, v = iR = 0 showing that the voltage is zero but the current could be anything. In practice, a short circuit is usually a connecting wire assumed to be a perfect conductor.
A short circuit is a circuit element with resistance approaching zero.
Similarly, an element with R = ∞ is known as an open circuit,
For an open circuit, i

R lim
   v
R

0 indicating that the current is zero though the voltage could be anything.
An open circuit is a circuit element with resistance approaching infinity.
It should be pointed out that not all resistors obey Ohm’s law. A resistor that obeys Ohm’s law is known as a linear resistor. It has a constant resistance and thus its current-voltage characteristic is a straight line passing through the origin.
A nonlinear resistor does not obey Ohm’s law. Its resistance varies with current and its i-v characteristic are not linear.
Examples of devices with nonlinear resistance are the lightbulb and the diode. Although all practical resistors may exhibit nonlinear behavior under certain conditions, we will assume in this book that all elements actually designated as resistors are linear.
A useful quantity in circuit analysis is the reciprocal of resistance
R, known as conductance and denoted by G:
G

1
 i
R v
The conductance is a measure of how well an element will conduct electric current. The unit of conductance is the mho (ohm spelled backward) or reciprocal ohm, with symbol

the inverted omega. Although engineers often use the mhos, in this book we prefer to use the siemens (S), the SI unit of conductance:
1 S = 1 mho = 1 A/V
Conductance is the ability of an element to conduct electric current; it is measured in mhos or siemens (S).
The same resistance can be expressed in ohms or siemens. For example, 10

is the same as 0.1 S. we may write i = Gv
The power dissipated by a resistor can be expressed in terms of R. p
 vi
 i
2
R
 v
2
R
The power dissipated by a resistor may also be expressed in terms of G as p
 vi
 v
2
G
 i
2
G
We should note two things:
1. The power dissipated in a resistor is a nonlinear function of either current or voltage.
2. Since R and G are positive quantities, the power dissipated in a resistor is always positive. Thus, a resistor always absorbs power from the circuit. This confirms the idea that a resistor is a passive element, incapable of generating energy.
Given an incandescent light bulb rated at 75 watts and 120 volts, find the “hot” resistance and “cold” resistance of the light bulb. The filament is made of tungsten
( 52 × 10 -8 Ωm )

Overview:
In this lab, we will continue to explore the characteristics of resistors. In this lab, we measure several combinations of voltage and current for a resistor and plot the resulting voltage-current characteristic curve measured for the resistor. The resistance of the resistor will be estimated from the slope of the voltage-current characteristic.
The slope of the curve will be estimated using linear regression techniques; MATLAB commands used to perform linear regression are provided in the background material associated with this lab assignment.
General Discussion:
We have previously noted that the resistance of a component is the slope of the current vs. voltage curve for the component. In this part of the lab assignment, we will measure a current-voltage characteristic curve for a resistor and estimate a resistance from this data. We will compare this resistance from the resistance measured by an ohmmeter.
In order to experimentally determine the current-voltage characteristic for our resistor, we will use the circuit shown schematically in Figure 1.
The arbitrary waveform generator will be used to apply the voltage v s
. We will measure the voltage across the resistor, v
R
, and the current through the resistor, i
R
, using our DMM. By varying v s
, we can measure a set of values for v
R
and i
R
and plot v
R
vs i
R
,

as shown in Figure 2(a). The slope of the line that “best fits” the measured data can then be used to estimate the component’s resistance, as shown in Figure 2(b).
Note:
Do not use the values displayed by the power supply as the resistor voltage and current, v
R
and i
R
. The values displayed by the power supply may differ from the resistor’s voltage and current due to non-ideal power supply effects, such as the power supply internal resistance.
1.
2.
3.
Connect the circuit shown in Figure 1. Use a 100 resistor and one of the AWG channels for the variable supply. Use an ohmmeter to measure the actual resistance of your resistor and record the value in your lab notebook.
Vary the supply voltage v s
from 0V to approximately 2V. Measure v
R
and i
R
for at least 10 different values of v s
over this range of applied voltage (e.g. measure v
R
and i
R
at approximately 0.2V increments in v s
. Tabulate the measured values of v
R
and i
R
in your lab blog. Note: since we only have one DMM, the voltage and current measurements will have to be performed separately. If you have access to two DMMs, the two measurements can be made simultaneously.
Take a picture and a screen shot of your circuit and data for your blog.
Related information:
Resistance is estimated in the post-lab exercises using linear regression of these data. Linear regression is discussed in Appendix A of this lab
Post-lab Exercises:
Determine a least-squares curve fit of the v
R
vs. i
R
. Plot the resulting line and the measured v
R
vs. i
R
data on the same graph. Comment on your results. Calculate a correlation coefficient for the data. Comment on your correlation coefficient relative to the qualitative agreement between the line and the data as shown on your plot. Include this in your blog.

Include the following in your blog:
1.
Circuit diagram; measured resistance value of resistor.
2.
Attach a table providing your measured v
R
and i
R
over range of v s
from 0V – 2V (minimum 10 data points)
3.
Provide below an equation for least-squares best fit line to the data and the correlation coefficient between the data and the curve fit. Attach to this worksheet a plot of data vs. best fit line.
NODES, BRANCHES AND LOOPS
Since the elements of an electric circuit can be interconnected in several ways, we need to understand some basic concepts of network topology. To differentiate between a circuit and a network, we may regard a network as an interconnection of elements or devices, whereas a circuit is a network providing one or more closed paths. The convention, when addressing network topology, is to use the word network rather than circuit. We do this even though the words network and circuit mean the same thing when used in this context. In network topology, we study the properties relating to the placement of elements in the network and the geometric configuration of the network. Such elements include branches, nodes, and loops.
A branch represents a single element such as a voltage source or a resistor.
In other words, a branch represents any two-terminal element.
The circuit shown has five branches, namely, the 10-V voltage source, the 2-A current source, and the three resistors.
A node is the point of connection between two or more branches.
A node is usually indicated by a dot in a circuit. If a short circuit (a connecting wire) connects two nodes, the two nodes constitute a single node. The circuit at above has three nodes a, b, and c. Notice that the three points that form node b are connected by perfectly conducting wires and therefore constitute a single point. The same is true of the four points forming node c. We demonstrate that the circuit has only three nodes by redrawing the circuit.
A loop is any closed path in a circuit.

A loop is a closed path formed by starting at a node, passing through a set of nodes, and returning to the starting node without passing through any node more than once. A loop is said to be independent if it contains a branch which is not in any other loop. Independent loops or paths result in independent sets of equations.
For example, the closed path abca containing the 2-

resistor is a loop. Another loop is the closed path bcb containing the 3-

resistor and the current source. Although one can identify six loops in the circuit, only three of them are independent.
A network with b branches, N nodes, and L independent loops will satisfy the fundamental theorem of network topology: b = L + N − 1
As the next two definitions show, circuit topology is of great value to the study of voltages and currents in an electric circuit.
Two or more elements are in series if they are cascaded or connected sequentially and consequently carry the same current.
Two or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them.
Elements are in series when they are chain-connected or connected sequentially, end to end. For example, two elements are in series if they share one common node and no other element is connected to that common node. Elements in parallel are connected to the same pair of terminals. Elements may be connected in a way that they are neither in series nor in parallel. In the circuit, the voltage source and the 5-

resistor are in series because the same current will flow through them. The 2
resistor, the 3-

resistor, and the current source are in parallel because they are connected to the same two nodes (b and c) and consequently have the same voltage across them. The 5-

and 2-

resistors are neither in series nor in parallel with each other.
Determine the number of branches, loops and nodes in the circuit shown below.
Check Fundamental Network Thm.

Kirchhoff’s LAWS
Ohm’s law by itself is not sufficient to analyze circuits. However, when it is coupled with
Kirchhoff’s two laws, we have a sufficient, powerful set of tools for analyzing a large variety of electric circuits. Kirchhoff’s laws were first introduced in 1847 by the German physicist Gustav Robert Kirchhoff (1824–1887). These laws are formally known as
Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL).
Kirchhoff’s first law is based on the law of conservation of charge, which requires that the algebraic sum of charges within a system cannot change.
Kirchhoff’s current law (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero.
Mathematically, KCL implies that
N  n

1 i n

0 where N is the number of branches connected to the node and i n
is the nth current entering (or leaving) the node. By this law, currents entering a node may be regarded as positive, while currents leaving the node may be taken as negative or vice versa.
To prove KCL, assume a set of currents i k
(t), k = 1, 2, . . . , flow into a node. The algebraic sum of currents at the node is i
T
(t) = i
1
(t) + i
2
(t) + i
3
(t) + · · ·

Integrating both sides gives where q k
(t) =

i k q
T
(t) = q
1
(t) + q
2
(t) + q
3
(t) + · · ·
(t) dt and q
T
(t) =

i
T
(t) dt. But the law of conservation of electric charge requires that the algebraic sum of electric charges at the node must not change; that is, the node stores no net charge. Thus q
T
(t) = 0 → i
T
(t) = 0, confirming the validity of KCL.
Applying KCL to the node at right gives i
1
+ (−i
2
) + i
3
+ i
4
+ (−i
5
) = 0 since currents i
1
, i
3
, and i
4
are entering the node, while currents i
2
and i
5
are leaving it.
By rearranging the terms, we get i
1
+ i
3
+ i
4
= i
2
+ i
5
The sum of the currents entering a node is equal to the sum of the currents leaving the node.
Note that KCL also applies to a closed boundary. This may be regarded as a generalized case, because a node may be regarded as a closed surface shrunk to a point. In two dimensions, a closed boundary is the same as a closed path. As typically illustrated in the circuit at right, the total current entering the closed surface is equal to the total current leaving the surface.
A simple application of KCL is combining current sources in parallel. The combined current is the algebraic sum of the current supplied by the individual sources. For example, the current sources shown can be combined. The combined or equivalent current source can be found by applying KCL to node a.
I
T
+ I
2
= I
1
+ I
3 or
I
T
= I
1
− I
2
+ I
3
A circuit cannot contain two different currents, I
1
and I
2
, in series, unless
I
1
= I
2
; otherwise KCL will be violated.
Kirchhoff’s second law is based on the principle of conservation of energy:
Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around a closed path (or loop) is zero.

Expressed mathematically, KVL states that
M  m

1 v m

0 where M is the number of voltages in the loop (or the number of branches in the loop) and v m
is the mth voltage.
To illustrate KVL, consider the circuit at right. The sign on each voltage is the polarity of the terminal encountered first as we travel around the loop. We can start with any branch and go around the loop either clockwise or counterclockwise. Suppose we start with the voltage source and go clockwise around the loop as shown; then voltages would be −v
1
, +v
2
, +v
3
, −v
4
, and
+v
5
, in that order. For example, as we reach branch 3, the positive terminal is met first; hence we have +v
3
. For branch 4, we reach the negative terminal first; hence, −v
4
. Thus, KVL yields
−v
1
+ v
2
+ v
3
− v
4
+ v
5
= 0
Rearranging terms gives v
2
+ v
3
+ v
5
= v
1
+ v
4 which may be interpreted as
Sum of voltage drops = Sum of voltage rises
This is an alternative form of KVL. Notice that if we had traveled counterclockwise, the result would have been +v
1
, −v
5
, +v
4
, −v
3
, and −v
2
, which is the same as before except that the signs are reversed.
When voltage sources are connected in series, KVL can be applied to obtain the total voltage. The combined voltage is the algebraic sum of the voltages of the individual sources. For example, for the voltage sources shown, the combined or equivalent voltage source is obtained by applying KVL.
−V ab
+ V
1
+ V
2
− V
3
= 0
To avoid violating KVL, a circuit cannot contain two different voltages V
1
and V
2
in parallel unless V
1
= V
2
.

Find I and V ab
in the circuit
Overview:
In this lab assignment, a qualitative discussion of dependent sources is presented in the context of MOSFETs (Metal Oxide Semiconductor Field Effect Transistors). A simple voltage controlled current source is constructed and tested.

General Discussion:
Many common circuit elements are modeled as dependent sources – that is, the mathematics describing the operation of the element is conveniently described by the equations governing a dependent source. In this portion of the lab assignment, we will build and test a circuit which acts as a Voltage Controlled Current Source
(VCCS).
The primary circuit element used in this assignment is a Metal Oxide Semiconductor Field
Effect Transistor (MOSFET). There are two basic types of MOSFETs: n-channel and pchannel; the discussion presented here is for n-channel MOSFETs, though similar concepts apply to p-channel MOSFETs. A MOSFET is a three-terminal device; the symbol commonly used to represent a MOSFET in circuit diagrams is shown in Figure 1(a). The three terminals of the device are called the source (S), the drain (D) and the gate (G). Our circuit will employ a ZVN2210A MOSFET; the physical appearance of this MOSFET is shown in Figure 1(b), along with the relative locations of the drain, gate and source for that model MOSFET.
An extremely simplified discussion of a MOSFET’s operation is as follows: A “channel” is opened in the MOSFET by application of a voltage at the gate of the MOSFET. This channel allows current to flow from the drain to the source of the MOSFET (i
D
in Figure 1(a)). Thus, if a power supply is connected to the drain of the
MOSFET, the MOSFET can be used to control the power supply’s current: increasing the gate voltage increases the current out of the power supply. A rough analogy to this process is a valve placed at the base of a water tank – opening the valve allows water to flow out of the tank. Likewise, increasing the gate voltage allows current to “flow” out of the power supply. A
MOSFET, therefore, in conjunction with a power supply, can act as a voltage controlled current source in which the drain current is controlled by the gate voltage.
Lab Procedures:
1.
onnect the circuit shown in Figure 2. Two power supplied are used in the circuit. Use channel 1 of your
Arbitrary Waveform Generator (W1) to apply the
(variable) gate voltage, V
G
. Use of the waveform generator to apply constant voltages is presented in
Appendix A of this assignment. Use the positive power supply (VP+) to provide a constant 5V to the
MOSFET drain; this power supply provides the drain current I
D
.
The 100 resistor in Figure 2 is used to limit the amount of current flowing through the MOSFET. If no resistor is used between the power supply
C

and the MOSFET, an excessive amount of current can flow through the MOSFET resulting in damage to the MOSFET and/or the rest of the circuit. The 100 resistors in your parts kit can be identified by the color bands on the side of the resistor – they will be as shown in Figure 3. We will discuss resistors in detail in later modules. Use an ohmmeter to measure the resistance of the resistor and record this value in your blog
(the actual resistance will most likely be slightly different from 100 ).
Connect your DMM as shown in Figure 2 to measure the current I
D
as indicated in the schematic above. Record this value
2.
MOSFETs have a threshold voltage, below which essentially no current passes through the MOSFET. To determine the threshold voltage for our MOSFET, begin with zero voltage applied at the gate by the variable voltage source V
G
(V
G
= 0V).
The drain current, with no voltage applied at the gate, should be essentially zero.
Gradually increase the MOSFET gate voltage while monitoring the MOSFET drain current I
D
. Record in your lab notebook the voltage at which the drain current begins to increase significantly. This is the MOSFET’s threshold voltage.
3.
Now characterize the MOSFET’s relationship between gate voltage and drain current. Starting at the threshold voltage, continue to increase the gate voltage at increments of about 0.3V up to a maximum of about 5V. Record the gate voltages and their corresponding drain currents in your blog. Plot the gate voltage vs. drain current data in your lab notebook. Comment on your observations relative to the data, especially relative to how the circuit behaves like a dependent source.
4.
The parameter g of a VCCS provides a relationship between the rate of change between the applied voltage and the resulting current. This is essentially the slope of the data you plotted in part 3 above. Use the curve of part 3 to estimate the value of g for the circuit you built. Note:
Your curve will most likely not be a straight line. Do your best to fit a straight line to the data you acquired in part 3 for your estimate of g.
5.
Take a picture of your circuit and a screen shot for inclusion in your blog.
Include the following in your blog:
Diagram of circuit(from your whiteboard), including measured resistance value.
1.
MOSFET threshold voltage

2.
Attach a table providing your measured gate-to-source voltage vs. drain current values and a plot of data.
3.
What type of dependent source is the transistor behaving like? Why?
4.
Estimated value of g for circuit. Annotate the plot, indicating how the value of g was determined.