4. INTRODUCTION

PROCEDURE.

1. Select one of the first four laboratory reports (up to Wheatstone Bridges.)
2. Select an extension to the laboratory report. This will require that you spend additional laboratory time on your own. Possible extensions for each lab are outlined below.

Lab 1: STANDARD ELECTRICAL LABORATORY INSTRUMENTATION

· A possible expansion of this experiment is to investigate the temperature coefficient for carbon resistors and how a function for resistance versus temperature can be derived from empirical data.

Lab 2: KIRCHHOFF'S CIRCUIT LAWS

· Expand this lab by trying other circuit configurations of your choice (text book circuits are a good source) and repeat the analysis and measurement steps above. Note: the complexity of the circuit will effect your grade (i.e., trivial circuits will lower your grade.)

Lab 3: TEE-PI EQUIVALENT CIRCUITS

· Possible expansions for this lab are to study Tee and Pi circuits of problems from your textbook. Note: the complexity of the circuit will effect your grade (i.e., trivial circuits will lower your grade.)

Lab 4: THE WHEATSTONE BRIDGE

· Choose different values for R1 and R2;
· Measure the current flow from point `a' to `b' instead of the voltage;
· Solve the circuit (find all voltages and currents) for the unbalanced case and verify them by experiment.

3. Prepare a rough draft in Microsoft Word using the guidelines below.

· The length of the report should be approximately 1,500 words (6 pages double spaced) not including illustrations or appendices.
· Illustrations should be embedded in the text with figure or table numbers and titles. Full page illustrations should be put in an appendix.
· Follow the guidelines for figures, equations, tables, etc. in the writing guide.
· Use numbered sections including 0.0 Introduction, 1.0 Analysis, 2.0 Experiment (Verification), 3.0 Comparison and Discussion, 4.0 Conclusion, Appendices.
· Refer to the attached example report.
· This paper is for an educated reader, so the point of view should be selected carefully. Passive voice is the most common choice (i.e., avoid I, we, our, it, etc.)

4. Submit the rough draft report February 28th, 2000 and make an appointment for a review session with your laboratory instructor.
5. Attend the review session appointment. Your laboratory instructor will review your rough draft with you.
6. Make revisions to the rough draft and submit the final report within 1 week after your review meeting.

INTRODUCTION

The multimeter is the most commonly used instrument for engineers, technicians and electricians making electrical measurements. Originally electrical work required multiple instruments for voltage, current and resistance. These instruments have been combined into a "multi"meter.

Earlier multimeters used analog needles to indicate values, newer models use digital displays. These meters measure values such as:

voltage (voltmeter)
current (ammeter)
resistance (ohmmeter)

An inexpensive multimeter with limited capabilities can be purchased for $10. High precision multimeters with special features can cost over $1000. The precision of a multimeter is a function of the voltage ranges that can be measured (from nV to KV) and the number of digits on the readout (typically 3 to 7).

Resistors are the simplest and most common circuit component. Resistors are mass produced and typically sell for under 1 cent. But, the production process is not precise, so the resistor values have tolerance values. Typical tolerance values are ±1% ($0.05), ±5% ($0.01), ±10% (<$0.01). The value indicated on the resistor is the nominal value, and the actual value will vary statistically about the nominal value. The tolerance band should contain approximately 3 standard deviations of the values.

Resistor values are indicated using colored bands read from one side to the other. To read the code start from the side that the color bands are closer to. Each color corresponds to a number, then a multiplier, then a tolerance. There is often a gap between the tolerance band, and the other color bands. The color is shown in Figure 1.

Figure 1 - Resistor Color Code

When first looking at resistor values the sequence does not seem rational. But, the sequence is devised so that any resistance value can be obtained by adding two or more resistors together. The spacing of the resistor values is determined by their tolerance (a larger tolerance gives a larger spacing between resistor values. A typical series of resistor values for a 5% tolerance is:

1.0, 1.1, 1.2, 1.3, 1.5, 1.6, 1.8, 2.0, 2.2, 2.4, 2.7, 3.0, 3.3, 3.6, 3.9, 4.3, 4.7, 5.1, 5.6, 6.2, 6.8, 7.5, 8.2, 9.1, 10.0, etc...

Batteries are typical voltage sources that use chemical reactions to create an electrical potential. They are also prone to variations in production and the limitation of the battery life. As a result new batteries will not produce identical voltages, and the voltage will decrease with use.

VERIFICATION PROCEDURE

1. Locate the Fluke 8050 Digital MultiMeter (DMM) and familiarize yourself with its operation by studying the operation manual attached. Setup the DMM for making DC voltage measurements.

2. Starting with the highest scale on the DMM, measure and record the voltage of a 1.5V battery as displayed on the DMM while the voltage ranges are selected in descending order. Observe and identify overload on the DMM and determine the scale that would give the most accurate reading. The term "record" refers to making log book entries (in Mathcad) of calculations and measured data.

3. Ask the instructor to provide two resistor values for you to measure. Obtain 10 identical resistors from the resistor cabinet, and verify these values using the color codes. These resistors will be plugged into the protoboards on the trainers. Connect two wires to either end of the resistors, and connect these to the multimeter. The basic layout of the protoboards is shown below.

Figure 2 - Conductor Layout of the Protoboard

4. Set the DMM to measure resistance, and verify that the scale gives the highest accuracy for the resistors. Measure resistor values for both sets of resistors, and record these in Mathcad. Calculate the average and standard deviations for each of the resistor values.

COMPARISON PROCEDURE

1. Compare the resistor values in the parts cabinet to the resistor series calculated using tolerances.

2. Examine the resistor values to determine how many standard deviations are between the tolerance limits. Also determine the percentage deviation of the average resistor values from the nominal values. (Note: be careful to calculate the percentage from the nominal, not from the experimental.)

3. Compare the two battery voltage values.

4. Write a discussion and conclusion values for the results in general and discuss the value of the multimeter as an instrument. (Note: State facts and details, avoid vague and rambling statements.)

INTRODUCTION

Kirchhoff's voltage and current laws are two statements that relate voltages and currents in an electric circuit. These two laws are all inclusive in that (1) they apply to all electrical circuits, and (2) they are the only laws needed, aside from the behavior of the electrical devices themselves and Ohm's law, to solve for all the voltages and currents in a circuit.

Kirchhoff's Voltage Law (KVL) states that

"The algebraic sum of circuit element voltages around any closed path is zero."

Kirchhoff's Current Law (KCL) states that

"The algebraic sum of all currents at a node is zero."

The phrase "algebraic sum" implies that there can be positive and negative quantities in forming the sum of voltages or currents. In other words, there is a dependency on an assumed direction to the currents and voltages. To establish the proper sign of a voltage or current, positive reference directions need to be established before a circuit analysis is started. In this course these positive reference directions are as follows:

1.Currents leaving a node are positive.
2.The voltage drop across a branch circuit element is positive at the tail of the arrow that represents the direction of current flow through the branch circuit element (see Fig. 1).

Figure 1. Sign conventions used for current flow through, and voltage drop across a circuit element.

Voltage Divider Rule

Kirchhoff's voltage law (KVL) will be demonstrated with the simple series circuit shown in Figure 2.

Figure 2. A simple series circuit.

Applying Kirchhoff's voltage law to the circuit in Fig. 2 yields:

Since the current is the same through each branch of the circuit and is equal to iS (from KCL), applying Ohm's law we have

Therefore

and

where

In general, the voltage drop across any branch, n, can then be found from

This is the voltage divider rule. The voltage divider rule can be used to find v2 in Figure 2 from

Current Divider Rule

The circuit "duals" of Kirchhoff's voltage law and the voltage divider rule are Kirchhoff's current law and the current divider rule. The voltage law was applied to resistances in series to determine the voltage divider rule. The current law will now be applied to resistances in parallel, as shown in Figure 3, to determine the current divider rule.

Figure 3 A simple parallel circuit.

Applying the current law to the circuit of Fig. 3, we have at Node 1,

and at Node 0,

Using Gi as the conductance of the ith branch, and since the voltage, v, is the same across each branch, the branch currents are

and therefore

thus

where

The current through any branch, n, can be found from

This is the current divider rule. The current divider rule can be used to find current i2 in Figure 3 from

VERIFICATION PROCEDURE (Build and Measure)

1. Obtain the resistors for this experiment and then measure and record their values as accurately as you can. Let R1 = 1000, R2 = 2000 and R3 = 3000 nominal ohms, respectively. Refer to the DMM operation sheets, page 2-7. Note: when measuring resistance make sure that the resistor is not connected to the power supply. Never measure resistance when the resistor is energized.
2. Wire the circuit shown in Figure 2 onto the CDT. Use the +5V power supply on the CDT for the source.

Figure 5. Conductor Layout of the Protoboard

3. For the circuit in Figure 2, verify KVL by measuring the voltage drop across each resistor and the actual source voltage and demonstrate that the algebraic sum of the voltages is zero (or close to zero.)
4. For the circuit in Figure 2, replace the 5V power supply with the adjustable positive voltage supply. Connect the DMM to measure the current in the loop and adjust the supply voltage, vS,, until the current is is = 2.0 mA. Disconnect the DMM and measure the supply voltage. Note: see Figure 6 before measuring current with the DMM.)
5. Rewire the circuit on the CDT to be the circuit of Figure 4 using the previous resistors, R1, R2 and R3. Measure the voltage drops across each resistor, and the currents through each resistor.

COMPARISON PROCEDURE

1. For the circuit in Figure 2, compare the calculated and measured values for voltage drops across each resistor with the supply voltage of 5V.
2. For the circuit in Figure 2, compare the calculated and actual supply voltage needed to get a current of is=2mA.
3. For the circuit in Figure 4, compare the calculated and measured voltages accross and currents through each resistor.
Note: When comparing the calculated and measured values use percentages relative to the calculated values.

Figure 6. DMM Connection for Voltage and Current Readings

INTRODUCTION

Tee and Pi (or Y and Delta) networks occur often in electric and electronic circuits. Your ability to recognize these circuit structures and your ability to convert them from one form to the other are important skills for simplifying, understanding, and designing networks. A network often has to be simplified in order to understand its operation sufficiently well to be able modify its design.

For two networks to be equivalent, their terminal characteristics must be the same. By terminal characteristics we mean the voltage-current relationship at each terminal pair. Refer to Figure 1.

Figure 1. Equivalent Pi and Tee (or Delta and Y) networks.

For the circuits to be equivalent, we must have

We solve the above equation to obtain the Tee to Pi transformation equations:

DESIGN AND ANALYSIS PROCEDURE

1. Given a Tee network where R1 = 1K, R2 = 4.7K and R3 = 2.2K, find the equivalent Pi network.
2. Engineers create designs to meet specifications. For example, telephone systems are often based on 600 Ω standards. What follows is a set of design requirements that need to be satisfied using a resistor network. Determine the values for each resistor in a Tee circuit (R1, R2, R3) such that its terminal resistance Rac is equal to 600 Ω, the output-to-input voltage ratio from left to right (that is vbc/vac) equals 0.65 with the terminal current ib = 0, the terminal resistance Rbc is equal to 600 Ω, and the output-to-input voltage ratio from right to left (vac/vbc) equal to 0.65 with the terminal current ia = 0. Note that the network must be symmetrical because the terminal resistances are equal. Note: don't forget to write down the given information before starting to find resistor values.

Figure 2. A Tee Network for ib=0A.

3. Calculate the resistors Ra, Rb and Rc for a Pi network that is equivalent to the Tee you designed in step 2.

BUILD AND MEASURE PRODEDURE

1. Set up a Tee network using the resistor values R1 = 1K, R2 = 4.7K and R3 = 2.2K. Measure the resistances Rac, Rbc, Rab.
2. Set up a Pi network using the resistor values calculated in Analysis Procedure Step 1. Measure the resistances Rac, Rbc, Rab.
3. Select the resistors needed to implement the Tee circuit of the Analysis Procedure Step 2. (You may have to search through a batch of resistors to select values that are "right on".) Set up the Tee circuit and measure Rac, Rbc, vbc/vac and vac/vbc for your Tee circuit. For example, to measure vbc/vac select a voltage value for vac using the variable voltage supply on the CDT (a value between 5 to 10V is reasonable.) Measure both voltages using the DMM, and divide the values to find the ratio.
4. Select three "pots" (electrical jargon for potentiometers or variable resistors) to cover the resistance range needed for Ra, Rb, and Rc, respectively in Analysis Step 3. Adjust each pot to equal the resistance needed for Ra, Rb and Rc. Set up the Pi circuit with the pots and verify that it has the same parameter values as the Tee network. (See figures 3 and 4 before building the circuit.)

Figure 3. Potentiometers

Figure 4. Potentiometer Packages

INTRODUCTION

The Wheatstone bridge circuit (named after its inventor) is shown in Figure 1 below. The bridge is balanced, i.e. no current flow between points a and b, when the products of the

Figure 1. The Wheatstone bridge circuit.

cross-arm resistances are equal. That is, the bridge balance equation is

If R1 is made equal to R2, then when the bridge is balanced, Rx = R3. This is a convenient way to make "in circuit" measurements unknown of impedances without having to disconnect components to connect the DMM. In this laboratory experiment only DC measurements will be made. However, these same techniques apply to measurements of complex AC impedance as well.

This bridge is common in most engineering disciplines. For example, it will be used in EGR 309 for strain gauge measurements.

BUILD AND MEASURE PROCEDURE

1. Obtain several 1000 Ω resistors and start measuring them with the DMM. Use masking tape on the resistor to identify the value of each resistor and record these values in your log book. After several resistor values have been recorded a pattern should emerge that should allow you to decide how close the values of your sample can be for the resistors to be considered "matched". You may be able to match them perfectly (within the measurement capability of the DMM), but don't spend too much time trying. Make a judgment and select a matched pair after a reasonable effort has been made to find the "perfect" match.

2. Set up the circuit of Figure 1. Use a resistor with a nominal value in the range 100 Ω to 1000 Ω as the unknown resistor Rx. Use a potentiometer for variable resistor R3.

Figure 2. Pinout diagram of a miniature, multiturn potentiometer.

3. Measure the resistance of Rx by adjusting the variable resistor R3 until the bridge is balanced. Use the DMM to measure the voltage from point a to point b in the circuit and adjust R3 for a zero reading on the DMM.

4. Once balance has been achieved, remove R3 and measure its resistance with the DMM. Calculate the value of Rx using the bridge balance equation.

5. Repeat steps 3 and 4 for another "unknown" resistor.

COMPARISON PROCEDURE

1. Given the resistor values measured in Build step 1, and the resistor Rx selected in Build step 2, calculate the value of R3. Compare this to the actual resistor value measures in Build step 3.

2. Repeat the previous step for Build step 5.

INTRODUCTION

The node-voltage method of circuit analysis consists of the following sequential steps:

1) All nodes of the circuit are identified and one node is selected as the "datum" or reference node.
2) All node voltages are defined with respect to the datum node.
3) All of the branch currents are expressed in terms of the node voltages using the positive reference direction: "the positive voltage drop across any branch is at the tail of the branch current arrow".
4) Kirchhoff's current law (KCL) is written at each node using the positive reference direction: "branch currents out of a node are positive".
5) The resulting set of node-voltage equations is then written in matrix form. The resulting matrix of coefficients is called the admittance matrix for the circuit.
6) The node voltages are found by solving the set of linear equations using mathematical techniques you have previously learned. By this lab period, you are expected to be familiar with the following techniques for solving simultaneous linear equations:

· substitution and variable elimination by hand.
· matrix methods using Mathcad or a calculator.

· Cramer's rule using determinants.
· the inverse matrix method.
· Gauss-Jordan row reduction

The node-voltage method of circuit analysis will be verified in this lab by comparing calculated node-voltage values to measured values.

BUILD AND MEASURE PROCEDURE

1. You will study the circuit shown in Figure 1. Select the source voltage from the range of voltages available on the CDT and seven 1/4 W resistors of various values. Select resistor values so that for the source voltage selected, the power dissipation in each resistor does not exceed 1/4 W. A "worst case" design criteria is to limit the total power from the source to be less than 1/4 W. Therefore, if each of the seven resistors chosen has a value greater than V/0.25, then the power dissipated in each resistor will not exceed resistor's rating. For example, if 10 V is chosen for the supply voltage, then a minimum resistance of 100/.25 = 400 Ω for each of the resistors keeps the power dissipated in each resistor less than 1/4 W.

2. Measure the values of the resistors you have chosen as precisely as you can and record their values. Set or measure the power supply to the value you have chosen.

3. Set up the circuit of Figure 1 on the CDT using your measured resistors and experimentally analyze the circuit. (You must decide which currents and/or voltages to measure.) To measure the node voltages, first connect the black "common" lead of the DMM to the circuit datum node and leave it there. Then probe each node with the red lead to measure the node's voltage.

INTRODUCTION

The mesh-current method of circuit analysis requires the following sequential steps:

1. All meshes are identified by a mesh current. The mesh current direction is determined at the time the meshes are assigned. The choice of mesh current direction is arbitrary. A popular choice is to take the mesh current "out of" the "+" terminal of the voltage source. This usually results in a clockwise current direction that is used for all the remaining meshes of the circuit.
2. All the branch voltage drops are expressed in terms of the mesh currents using the positive reference direction: "the positive voltage drop across any branch is at the tail of the branch current arrow".
3. Kirchhoff's voltage law (KVL) is written around each mesh following the mesh current direction. The positive reference direction used to determine the algebraic sign of the branch voltage drops is the same as that in step 2.
4. The resulting set of mesh current equations is then written in matrix form. The resulting matrix of coefficients is called the resistance matrix for the circuit.
5. The mesh currents are found by solving the set of linear equations using mathematical techniques you have previously learned. By this lab period, you are expected to be familiar with the following techniques for solving simultaneous linear equations:

· substitution and variable elimination by hand.
· matrix methods using Mathcad or a calculator.

· Cramer's rule using determinants.
· the inverse matrix method.

The mesh-current method of circuit analysis will be verified in this lab by comparing calculated current values to measured values.

BUILD AND MEASURE PROCEDURE

1. You will study the circuit shown in Figure 1. Select the source voltage from the range of voltages available on the CDT and seven 1/4 W resistors of various values. Select resistor values so that for the source voltage selected, the power dissipation in each resistor does not exceed 1/4 W. A "worst case" design criteria is to limit the total power from the source to be less than 1/4 W. Therefore, if each of the seven resistors chosen has a value greater than V/0.25, then the power dissipated in each resistor will not exceed resistor's rating. For example, if 10 V is chosen for the supply voltage, then a minimum resistance of 100/.25 = 400 Ω for each of the resistors keeps the power dissipated in each resistor less than 1/4 W. (Note: the resistors you select should be much larger than the minimum.)
2. Measure the values of the resistors you have chosen as precisely as you can and record their values. Set or measure the power supply voltage to the value you have chosen.
3. Set up the circuit of Figure 1 on the CDT using your measured resistors and experimentally analyze the circuit. (You must decide which currents and/or voltages to measure.) Measure each mesh current by inserting an ammeter at an appropriate point in the mesh.

INTRODUCTION

The superposition principle is a very important analysis tool that is applicable to all linear systems. In electric circuits superposition is usually applied in two ways:

1. Scaling. If the single independent source of an electrical network is multiplied by a constant, then each response of the network is multiplied by the same constant.
2. Additive. If two or more independent sources are used to excite an electrical network, then the response of the network can be found by adding the responses of that network to each independent source individually with every other source deactivated (turned off).

In this lab the additive superposition property will be investigated. Three independent voltage sources will be used in the circuit shown in Figure 1. The desired response for this circuit with all sources energized is node voltage v2. The objective of this experiment is to analyze the circuit of Figure 1 using superposition and then to verify the principle by measurement.

The first step in the analysis is to solve for v2 due to each source acting individually with the other sources deactivated. These voltages will be identified as

, and .

To deactivate a voltage source, do the following:

1) set the voltage to zero, and
2) replace the voltage source with a short circuit.

Therefore, to find for this circuit, the leads connecting nodes 3 and 4 to their respective sources must be disconnected from the sources and reconnected to the circuit datum. This deactivates sources 3 and 4 and replaces them with a short so that the voltage at node 2 is due to source 1 alone. This procedure is then repeated for each of the other sources. The total response, by superposition, is the sum of the individual responses. That is,

Figure 1. A circuit for illustrating superposition.

INTRODUCTION

Thevenin's theorem states that "Any linear network that contains branch circuit elements and sources (voltage, current, independent as well as controlled sources) can be reduced to an equivalent circuit that contains a single independent voltage source and series impedance." This equivalent circuit is called a "Thevenin's equivalent circuit". This is quite a remarkable theorem since it implies that no matter how complicated a circuit may be, it can always be reduced to an equivalent circuit consisting of a single independent voltage source and a single series impedance. Therefore, a complicated circuit can be thought of as a simple Thevenin equivalent circuit.

For this experiment the only circuit considered will be resistive with a single independent voltage source. For a circuit with "output" terminals designated a and b, the Thevenin equivalent will replace the circuit at these terminals as shown in Figure 1.

Figure 1. Example Thevenin's equivalent circuit.

In this lab we will determine the Thevenin equivalent circuit using both analytical and experimental methods. Analytically, the Thevenin's equivalent circuit can be found as follows:

1. Find the open circuit voltage at the terminals a and b.
2. Deactivate (that is, de-energize) all independent sources (but not the controlled sources) in the circuit. Independent voltage sources are replaced with shorts and independent current sources are replaced with opens to deactivate them.
3. With all independent sources deactivated, "look back" into terminals a and b and calculate the equivalent resistance, RT.
4. The Thevenin's equivalent circuit is then an independent voltage source with a voltage equal to the open circuit voltage in series with the Thevenin's equivalent resistance.

We apply the above steps for the circuit shown in Figure 1. The open circuit voltage can be found by applying the voltage divider rule

The Thevenin's equivalent impedance can be found by deactivating the source and looking back into terminals a and b as shown in Figure 2. The Thevenin's equivalent resistance can then be calculated by simply adding the resistance of R3 to the resistance of the parallel combination R1 and R2.

Figure 2. Finding the Thevenin's equivalent resistance by turning off independent sources.

Part of this experiment is to verify that the two parameter Thevenin's equivalent circuit is indeed equivalent to the original circuit. The test for equivalency is to determine if the v-i relationship matches at the a and b terminals. This can be accomplished by plotting the output voltage versus current as in Figure 3 for both circuits and comparing the two plots. Since the circuits are linear, the v-i curve for the a and b terminals will be a straight line with a negative slope because the terminal voltage must decrease as the load current increases. Therefore, only two points are required to define the line. For practical circuits, there are two obvious points to use. These points are (1) at no-load or open circuit where the source current is zero, and (2) at full load where the source current is the maximum the source can supply without burning up. Actually, any load resistance can be used so long as the load current does not exceed that which the circuit's power sources can supply. Circuit loads are just resistances that are put across the output terminals of a circuit in order to cause a current flow.

Figure 3. Output voltage droop as a function of load current.

Once a current has been established for the Thevenin equivalent circuit, there will be a voltage drop ΔV from the no-load value as shown in Figure 3. From Figure 3

Therefore

This relationship for determining RT can be applied to any circuit no matter what the circuit may consist of. In most instances the circuit may be a box with two terminals and no knowledge of the circuit inside the box. Thevenin's equivalent circuit for this box may be found by measuring VNL, VL and iL and then calculating ΔV and Δi.

INTRODUCTION

An ideal operational amplifier (or op-amp) has infinite gain, infinite input impedance and zero output impedance. With the development of integrated circuit technology, current state-of-the-art op-amps very nearly approach this ideal. Oddly enough, as a linear circuit element the ideal op-amp is useless. In order to make a linear amplifier with an op-amp, the op-amp must be used with an external negative feedback circuit. Negative feedback is required to control the gain of the amplifier as well as its frequency response (bandwidth). In this lab, only gain and input impedance will be controlled by design. The inverting and non-inverting amplifier circuits are shown in Figure 1.

Figure 1. Basic operational amplifier circuits.

For the inverting amplifier, Figure 1a, the input to the op-amp constitutes a current summing point so that

Now the closed-loop gain of a circuit, Av, is defined as the output voltage divided by the input voltage, that is

so that for the inverting amplifier, the closed-loop gain is

upon combining the above equations.

For the non-inverting amplifier, Figure 1b, we have

so that the closed loop voltage gain, Av, is

BUILD AND MEASURE PROCEDURE

1. Set up the 741 op-amp circuit you have designed on the CPS. Make sure the power is OFF as you build the circuit. Connect +VCC and -VCC to the +12 V and -12 V supplies, respectively, on the CPS. Sketch the circuit in your logbook. For your reference, the pin-outs of the 741 mini-DIP are shown in Figure 2.

Figure 2. 741op-amp pinouts (looking down on the DIP package).

2. Turn the CPS on and with the input voltage to the amplifier set to zero (vi = 0) check to see that the output voltage, vo, is also zero. If it isn't, check to make sure your circuit is connected correctly.
3. Put a small, less than 1 V, positive DC voltage into the input, (i.e., vi < 1 V). Measure and record the output voltage, vo. Calculate the voltage gain, Av, and compare this value to your design. Do the same with a voltage time function from the FG (sine, square or triangular wave form). Measure the output voltage, vo, on the SCOPE.
4. Repeat steps 1 through 3 for the non-inverting amplifier.

INTRODUCTION

A capacitor is a physical device for storing charge. Using a water analogy, a capacitor is like a bucket in which water is accumulated or stored. A capacitor accumulates or stores electrical charge.

A physical model of capacitor operation is that the amount of charge stored in the capacitor is proportional to the voltage across the capacitor. The proportionality constant between voltage and charge is the "capacitance" value of the capacitor. This relationship is stated as

(1)

where q is the charge in coulombs, v is the voltage in volts and C is the capacitance in farads.

From the definition of current, i = dq/dt, the v-i relationship (that is, the branch element circuit model for a capacitor) can be found by taking the derivative of the above equation

In this experiment, only capacitors with constant values will be used so that dC/dt = 0. Thus, the v-i characteristic to be verified in this experiment is

. (2)

Currently, the most convenient instrument available to measure and display voltage time functions is the oscilloscope or "scope". The scopes used in EGR 214 require a repetitive wave form to clearly display a voltage time function. Therefore, for this experiment it is desirable to make either the voltage, v, or the current, i, a periodic time function with period, T, and display the other variable on the scope for measurement.

Since a scope is designed to display voltage, current will be selected as the periodic parameter and the voltage across the capacitor will be displayed on the scope. Because of Eq (1), the voltage displayed is a measure of the charge on the capacitor at any instant of time.

The time function used for the current will be a square wave with zero average value. That is, the area under the curve above the horizontal axis is equal to the area under the curve below the axis. The current wave form is shown in Figure 1.

Figure 1. Current square wave used to drive the integrator circuit.

The voltage across a capacitor can then be found by integrating Eq (2) to get

, . (3)

Since the current is constant during the time that charge is flowing into (+i) or out of (-i) the capacitor, the voltage across the capacitor has the time function shown in Figure 2

Figure 2. The time function of the voltage across the capacitor is a triangular wave when its current is a square wave.

and is described by the equations

for

and

for .

Since the current square wave has zero average value, the voltage across the capacitor must also have zero average value. This implies that the voltage wave form's amplitude, V, is symmetrical about the time axis and

V = v(T/2) = -v(0).

We can solve for the amplitude from the fact that

for the periodic wave form. Thus

(4)

since the period, T, is the reciprocal of the frequency, f. From the above relationships, we also have

PROCEDURE

Note: This week you must divide the procedure steps into analysis, verification and comparison sections.
1. In the expression for the peak or maximum voltage (and peak charge) across the capacitor (Eq (4)), there are three parameters: I, C, and f which determine v(T/2). In order for this experiment work, values for each of these parameters must be determined for permissible values of the peak voltage. If this were a pure analysis problem, then values for these parameters would have been specified in the problem or four independent equations would have been given so that we could solve for unique values for each of these parameters (a right answer type of problem). However, these parameters will not be specified for you in this experiment. Instead, you will have to design this experiment. That is, you will have to choose suitable values for these parameters.
In a typical design problem there are more parameters and variables that have to be specified than there are equations to determine them. In our problem we have a single equation that relates the three parameters to the maximum voltage amplitude. The values for the three parameters and voltage must be determined. Once any three values are specified, the value of the fourth can be determined from the peak voltage equation (i.e., Eq (4)).
For this experiment, a reasonable value for I is 1.0 mA since we can only draw a small current from the FG (due to the FG's design). The FG is also limited to a peak or maximum output voltage of 10 V. A constant current from the FG can be obtained by using an op amp and placing the capacitor in the op amp's feedback circuit. The current charging the capacitor is then determined by the input resistance, Ri, to the op amp and RS, the source resistance internal to the FG. The combined input and internal source resistance must be about 10 kΩ since 10V/.001A = 10,000 Ω. The voltage across the capacitor is limited to the supply voltage used to power the op amp. This means that v(T/2) < 12 V. Values have now been established for two of the four parameters. The next step is to choose values for either C or f and solve for the remaining parameter. Familiarize yourself with the types and values of the capacitors available in the lab and make a choice of C or choose a frequency greater than 500 Hz but less then 10 kHz and solve for C. See Appendix A for how to read coded capacitor values.
Using either approach, design the experiment by determining values for C and f. Select the capacitor and measure its actual value with the LCR Meter.
2. Setup the circuit shown in Figure 3, which is often referred to as an integrator circuit.
3. In the circuit of Figure 3, the voltage between the op amp inputs must always be approximately zero. That is, the voltage ε 0. For the circuit of Figure 3, with the "+" input connected to the datum, the "−" input is sometimes referred to as a "virtual ground". Therefore, the output voltage, vo, is the voltage across the capacitor. The 1 MΩ resistor across the capacitor allows a little bit of current to bypass the capacitor in either direction to balance the positive and negative voltage swings so that the average value of the voltage output is zero.

Figure 3. Op amp integrator circuit.

4. Analyze, measure and compare the voltage across the capacitor for the design values you have chosen.

Background Reading for Interest

Physically, an inductor is simply a coil of wire. Current flowing through a wire creates a magnetic field. The magnetic field created by a section of wire through which current is flowing interacts with adjacent sections of the same wire and induces voltages in those sections of the wire. The total voltage induced across the inductor, v, is proportional to the rate of change of magnetic flux, Φ, by Faraday's law

where N is the number of turns for the coil of wire through which current is flowing. If N is assumed to be constant, then the voltage expression can be written as

The term NΦ is called the flux linkage and is related to the magnetic field that is made by current flowing through one turn linking with all other turns in the coil. The magnetic flux, Φ, is proportional to current or Φ ≈ i which is Ampere's law. The proportionality constant is N/ℜ where N is the number of turns and ℜ is a term that is dependent on the physical properties of the coil and is called the reluctance. Therefore,

By combining Faraday's law with Ampere's law, the v-i relationship for an inductor can be derived as

The term N2/ℜ is a constant that is called inductance and given the symbol L. Inductance has the unit henry named after the American physicist Joseph Henry. Therefore, the equivalent branch circuit v-i relationship for an inductance is

Notice that from Faraday's law, Eq (2),

so that the flux linkage NΦ = Li. This means that flux linkage in an inductor can be thought of as being analogous to charge in a capacitor by comparing

q = Cv

with

NΦ = Li.

Capacitance and inductance are circuit element "duals" in that the roles of v and i are interchanged in their v-i relationships.

The Experiment

In order to verify the v-i relationship for the inductor, the same type of experiment used in Experiment 10 for the capacitor will be applied to the inductor. In verifying the capacitor's v-i relationship, a constant current square wave was used to drive the capacitor and the capacitor's voltage was displayed and measured. In this inductor experiment, a constant current will also be used to drive the inductor and the voltage across the inductor will also be displayed and measured. However, since the capacitor and inductor are duals of each other, if the current time function selected to drive the inductor is a triangular wave instead of a square wave, then the output will be a voltage square wave. This is just the opposite behavior of the capacitor.

There is, however, a significant difference between the physical operation of these devices. Charge is a material that can be stored like water so that in the capacitor experiment, the capacitor started with an initial state of zero charge and was then alternately charged and discharged periodically the same amount so that the charge always returned to zero at the end of the period. Flux linkage cannot be stored like charge since it is a derivative away, i.e. i = dq/dt, from charge. In order to "store" a state of flux linkage, a continuous (DC) current must flow through the inductor.

There will be another change from Experiment 10. In this experiment you will use peak-to-peak voltages rather than amplitude voltages to describe the voltage waveforms. This change will give you experience in dealing with a commonly used way to specify and make scope voltage readings.

DESIGN AND ANALYSIS PROCEDURE

1. Analyze the circuit of Figure 1. Assume the inductor is ideal to start your analysis. That is, assume that the resistance of the inductor, RL, is zero (i.e., the inductor is lossless).

Figure 1. An op amp differentiator circuit.

Select vS(t) to be a triangular wave with peak-to-peak voltage of, V, volts and a frequency, f, or period, T = 1/f, as shown in Figure 2. Thus, the source voltage is given by

Figure 2. Voltage triangular wave used to drive the differentiator circuit.

and the current through the inductor is

Since the voltage function described is peak-to-peak, the current through the inductor is also a peak-to-peak value. Therefore, the output voltage induced across the inductor is also a peak-to-peak value and equal to

For the positive slope of the triangular wave, the output voltage is a negative constant value and for the negative slope the output is a positive constant value. Therefore, the output voltage of the ideal inductor is a square wave with a peak-to-peak value of 4VfL/R, as shown in Figure 3.

Figure 3. The time function of the differentiator's output voltage for an ideal inductor.

The actual output voltage, v0(t), is the voltage across the real inductor. From the model in Figure 1, the real inductor is made up of an inductance in series with a resistance, RL. The inductor resistance, RL, not only includes the DC resistance of the coil wire that makes the inductor, but also all the core losses on which the coil is wound. Therefore, RL is always larger than the DC resistance measured with a DMM.

APPENDIX A - PRACTICAL INDUCTOR ISSUES

The inductor's resistance, RL, can be estimated from the inductor's quality factor (Q value). The Q value is a measure of the energy stored per cycle compared to the energy input per cycle. Later work in a more advanced course will show that

so that the inductor's resistance can be determined from

The inductor's Q value can be measured using an LCR meter available in the laboratory.
The real output voltage, vo(t), is then the sum of two components, the voltage induced by the inductor, Ldi/dt and the voltage drop across RL. Hence,

The term contributed by the voltage drop across RL will force the output voltage waveform to slope rather than be flat (See Figure 4 and compare with Figure 3.). The amount that the voltage will deviate from flatness over a half period is

Figure 4. The time function of the differentiator's output voltage for a real inductor.

Therefore,

INTRODUCTION

You have already learned many techniques to analyze DC circuits. These same techniques can be used to analyze AC circuits using phasors. A phasor represents a complex resistance that is called an impedance. Figure 1 shows the impedances for resistors, capacitors and inductors. For a resistor the impedance is real. Both the inductor, and capacitor, have complex impedances, indicating a phase shift.

Figure 1 - Impedances for Passive Circuit Components

Consider the voltage divider in Figure 2. The output sine wave has a lower amplitude than the input, but it is also shifted (offset) in time. The offset between the waves is usually expressed as an angle, instead of time. Equations 1 and 2 represent the waves seen. The values for Vs are determined by the settings on the signal generator. We can calculate the values of B and θ using the voltage divider rule, and the phasor representation. In the figure the output leads, or is ahead of the input, by θ.

Figure 2 - A Voltage Divider

As you learned in class, a phasor has a magnitude and angle. The magnitude is the value from the center line, to the peak voltage. The phase shift is the angular displacement from the reference wave. The voltages in Figure 1 can be rewritten using phasors as shown in equations 4 and 5.

Figure 2 - Phasor Representation of Voltages

Notice that the frequency ω is not included in equations 4 and 5. But, it does influence the impedance values in the example in Figure 3. These values can be used to calculate the output voltage based on the input voltage from the signal generator. The example substitutes the impedance values into the voltage divider equation, then assumed values are used to solve for the output, VL. The resulting complex number can be converted to polar notation, to give the angle and phase shift for the output.

Figure 3 - Phasor Analysis of Circuit

INTRODUCTION

In most electric circuit applications, the circuit has been connected to a sinusoidal voltage source for a relatively long time (for example, appliances plugged into the 60 Hz power distribution system). This implies that for any response in the circuit (voltage or current), the transient part of the total response has died away and only the steady-state part remains. If the forcing function is a sinusoidal voltage at frequency, f, every response in the circuit is also sinusoidal at frequency f. Analysis of an LCR circuit with a sinusoidal forcing function in the steady-state must be concerned with the time delay or phase as well as the amplitude of the response at every point in the circuit.

In this lab techniques for calculating and measuring the phase and amplitude of the sinusoidal steady-state response will be investigated. Since the frequency is the same throughout the circuit, the approach used is to disregard the function part (i.e., the sinusoidal part) and to use complex reactive or phasor analysis to keep track of time delays and amplitudes throughout the circuit. The first circuit analyzed in this way will be the series LR circuit of Figure 1.

Figure 1. A series LR circuit.

In this circuit current through the inductor must lag behind the voltage across it by 90. Voltage and current are in phase through the resistor and the voltage drops across the resistor and inductor must add to equal the applied voltage. This means that the voltage drops across the inductor and resistor are 90° out of phase and the voltage drop across the resistor must lag behind the voltage drop across the inductor. These phase relationships are illustrated in Figure 2.

Figure 2. Resistor voltage lags the inductor voltage in a series LR circuit.

Let the current through the series LR circuit be I. If the input voltage source, VS, is taken as phase reference, then I is

and its magnitude is

at a phase angle Θ = -tan-1(XL/RT)

where RT = 50 + RL + R.

For L = 0.4H, RT = 2200 Ω, VS = 10 VP-P and f = 1000 Hz, the inductor's reactance is XL = 2π(1000)(0.4) = 2513.3 Ω, so that the magnitude and phase of the current are,

This means that the voltage across R is VR = R x (3.0 mA) and its phase is the same as that of I.

Also, the voltage across the inductor is VL = (2513.3 Ω) x (3.0 mA) = 7.54 V and its phase is 90° greater than that of I.

If the inductor is replaced with a capacitor, the analysis is the same except that current through the capacitor now leads the voltage across it. This means that with the voltage source taken as reference (θ = 0), the current will have a positive rather than negative phase angle.

and

at phase angle Θ = -tan-1(XC/RT)

PROCEDURE

1. Measure the series inductance on the LCR meter and adjust the inductor for a value of L = 0.4 H. Read the value of Q at f = 1 kHz and calculate RL from the formula
RL = 2πfL/Q
Measure and record the exact value of the capacitor.
2. Analyze the series circuit of Figure 1 and determine the value of R required to have its voltage lag behind the source voltage by 450 with a source voltage of 10 VP-P at 1 kHz.
3. Set up the circuit of Figure 1 and drive it with a 10 VP-P sine wave at 1 kHz. Display the FG output on Channel 1 of the SCOPE and trigger from Channel 1. Connect Channel 2 to the FG output and adjust both displays until the sine wave amplitude extends from top to bottom of the display. Make sure both channels are matched by inverting Channel 2 and displaying the difference of the two channels. If any signal remains, then readjust the amplitude and phase of the probe to one of the channels until they are matched.
4. Measure the phase of the output sine wave relative to the input sine wave. Put the scope in X-Y mode and observe a diagonal trace extending from the upper right corner to the lower left corner of the display. This represents a Lissajous pattern for two sine waves that have zero phase difference between them. Refer to Figure 3a. Two sine waves that have a 90° phase difference between them will produce a circular Lissajous pattern. For any other angle between 0° and 90°, the Lissajous pattern is an ellipse. The Lissajous pattern on the SCOPE is used to measure the phase difference between two sine waves.

Figure 3. Lissajous pattern for two sine waves that have (a) 0° phase difference between them, and (b) a 90° phase difference.

From Figure 4 observe that if the signals to Channels 1 and 2 of the SCOPE are matched in amplitude, then the ratio of the vertical intercept of the elliptical Lissajous pattern to the vertical scale is the sine of the phase angle, sin(θ). Let the peak display be A and the vertical intercept be B. Then, sin(θ) = B/A.

Figure 4. Phase determination from an elliptical Lissajous pattern.

Another technique for measuring phase that is good for small angles is to measure the amplitude difference between two equal amplitude sine waves. That is

Bsin(ωt + φ) = Asin(ωt) - Asin(ωt + θ) (1)
Bsin(ωt)cos(φ) + Bcos(ωt)sin(φ) = Asin(ωt) - Asin(ωt)cos(θ)- Acos(ωt)sin(θ) (2)
Bsin(ωt)cos(φ) + Bcos(ωt)sin(φ) = A(1 - cos(θ))sin(ωt) - Acos(ωt)sin(θ) (3)
So that, Bcos(φ) = A(1 - cos(θ)) and Bsin(φ) = -Asin(θ)
B2 = A2(1 - cos(θ))2 + A2sin2(φ)
(B/A)2 = 2(1 - cos(θ)) or cos(θ)= 1 - (B/A)2/2

However, cos(θ) can be expanded in a power series as: cos(θ) = 1 - θ2/2 + θ4/24 - Therefore for θ small, 1 - θ2/2 ≈ 1 - (B/A)2/2 or θ ≈ (B/A) where B is the amplitude of the difference of two equal amplitude sine waves with amplitude, A and θ is in radians.
5. Connect the SCOPE so that the voltage from the source drives the x input while the voltage across the variable resistor drives the y input.
6. Adjust the variable resistor until the Lissajous pattern measurement of phase for the variable resistor measures 45°. As the variable resistance is changed, its voltage drop will also change. Keep adjusting the SCOPE's variable attenuator so that the x and y amplitudes remain equal. This is essential for correct angle measurements.
7. Remove the variable resistor from the circuit. Measure the resistance of the variable resistor and compare its value to the value calculated in step 2.
8. Replace the inductor with the 0.068 μF capacitor and repeat steps 2 through 7. Since RL for a capacitor is zero, so there is no need to do step 1.
9. Pick other values of phase angle and/or values of resistance and repeat steps 1 through 7. Record your results in the log book.

INTRODUCTION

The descriptor rms (short for root-mean-square) is a designation that is used to describe power in time varying electrical systems. It's a way of defining the amplitude of a time varying voltage or current so that the same power is delivered as an non-time varying voltage or current.

The rms amplitude can be found by determining the average power delivered to a 1 Ω resistor. We know that the average power delivered to the 1 Ω resister is

This type of average is called a root-mean-square average or rms average.

For a sine wave, i(t) = Im sin(ωt), the average power delivered to the 1 Ω resistor is

while for a DC signal, the average power delivered to the 1 Ω resistor is (Idc)2. Equating the AC and DC average powers, we have

(Idc)2 = (irms)2 = Im2/2.

Therefore, for a sine wave

The equivalent Idc value is called the rms value of the sine wave, and we say that the peak to rms conversion factor is one over the square root of two or 0.707. This implies that if a time varying voltage with an amplitude of Vm is applied to a resistance, R, then the amount of power dissipated (converted to heat) in the resistance is Vrms2/R = Vm2/2R.

There is a unique conversion factor for every voltage or current waveform. An objective of this laboratory exercise is to calculate conversion factors for waveforms produced by most function generators.

PROCEDURE

1. Design and build a voltage follower using a 741. Use the SCOPE to display its response to a square wave with zero DC offset and ≈50% duty cycle. Measure both its input and output slew rates and compare them with the manufacturer's specified values given in the laboratory.
2. Design and build a difference amplifier whose output is v0 = 0.5(v1 - v2). Test it by applying the same square wave to both inputs. Accurately measure the output, which will not be zero. There should be a small square wave with monster spikes. Explain what you see. (Explain means to tell why, not what.)
3. Design and build an analog adder which will produce the output v0 = -2v1 - 3v2. Use input and feedback resistors in the kilohm range. Predict and record the performance of your analog adder. Next, investigate what happens when the feedback and input resistors are made low (≈tens of ohms). Describe what you see when you apply a 40 mV sine wave. Now describe what happens as you gradually increase the amplitude of the sine wave. Explain.

4. INTRODUCTION

4.1 PURPOSE OF THE COURSE

4.2 RESOURCES

4.3 EVALUATION

OBJECTIVE

PROCEDURE.

OBJECTIVES

INTRODUCTION

MATERIALS

ANALYSIS PROCEDURE

VERIFICATION PROCEDURE

COMPARISON PROCEDURE

OBJECTIVES

INTRODUCTION

Voltage Divider Rule

Current Divider Rule

MATERIALS

ANALYSIS PROCEDURE

VERIFICATION PROCEDURE (Build and Measure)

COMPARISON PROCEDURE

OBJECTIVES

INTRODUCTION

MATERIALS

DESIGN AND ANALYSIS PROCEDURE

BUILD AND MEASURE PRODEDURE

COMPARISON PROCEDURE

OBJECTIVES

INTRODUCTION

MATERIALS

DESIGN AND ANALYSIS PROCEDURE

BUILD AND MEASURE PROCEDURE

COMPARISON PROCEDURE

OBJECTIVES

INTRODUCTION

MATERIALS

DESIGN AND ANALYSIS PROCEDURE

BUILD AND MEASURE PROCEDURE

COMPARISON PROCEDURE

OBJECTIVES

INTRODUCTION

MATERIALS

DESIGN AND ANALYSIS PROCEDURE

BUILD AND MEASURE PROCEDURE

COMPARISON PROCEDURE

OBJECTIVE

INTRODUCTION

MATERIALS

DESIGN AND ANALSYS PROCEDURE

BUILD AND MEASURE PROCEDURE

COMPARISON PROCEDURE

OBJECTIVES

INTRODUCTION

MATERIALS

DESIGN AND ANALSYS PROCEDURE

BUILD AND MEASURE PROCEDURE

COMPARISON PROCEDURE

OBJECTIVE

INTRODUCTION

MATERIALS

DESIGN AND ANALSYS PROCEDURE

BUILD AND MEASURE PROCEDURE

COMPARISON PROCEDURE

OBJECTIVES

INTRODUCTION

MATERIALS

PROCEDURE

OBJECTIVES

INTRODUCTION

Background Reading for Interest

The Experiment

MATERIALS

DESIGN AND ANALYSIS PROCEDURE

BUILD AND MEASURE PROCEDURE

COMPARISON PROCEDURE

APPENDIX A - PRACTICAL INDUCTOR ISSUES

OBJECTIVES

INTRODUCTION

MATERIALS

DESIGN AND ANALYSIS PROCEDURE

BUILD AND MEASURE PROCEDURE