List of Tables List of Figures 1

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Contents

List of Tables iii

List of Figures iv

1 Introduction to Basic Laboratory Test and Measurement Equipment

Part A 2

1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The DC Power Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The Digital Multimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 The Function Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Intro duction to Basic Lab oratory Test and Measurement Equipment - Part B 4

2.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The Oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Theory of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Oscilloscope Blo ck Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Introduction to the experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Resistors, Potentiometers, and Rheostats 8

3.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.1 Resistance Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Potentiometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3.1 Potentiometer Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Rheostats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4.1 Rheostat Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 DC Circuit Measurements - Part A 124.0.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.0.3 Series Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.1 Parallel Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Series-Parallel Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 DC Circuit Measurements - Part B 14

5.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Current-Limited Power Supply

I-V Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 Circuit Loading by Measuring Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5.3.1 Ammeter Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.3.2 Voltmeter Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3.3 Oscilloscope Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3.4 Function Generator Equivalent Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 DC Circuit Analysis 17

6.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.2 Mesh and Nodal Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.3 Superposition Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.4 Thevenin Equivalent and Maximum Power Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.4.1 Thevenin Equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.4.2 Maximum Power Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.5 Source Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18


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7 Inductance, Capacitance I-V Relations and Transients in RL and RC Circuits 20

7.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.2 Inductance and Capacitance: Voltage-Current Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.2.1 Inductor Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7.2.2 Capacitor Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.2.3 RL and RC Circuit Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.3 RL-Circuit Transient Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.4 RC-Circuit Transient Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8 Transients in RLC Circuits 25

8.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.2 RLC Circuit Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8.2.1 The Underdamped Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.2.2 The Critically-Damped Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2.3 The Overdamped Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

9 Sinusoidal AC Circuit Measurements 29

9.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.2 Phase-Angle Measurements and Average Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9.2.1 Time-Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.2.2 Ellipse Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9.3 Current and Voltage Phasor Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.4 Thevenin Equivalent and Maximum Power Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

10 Series and Parallel Resonance 34

10.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.2 Series Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.3 Parallel Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

11 Filter Frequency Response and Bode Diagrams 37

11.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.2 Frequency Response and Bode Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.3 Lowpass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.4 Highpass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.5 Bandpass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.6 Bandstop Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


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List of Figures

1 Oscilloscope Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Sawtooth Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Axial-Lead Resistor, Color-Coded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Voltage and Current Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Bridge Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Potentiometer Schematic Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Circular and Straight line Potentiometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Potentiometer Voltage Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Rheostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110 Series Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1211 Parallel Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312 Series Parallel Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313 DC Mesh and Nodal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714 Source Transformation Figure 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815 Source Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916 Source Transformation Figure 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1917 Circuit Models for Practical Inductors and Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018 Inductor i-v Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2119 Capacitor i-v Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2120 RL Circuit and Transient Responses. Left (a), Upper (b), Lower (c) . . . . . . . . . . . . . . . . . . . 2221 RC Circuit and Transient Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2322 RLC Circuit (a) and Transient Responses: (b) Underdamped, (c) Critically-Damped, (d) Overdamped 2623 Time-Difference Method for Phase-Angle Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 2924 Ellipse Method for Phase-Angle Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3125 Different Shapes of the ellipse for different phase differences . . . . . . . . . . . . . . . . . . . . . . . . 3126 Circuit for Measuring Phase Angle between Voltage Across and Currents through RL and RC Impedances 3227 Phasor Measurement-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

28 Thevenin Equivalent and Maximum Power Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3329 RLC Circuit for Series Resonance Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3430 RLC Circuit for Parallel Resonance Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3631 Lowpass-Filter Bode Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3832 Lowpass RLC Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3933 Highpass RLC Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4034 Bandpass RLC Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4035 Bandstop RLC Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


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Jordan University of Science and TechnologyEE316

Electric Circuits Laboratory

Dr. Ihsan Babaa

1992

Revised 2006

1


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Part A

1.1 Objective

This experiment is intended to give the student a quick exposure to the laboratory equipment which will be used inthis course: this part concerns the dc power supply, the function generator (FG), and the digital multimeter (DMM).

1.2 The DC Power Supply

Generally, this is a dual power supply with (+) and (-) voltage terminals, and a ground (common) terminal. Panelmeters are usually provided for voltage and current reading. There are also controls for Coarse and Fine adjust-ments of output voltage, a control to adjust the output current limit, a switch to select the voltage-reading meteror the current-reading meter, and an Output On/Off switch. There may also be indicator lights for Input On andCurrent Limit.

Before performing the following procedure, make a neat sketch of the power-supply front panel; mark the variouscontrols and meters.

Apply input power; turn the Current Limit control to maximum (clockwise); turn the Coarse and Fine voltagecontrols to maximum (clockwise); select the voltage meter, and record the indicated value.

Turn both Coarse and Fine voltage controls to minimum; select the current meter; place a short circuit betweenthe (+) and (-) output terminals; turn the Fine control slowly up until maximum current is indicated; recordthis value. Does this value change when the voltage controls are turned up any further? Is the Current Limitindicator light on?

Select the voltage meter and record its reading. Select the current meter; gradually turn the Current Limit Control down (counterclockwise), and record what

you observe.

1.3 The Digital Multimeter

Most digital multimeters are designed to measure DC resistance, direct current and voltage, and the RMS value ofsinusoidal current and voltage. Some meters measure the true RMS (TRMS) value of any waveform. In the followingprocedure, use the TRMS DMM for voltage and current measurements.

Before performing the following procedures, make a neat sketch of the DMM front panel; mark the various controls,switches, etc.

(A) Resistance Measurements

Obtain a resistance decade box; prepare the DMM for resistance () measurement; connect the DMMprobes to the box and set all box switches to zero; select the DMM Auto range and record its reading;repeat with the smallest range setting.

Select one arbitrary value of resistance in each of the following box ranges; record its marked value and itsmeasured value using all possible ranges including Auto.

1 10 1 10k 100 500k 2 5M(B) Direct-Current Measurements

Set the Coarse and Fine voltage controls on the power supply to minimum; select the panel current meter;turn the Current Limit control to maximum; place a short circuit between the (+) and (-) terminals;slowly move the Fine voltage control to maximum; adjust the Current Limit control for a reading of 0.5 A;disconnect the short circuit.

Prepare the DMM for current measurement at the highest range (2 A), and connect its probes where the

short circuit was. Record its reading. Without looking at the DMM, adjust the Current Limit control for a reading of 1.0 A on the panel meter;

record the DMM reading. Repeat for 1.5 A. Disconnect the DMM from the power supply.


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(C) Alternating-Current Measurements

This will be done in future experiments.

(D) Direct-Voltage Measurements

Adjust the power supply voltage until its panel meter reads 10 volts. Now use the DMM to measure voltageas follows: (+) to (-), (+) to Gnd, and (-) to Gnd.

Connect the (-) and Gnd terminals of the power supply; measure the voltage between Gnd and the (+)terminal with the DMM. Similarly, connect the (+) and Gnd terminals and measure the voltage from Gndto (-).

Remove all connections from the power supply. Adjust its voltage for a panel-meter reading of 25 V.Measure the (-) to (+) output voltage with the DMM.

(E) Alternating-Voltage Measurements

This is done in conjunction with the FG in the coming sections.

1.4 The Function Generator

A FG provides voltages of different forms. These may include: sinusoidal, triangular, square, and ramp. An adjustablelevel of DC offset (+ or -) may also be available. In addition, a control may be present to vary the waveform symmetry.Output-voltage frequency and amplitude may have a wide dynamic range, e.g., 108 to 1 for frequency.

Before performing the following procedure, make a neat sketch of the FG front panel; mark the various controlsand adjustments.

Turn the power on; turn DC Offset to Off, Symmetry to Normal, and set the frequency to 1000 Hz. Set the function selector to sinusoidal output, and the amplitude to maximum value. Measure the RMS value

of the output with the DMM.

Set the output amplitude to minimum; again measure the output with the DMM. Calculate the dynamic rangeas a simple ratio: VmaxVmin , and in dB: 20 log10(

VmaxVmin

).

Return to maximum-amplitude setting; increase the frequency to 10 kHz, and record the DMM reading. Repeatwith f=100 kHz, and f = 1 MHz.

Other measurements and observations using different waveforms will be made next in conjunction with the Scope.

Write your report in a concise form using clear and short forms for data presentation tables and graphs.Discuss the not-so-obvious observations. Some results are related to basic laws or formulas you learned inprevious courses. Clarify this and discuss where appropriate. In short, write an intelligent report .. now and inthe future.


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Figure 1: Oscilloscope Block Diagram


- Part B

2.1 Objective

This experiment is intended to give the student a quick exposure to the laboratory equipment which will be used inthis course: this part concerns the Oscilloscope (Scope).

2.2 The Oscilloscope

2.2.1 Theory of Operation

As shown in Figure 1 the cathode ray is a beam of electrons which are emitted by the heated cathode (negativeelectrode) and accelerated toward the fluorescent screen. The assembly of the cathode, intensity grid, focus grid,

and accelerating anode (positive electrode) is called an electron gun. Its purpose is to generate the electron beamand control its intensity and focus. Between the electron gun and the fluorescent screen are two pair of metal plates- one oriented to provide horizontal deflection of the beam and one pair oriented to give vertical deflection to thebeam. These plates are thus referred to as the horizontal and vertical deflection plates. The combination of these twodeflections allows the beam to reach any portion of the fluorescent screen. Wherever the electron beam hits the screen,the phosphor is excited and light is emitted from that point. This conversion of electron energy into light allows us towrite with points or lines of light on an otherwise darkened screen.

In the most common use of the oscilloscope the signal to be studied is first amplified and then applied to thevertical (deflection) plates to deflect the beam vertically and at the same time a voltage that increases linearly withtime is applied to the horizontal (deflection) plates thus causing the beam to be deflected horizontally at a uniform(constant) rate. The signal applied to the vertical plates is thus displayed on the screen as a function of time. Thehorizontal axis serves as a uniform time scale.

The linear deflection or sweep of the beam horizontally is accomplished by use of a sweep generator that is

incorporated in the oscilloscope circuitry. The voltage output of such a generator is that of a sawtooth wave as shownin Figure 2. Application of one cycle of this voltage difference, which increases linearly with time, to the horizontalplates causes the beam to be deflected linearly with time across the tube face. When the voltage suddenly falls to


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zero, as at points at the end of each sweep - the beam flies back to its initial position. The horizontal deflection of thebeam is repeated periodically, the frequency of this periodicity is adjustable by external controls.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Sawtooth waveform

Time

Voltage

Figure 2: Sawtooth Waveform

To obtain steady traces on the tube face, an internal number of cycles of the unknown signal that is appliedto the vertical plates must be associated with each cycle of the sweep generator. Thus, with such a matching ofsynchronization of the two deflections, the pattern on the tube face repeats itself and hence appears to remainstationary. The persistence of vision in the human eye and of the glow of the fluorescent screen aids in producing astationary pattern. In addition, the electron beam is cut off (blanked) during fly back so that the retrace sweep is notobserved.

CRO Operation: A simplified block diagram of a typical oscilloscope is shown in Fig. 3. In general, the instrumentis operated in the following manner. The signal to be displayed is amplified by the vertical amplifier and applied tothe vertical deflection plates of the CRT. A portion of the signal in the vertical amplifier is applied to the sweep triggeras a triggering signal. The sweep trigger then generates a pulse coincident with a selected point in the cycle of thetriggering signal. This pulse turns on the sweep generator, initiating the sawtooth wave form. The sawtooth waveis amplified by the horizontal amplifier and applied to the horizontal deflection plates. Usually, additional provisionssignal are made for applying an external triggering signal or utilizing the 60 Hz line for triggering. Also the sweepgenerator may be bypassed and an external signal applied directly to the horizontal amplifier.

CRO ControlsThe controls available on most oscilloscopes provide a wide range of operating conditions and thus make the

instrument especially versatile. Since many of these controls are common to most oscilloscopes a brief description ofthem follows.

2.3 Oscilloscope Block Diagram

The heart of the oscilloscope is the cathode ray tube, which generates the electron beam, accelerates the beam toa high velocity, deflects the beam to create the image, and contains the phosphor screen where the electron beameventually becomes visible. To accomplish these tasks, various electrical signals and voltages are required, and theserequirements dictate the remainder of the blocks of the oscilloscope outline as shown in Figure 1. The power supplyblock provides the voltages required by the cathode ray tube to generate and accelerate the electron beam, as wellas to supply the required operating voltages for the other circuits of the oscilloscope. Relatively high voltages arerequired by cathode ray tubes, on the order of a few thousand volts, for acceleration, as well as a low voltage for theheater of the electron gun, which emits the electrons. Supply voltages for the other circuits are various values, usuallynot more than a few hundred volts. The laboratory oscilloscope has a time base which generates the correct voltageto supply the cathode ray tube to deflect the spot at a constant time-dependent rate. The signal to be viewed is fed toa vertical amplifier, which increases the potential of the input signal to a level that will provide a usable deflection ofthe electron beam. To synchronize the horizontal deflection with the vertical input, such that the horizontal deflectionstarts at the same point of the input vertical signal each time it sweeps, a synchronizing or triggering circuit is used.This circuit is the link between the vertical input and the horizontal time base.


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2.4 Introduction to the experiments

This is one of the most important pieces of laboratory test equipment. It is basically a voltage sensing and displaydevice; it cannot measure current directly. However, it can be used to measure a voltage proportional to a desired

current, e.g., across a small sampling resistance.Most modern Scopes have two input channels with adjustable, calibrated, gain. Two signals can thus be viewedseparately, or simultaneously if they are synchronized. Calibrated gain settings enable the measurement of voltageamplitude.

A horizontal Time axis is provided by an internal generator. This generator produces a calibrated variable-frequencyvoltage the amplitude of which varies linearly with time. Thus, a voltage waveform applied to either input channelcan be viewed as a function of time.

Instead of the internal Time generator, one input-channel signal, say Vx(t), may be used to provide the horizontaldeflection x. In this case, a signal applied to the other channel, say Vy(t) = y, may be viewed as a function of Vx(t).For example, if a sinusoidal signal is applied to both channels such that x = A sin(t) and y = B sin(t), a straightline appears, i.e., y = B

Ax.

Another important Scope function is the Trigger. Circuits in this subsection enable the selection of the amplitudeof the input signal at t = 0 relative to its peaks. This corresponds to having a selectable phase angle.

Finally, there are controls which affect the quality of the display. A Focus control adjusts the sharpness of thedisplayed image, and an Intensity control adjusts its brightness. There may also be an Illumination control whichadjusts the lighting of the axes and grid lines drawn on a glass plate in front of the screen.

Before performing the following procedure, make a neat sketch of the front panel of the Scope, and mark the basicinformation on the various controls and adjustments.

2.5 Procedure

Apply power to the Scope, and set switches and controls as follows:Time base 0.1 ms/div, Calib.Trigger Int, Auto, and Level to mid positionInput channels Vertical Sensitivity to 0.1V/div, Calib.; Mode to Ch1, Gnd coupling, Int Trig Ch1

Adjust the Vertical Position to center the beam on the screen. Adjust the Focus for a sharp line, and theIntensity to a medium level. Repeat for Ch2.

Select the sinusoidal output from the FG at 1000 Hz, and apply it to both Scope channels. - Select Scope Ch1 with AC coupling. Adjust the FG frequency dial until exactly one period appears on the

screen, and the output amplitude for a peak-to-peak deflection of exactly 4 divisions. Measure the rms value ofthe FG output with the TRMS DMM.

- Change the Vertical Sensitivity to 50 mV/div and record the peak-to-peak deflection. Repeat for 0.2 V/div. - Repeat the previous two steps with Ch2 used instead of Ch1.

- Return to 0.1 V/div; select the FG triangular output; center the 4-div peak-to-peak signal on the screen;measure its TRMS value. Repeat for a square-wave output.

- Place the time selector on the Scope in the 0.5 ms/div. Move the FG Symmetry control from Normal to twodifferent points between its extreme positions. Sketch the displayed waveform at each point, and measure itsTRMS value. Repeat for the triangular and sinusoidal waveforms.

- Return the FG Symmetry control to Normal; select the sinusoidal output, and adjust its amplitude for apeak-to-peak deflection of exactly 4 divisions with a Scope sensitivity of 1 V/div and DC coupling. Add +2V DC Offset from the FG. Record the maximum and minimum values of the displayed signal, and measure itsTRMS value. Repeat with a -2 V DC Offset. Turn the Offset control to the Off position.

Record the number of periods displayed on the Scope with time/div set to: .5, .2, .1 ms, and 50 ms.

Without changing the dial setting of the FG increase the frequency multiplier by a factor of 10, i.e., to 10 kHz.Change the Scope time/div setting to display 1, 2, 5, and 10 periods; record the values of these settings. Repeatwith the FG frequency multiplier decreased by a factor of 100, i.e., to 100 Hz.


11/44

7

Place the Scope trigger in Ext position. Can you stop the display completely from moving by adjusting theLevel control? Apply the FG output to the Trigger Input. Can you stop the display now?

Place the channel selector in Alt mode and both channel sensitivities in 0.1 V/div. Record the peak-to-peakvalues for Ch 1 and Ch2. Repeat with the selector in Chop mode. Finally, select the Add mode and record thepeak-to-peak values of the displayed signal.

Place the time/div selector in the XY position. Adjust the signal amplitude for a 4-division horizontal andvertical deflection. Change Ch1 sensitivity to 0.2 V/div, and record your observation. Return Ch1 to 0.1 V/div;set Ch2 to 0.2 V/ div and record your observation.


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8

3 Resistors, Potentiometers, and Rheostats

3.1 Objectives

1. Gain familiarity with available types of resistors, potentiometers, and rheostats.2. Determine the nominal value of resistance using the color code, and the actual value using different types of

measurement.

3. Determine the linearity of a potentiometer, and use it as a voltage divider or control element.

3.2 Resistors

As discrete components, resistors come in various sizes and shapes depending on their power rating and use. Theresistive element material may also vary, e.g., metallic wire, carbon, etc. The resistor most commonly used in thelaboratory is made of carbon encased in a tubular form with axial leads as shown in Figure 3.

n1 n2 n3 tol

boxes.m4

Figure 3: Axial-Lead Resistor, Color-Coded

Some resistors may have their nominal ohmic value stamped on the body of the resistor, e.g., 1100 or 2.2M. Moreoften, however, color code is used to indicate the nominal value. Three color bands are used for this purpose, eachhaving a numerical value between 0 and 9, as shown in Table 1.

Table 1. Numerical Values of Color Codes

Black Brown Red Orange Yellow Green Blue Violet Gray White0 1 2 3 4 5 6 7 8 9

Starting with the band closest to one end of the resistor, as shown in Figure 1, the three represented numbers, n , n, and n mean: R = (10n1+n2)10n3. For example, Orange-Blue-Black means 36100 = 36, and Gray-Red-Yellowmeans 82

104 = 820k.

The percent tolerance around the nominal value is indicated by a fourth band according to Table 2.

Table 2. Percent-Tolerance Color Code

Gold Silver No Color5 10 20

The physical size of a resistor depends on its power rating, and vice versa. To keep its temperature at a safe level,a resistor must be large enough to dissipate its rated power into the surrounding design environment.

To form an idea about size in relation to power rating, measure the dimensions of one resistor in each power rating:1/2, 1, and 2 W, and calculate the volumes and surface areas. What quantitative observations can you make?


13/44

9

VS+

aDM M1bIS +

R1+

c

R2d

DM M

exp2fig2.m4

Figure 4: Voltage and Current Measurements

3.2.1 Resistance Measurements

Several methods will be used to measure resistance. Their results will be compared with each other and with thenominal color-code value. For this purpose, obtain three resistors having arbitrary values between 100 ohms and 100kohms, and arbitrary power ratings. Tabulate their color codes, nominal values, percent tolerances, and power ratings.

1. Ohmmeter Measurements

Use the DMM to measure the value of each resistor directly, on the most sensitive range. Compare with thenominal values.

As an aside, measure and record your body resistance by holding the probes firmly, one with each hand. Recorda similar measurement made by your partner.

2. Voltage and Current Measurements

Construct a measurement circuit as shown in Figure 4, where R is the resistance to be determined using Ohmslaw:

Rx = Vx/Ix

Set each DMM to the highest possible range.

Within limits that are safe for the resistor R , increase V from zero to near the highest reasonable value. Decreasethe range setting of each DMM in steps, if necessary, to obtain a reading on the most sensitive range. Recordthe measured values of V and I . Calculate the R values by Ohms law.

For one of the resistors take three more readings at .25, .5, and .75 of the maximum V used. Plot I vs V .

3. Bridge Measurements

A Wheatstone bridge for measuring resistance is shown in Figure 5. When the bridge is balanced, i.e., Ib = 0,the following relation holds:

Rx = R2 R3/R1Derive this formula in your report.

VS

+

R1

R3

R2

Rx

DM M

Ib

exp2fig3.m4

Figure 5: Bridge Measurements

Generally, a good measurement is obtained when all resistor values are not too far from each other; for example,within a factor of 3 or less.


14/44


15/44

11

VS

+

RS

Ro

DM M

exp2fig7.m4

Figure 9: Rheostat

3.3.1 Potentiometer Measurements

Obtain one potentiometer of each type available in the laboratory. Estimate and mark the end points of the resistiveelement using light pencil marks. Divide the distance between the end points (as carefully as possible) into 4 equalparts. Again use light pencil to mark these divisions. Now perform the following measurements on each potentiometer.

1. With reference to Figure 7, use the DMM to measure resistance directly between points a and b with the movablecontact stopped at a, at c, and the three other marks in between. Do these measurements indicate that thepotentiometer is linear or nonlinear?

2. Connect the circuit shown in Figure 8 with VS = 5 volts. Measure V with the movable contact stopped at thepoints

specified in the previous part. Do these measurements indicate that the potentiometer is linear or nonlinear?

3.4 Rheostats

A rheostat is similar to a potentiometer in structure. However, it differs in its intended use. It is used as a serieselement to control current as shown in Figure 9. Thus, it is usually a higher-power device. To demonstrate its principle,one of the potentiometers you tested may be used in the following measurements.

3.4.1 Rheostat Measurements

For the circuit shown in Figure 7, use one of the potentiometers you just tested. Select Ro such that the maximumvariation in the current Io is 5 to 1. Measure and record the value of Ro. Construct the circuit using 10 V for VS .Starting at a safe DMM current range measure Io on the lowest possible range using the 4 marked sections of thepotentiometer for RS , i.e., 0, 25, 50, 75, and 100 percent.

Plot Io vs RS. What functional relation does this plot indicate?


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12

VS +

DM M1a b + R1

IS

+

c

R2

d

DM M2

+

R3

e

exp3afig1.m4

Figure 10: Series Circuits

4 DC Circuit Measurements - Part A

4.0.2 Objectives

In this part, the objective is to verify Kirchhoffs voltage and current laws and some of their consequences by mea-surements on dc circuits.

4.0.3 Series Circuits

Kirchhoffs Voltage Law (KVL) states that the sum of voltages around a closed path is zero. This can be verified bymeasurements on simple series circuits as illustrated in the following procedure.

Construct the circuit shown in Figure 10 with R1 = 330, R2 = 1k, and R3 = 2.2k. Ensure safe resistancepower ratings with VS up to 30V.Use DMM1 as an ammeter to measure the circuit current IS . Use DMM2 as a voltmeter to measure voltagessuch as Vcd across R2. Always start at the highest meter range, then gradually change to the most sensitiverange for the final measurement.

Connect DMM2 between points a and e, and adjust VS until its reading is 15 V. Record the value of IS measuredby DMM1.

Move the connection of DMM2 around the circuit to measure the voltages Vab, Vbc, Vcd, and Vde. Compare thesum of these voltages to VS . Calculate Vab as a percentage of VS .

Repeat the last two steps with VS = 27V. Disconnect the power supply from the circuit, and use DMM2 as an ohmmeter to measure the values of

R1, R2, and R3. Use these values and the measured value of IS to calculate the voltages Vbc, Vcd, and Vde byOhms law. Compare with the voltages measured previously.

A consequence of KVL is that the voltage across one resistance, Rk, in a series circuit can be calculated using

the Voltage Division Rule, viz,Vk = (Rk/Rt)Vt

where Rt is the sum of all resistances in series, including Rk , and Vt is the voltage across the circuit.

Calculate Vbc, Vcd, and Vde using this rule, and compare the results with the measured values.

4.1 Parallel Circuits

Kirchhoffs Current Law (KCL) states that the sum of all currents at any node in a circuit is zero. This can be verifiedby measurements on a simple parallel circuit as illustrated in the following procedure.

Construct the circuit shown in Figure 11 with R1 = 680 R2 = 1.2k, and R3 = 3.3k.

Adjust V until DM M1 reads 15 V. Record the value of IS as read by DM M2. Alternately place DM M2 in series with R1, R2, and R3 to measure I1, I2, and I3. Compare the sum of these

three currents with IS .


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13

VS DM M1

DM M2

IS

R1

I1

R3

I3

R2

I2

exp3afig2.m4

Figure 11: Parallel Circuits

VS

+ V1 R1

I1

+

V3

R3

I3

+V2

R2I2 +

V4

R4I4

+

V5

R5

I5

+V6

R6I6

exp3afig3.m4

0

1 2

3 4

Figure 12: Series Parallel Circuits

Repeat the last two steps with V = 27 V. Disconnect the power supply and use DM M1 as an ohmmeter to measure the parallel combination ofR1, R2, and R3.

Then disconnect and measure each resistance separately. Use the measured values of resistance and VS to cal-culate IS , I1, I2, and I3. Compare with the measured values of these currents.

A consequence of KCL is that the current through one conductance Gk(= 1/Rk) in a parallel circuit can be calculatedusing the Current Division Rule, viz,

Ik = (Gk/Gt)It

where Gt is the sum of all conductances in parallel, including Gk, and It the current into the circuit.Tasks

1. Calculate I1, I2, and I3 using this rule, and compare the results with the measured values.

4.2 Series-Parallel Circuits

Both KVL and KCL are now verified by measurements in a rather arbitrary circuit containing series and parallelcombinations of resistors.

Construct the circuit shown in Figure 12 using the same resistors you used before, whereR1 = 1.2 k R2 = 3.3 kWR3 = 330 R4 = 2.2 kR5 = 1.0 k R6 = 680

Use a DMM as a voltmeter to measure VS while you adjust its value to 25V. Then measure the voltages V1through V6 across the individual resistors as indicated.

Use a DMM as an ammeter to measure the currents I1, , I6 through the resistors.Use the measured values of voltages to verify KVL on all possible closed paths. Similarly, use the measured values

of currents to verify KCL at all nodes. Finally, use the measured values of resistances with Ohms law to calculatevoltages using measured currents, and vice versa. Compare all calculated and measured quantities.


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5 DC Circuit Measurements - Part B

5.1 Objectives

The objectives in this part of the experiment are:1. Measure the current-voltage (I-V) characteristic of a dc power supply with current limit.

2. Measure circuit loading caused by test equipment, viz, the Digital Multimeter (DMM) and the Oscilloscope(Scope).

3. Determine the output (source) resistance of the Function Generator (FG).

5.2 Current-Limited Power Supply

I-V Characteristics

A circuit used to determine the I-V characteristics of a dc power supply is shown in Figure 4. For any practical powersupply, there is a maximum value of the current IS , say Imax, that can be supplied when the output voltage VS is

set to some level Vso. As long as the current, IS, demanded by the load resistance Rl is not greater than Imax, theoutput voltage Vso remains constant. If Rl is less than the current-limiting value lmR = Vso/Imax, the supply voltageVS must decrease to Imax R1 for KVL to be satisfied.

An idealized power-supply I-V characteristic is shown by the the solid-line rectangle in Figure 5. The broken linesthrough the origin are the load lines which represent different values of the load resistance Rl. These lines intersectthe I-V characteristic at the operating points. Vso is called the open-circuit voltage and Imax the short-circuit current.

A dc power supply designed with a current limit that can be set as desired, automatically adjusts its output voltageto satisfy KVL as has been indicated. You are required to determine the laboratory power supply I-V characteristicfor two combinations of Vso and Imax using the following procedure.

Set the Coarse and Fine voltage controls and the Current Limit control on the power supply to minimum.

Construct the circuit shown in Figure 4. For Rl, select resistors having safe power ratings as dictated by the

following measurements.

With Rl = 1 and VS 2 V, carefully adjust the Current Limit control until DM M2 reads Imax = 0.6 A. Remove Rl and adjust the power supply for an open-circuit voltage Vso = 12 V. For each of the following values of Rl, record the voltage reading of DM M1 and the current reading of DM M2:

Rl = , 80, 40, 25, 20, 15, 10, 5, 0 Readjust the current limit to 0.4 A, and the open-circuit voltage to 16 V. For each of the following values of Rl, record the voltage reading of DM M1 and the current reading of DM M2:

Rl = , 160, 80, 50, 40, 30, 20, 10, 0Tasks: Plot the two I-V characteristics from these measurements.

1. On the first plot, show the load lines and the operating points for Rl = 80, 20, and 5.

2. On the second plot, show the load lines and the operating points for Rl = 80, 40, and 20.

5.3 Circuit Loading by Measuring Instruments

Ideally, a measuring instrument should have no effect on the quantity being measured. However, any practicalinstrument affects the quantity it measures to a certain degree. A voltmeter, for example, has a finite input resistance,although it may be very large. Therefore, it can change the measured circuit significantly if the equivalent circuitresistance is also very high. This is called circuit loading. Subsequent measurements are designed to demonstrate circuitloading caused by the ammeter, the voltmeter, and the oscilloscope. The equivalent resistance of each instrument will

be calculated from measurement data.


19/44

15

5.3.1 Ammeter Loading

When used as an ammeter, the equivalent resistance of the DMM may be different for each measurement range. Inthe following procedure, an evaluation of this resistance will be made for three different ranges.

Turn both power supply voltage controls to minimum. Construct the circuit shown in Figure 6. Choose R =2.2k with safe power rating at voltages up to 30V. Measure R using the DM M.

Adjust VS for a reading of 4 V on DM M1, then record the current read by DM M2 using the lowest possiblerange. Record the model numbers of both DM M1 and DM M2.

Move DM M1 to measure Va and Vr, and record the DM M2 readings in each case. Repeat the last measurements with VS = 30 V, again using the lowest possible ranges on DM M2. Repeat the measurements with VS = 30 V, but using the next higher current ranges on DM M2.

Tasks:

1. Calculate the equivalent resistance of DMM2 for each range used in the measurements, assuming the equivalentresistance of DM M1 to be infinite.

2. Calculate the percent error in circuit current, I, caused by DM M2 in each case.

5.3.2 Voltmeter Loading

Measurements are now made to determine the equivalent resistance of the DM M when used as a voltmeter, and howit affects the accuracy of experimental results.

Construct the circuit shown in Figure 7 with R1 = 470k and R2 = 1M. Measure the resistances directly withan ohmmeter.

Use a DM M to adjust VS to 30 V. Record the DM M model number, and use it to measure V2 on the lowest

possible range.

Repeat the V2 measurement with R2 = 220k and with R2 = 22k. Calculate the equivalent resistance of theDM M using the measured values of R1 and R2 in each case.

Tasks:

1. Calculate the percent errors in the measured values of V2 caused by the DM M.

5.3.3 Oscilloscope Loading

A measurement procedure is now used to determine the equivalent input resistance Rin of one of the Scope channels.The effect of this resistance on the accuracy of voltage amplitude measurements will then be evaluated.

Construct the circuit shown in Figure 8. Apply the voltage across R2 to Scope Ch 1 with DC coupling. Use afixed resistor for R1 and a decade box for R2.

Set the FG frequency to 1 kHz sinusoidal output and the open- circuit voltage (with R1 zero and R2 infinite) to8 V p-p.

With R1 = 1M, measure V2 p-p for R2 = 50k, 1 M, 2M, and 5M. Always use the highest possiblevertical sensitivity and record the settings used. Also, record the values of the resistors as measured with anohmmeter.

Tasks:

1. Calculate the equivalent Scope input resistance Rin using the measured values of R1 and R2. Also, calculate thepercent errors in the V2 measurements caused by the finite value of Rin.


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16

5.3.4 Function Generator Equivalent Resistance

Like any other practical voltage source, the FG has a nonzero equivalent resistance, Rg. Therefore, at any fixedamplitude setting, its output voltage changes with load. The extent of this change depends on the value of load

resistance relative to that of Rg. This is examined in the following procedure, and the FG source resistance isexperimentally determined.

Construct the circuit shown in Figure 9 using a decade box for Rl. Set R1 to 100 k and the FG output, Vo , to 1 V rms sinusoidal at 100 kHz. Without changing the amplitude setting, measure Vo with Rl = 10k, 1.0k, 500, 200, 75, and 50. With the same amplitude setting and Rl = 10k, measure Vo with the FG frequency changed to: 1 kHz, 10

kHz, 100 kHz, and 1 MHz. Repeat with Rl = 50.

Tasks:

1. Calculate Rg from every measurement made at 10 kHz. How does the output amplitude vary with frequency for

Rl = 10k and Rl = 50.


21/44

17

vb1

I2

16V

R1

3.3 Il

Rl

+

vl

R2

0.27

R4

0.68

vb224V

I3

I1

R3

1

meshandnodal.m4

1

a b

2

0 (Ref)

Figure 13: DC Mesh and Nodal Analysis

6 DC Circuit Analysis

6.1 Objectives

1. Verify the Mesh Analysis and the Nodal Analysis methods.

2. Verify the Superposition Principle.

3. Verify Thevenins and Maximum Power Transfer theorems.

4. Verify voltage-current Source Transformations.

6.2 Mesh and Nodal Analyses

Mesh and Nodal equations are verified using experimental data. Figure 13 shows the circuit used for this purpose,where the indicated resistances are in k.

Before coming to the laboratory, the student is required to write these equations using Rl = 150. The equationsshould be solved for the mesh currents I1, I2 , and I3 and the node voltages Va and Vb indicated in the figure.Measurements are made according to the following procedure:

Measure the actual resistance values used with the digital multimeter (DMM). Use a nominal 150- resistor forRl.

Adjust the two outputs of the dual power supply to 16 V and 24 V using a DMM. Measure the currents I1, I2 , and I3 using a DMM on the lowest possible range. Similarly, use another DMM to

measure the node voltages Va and Vb.

Tasks:

1. Compare the measured currents and voltages with the calculated nominal values.

2. Substitute the measured values of resistances, source voltages, and mesh currents into the mesh equations.Similarly, substitute the measured values of resistances and voltages into the nodal equations. Group all termsin every equation on one side and compare their sum with zero. Explain any discrepancies.

6.3 Superposition Principle

The circuit of Figure 13 is also used to verify the superposition principle using the following procedure:

Replace the 24-V source with a short circuit, but leave the 16-V source applied. Measure the mesh currents andthe node voltages; denote these by I1, I

2, I

3, V

a, and V

b .

Repeat the previous step with the 24-V source reapplied but the 16-V source replaced with a short circuit. Denotethese measurements by I1, I2, I3, Va, and Vb. Compare the sum of each two measurement components withthe corresponding total quantity measured, i.e., (I1 + I1) with I1 , (V

a + Va) with Va , etc.


22/44

18

Vs1

+

I1

R1

Io+

Vo

Ro

R2

Vs2

I2

exp4fig2.m4

Figure 14: Source Transformation Figure 2

6.4 Thevenin Equivalent and Maximum Power Transfer

6.4.1 Thevenin Equivalent

Again, the circuit of Figure 14 will be used to verify Thevenins and the Maximum Power Transfer theorems experi-mentally.

The Thevenin equivalent circuit wanted is that seen by the load resistance Rl. Different methods will be used to

determine this circuit, as follows:

With the 16-V and the 24-V sources applied, remove Rl and measure the open-circuit voltage Vao (o.c.). This isthe equivalent Thevenin voltage VTh.

Measure the short-circuit current Iao (s.c.). Replace both voltage sources with short circuits, and measure the Thevenin equivalent resistance, RTh, between

node a and the reference node 0 with an ohmmeter.

Calculate the Thevenin equivalent circuit using the measured values of the four resistors in Figure ?? and the16-V and 24-V sources.

Determine an experimental value for RTh as Vao (o.c.)/Iao (s.c.). Compare the measured and calculated values

of VTh. Also, compare the two experimental values of RTh with each other and with the calculated value.

6.4.2 Maximum Power Transfer

The value of Rl that will receive maximum power is determined experimentally as follows:

Use a decade box for Rl in Figure ??. Measure the voltage Vl across Rl using a DMM on the lowest possiblerange for Rl = 200, 300, 400, 500, 600, 800, 1000,and1500.

Calculate the power Pl = V2l /Rl. Plot Pl and Vl vs Rl. From the plot determine the value, Rmp, ofRl where Plis maximum, then find the corresponding value, Vmp, of Vl. Compare Rmp with RTh and Vmp with VTh/2.

6.5 Source Transformations

Although we do not have ideal sources, the availability of a power supply with adjustable current limit enables thesimulation of source transformations to some degree. For this purpose, the dual dc power supply will be used totransform the circuit shown in Figure 14. The procedure is as follows:

Set the short-circuit current limit on each supply to about 200 mA. Then set the open-circuit voltages Vs1 = 20V and Vs2 = 10 V.

Construct the circuit of Figure 14 using 2-Watt resistors R1 = 330 and R2 = 100. Use a decade box for Ro. Use two DMMs to measure Vo and Io for the following values ofRo : Ro = 0, 20, 50, 100, 200, 500, 1000, and 5000. Disconnect R1 and R2 and measure their values with an ohmmeter. Use these values to calculate equivalent

current sources Is1 = Vs1/R1 and Is2 = Vs2/R2.

Construct the equivalent (transformed) circuit shown in Figure 15 where Ise = Is1 + Is2. For Ise, use a short-circuit-current limited power supply. Set the open-circuit voltage of the supply to a value slightly above, say10% above, the value Ise R1.R2R1+R2 .


23/44

19

Ise Ro

+

Vo

Io

R2R1

exp4fig3.m4

Figure 15: Source Transformation

Vse

R1

R2Ro

+

Vo

exp4fig4.m4

Figure 16: Source Transformation Figure 4

Measure Vo and Io in this circuit for the same values of Ro used in the previous test. Finally, construct the equivalent (transformed) circuit with one voltage source as shown in Figure 16. open-circuit

output voltage set to the value Ise R1.R2R1+R2 , exactly. Measure Vo and Io in this circuit also for the same values of Ro used previously.Compare the results of the three sets of measurements made, and explain any discrepancies.


24/44

20

vt

+

iL

L

+

vL

RL

+vRL

exp5afig1.m4

vC

+

it

C

iC

RC

iRC

Figure 17: Circuit Models for Practical Inductors and Capacitors

7 Inductance, Capacitance I-V Relations and Transients in RL and RC

Circuits

7.1 Objectives

1. Measurement verification of current-voltage (i-v) relations for inductance and capacitance

2. Measurement verification of RL and RC circuit time constant.

7.2 Inductance and Capacitance: Voltage-Current Relations

Ideal inductors and capacitors can store energy, but their average power loss is zero. Practical components, however,lose a finite amount of energy. Therefore, in addition to inductance and capacitance, their electrical circuit modelsinclude resistance as shown in Figure 17.

From the figure,

vt = vL + vRL = LdiLdt

+ RLiL (1)

it = iC + iRC = CdvCdt

+1

RCvC (2)

For high quality components, RL is relatively small and RC is relatively large. Thus, ifdiLdt and

dvCdt are large

enough, then vRL


25/44

21

+Vs SquareWave

Rg

L RL

Rs

iL

FG

InductorBox

v1 to CH1 v2 to CH2

exp5afig2.m4

Figure 18: Inductor i-v Measurements

+Vs TriangularWave

Rg C

RC

Rs

iC

FG CapacitorBox

v1 to CH1 v2 to CH2

exp5afig3.m4

Figure 19: Capacitor i-v Measurements

Display the Function Generator (FG) output voltage, v1, and the voltage v2 across Rs together. Make anaccurate sketch of both signals showing values of time and amplitude.

Calculate (L/RL) and compare with (T /2), where T is the period of the input vs(t). Calculate an approximate expression for iL(t) = (1/L)

vLdt. Use the 400-mH nominal value of L and vL(t)

vs(t). Compare with the measured iL(t).

Use vL(t) vs(t) and diL/dt from measurements in the expression vL(t) = LdiL/dt to calculate an approximatevalue for L. Compare with the nominal value of 400 mH.

7.2.2 Capacitor Test

Obtain a capacitance decade box, and use a DMM to measure the dc resistance, RC , at the 0.02F setting.

Construct the circuit shown in Figure 19, where v is an 8-V p-p, 200-Hz, triangular wave, and Rs = 500. Display the FG output v1 and the voltage v2 across the sampling resistor Rs together, using dc coupling on both

scope channels. Sketch v1 and v2 accurately showing time and amplitude values. Calculate RCC and comparewith T /2, where T is the period of the input vs(t).

Calculate an approximate expression for iC(t) = CdvC/dt. Use the 0.02F nominal value for C and dvC/dt dvs/dt. Compare with the measured iC(t).In the expression vC(t) = (1/C)

iC(t)dt, use vC(t) vs(t) and the measured iC(t) to calculate an approximate

value for C. Compare with the nominal value of 0.02F.


26/44

22

+ Vmu(t)

R

+ VR

L

+

vL

iL

(a)

0 1 2 3 4 5 62

1

0

1

2x 10

3

t in multiples of

i L(t)

0 1 2 3 4 5 60

1

2

3

4

t in multiples of

vL(

t)

0.63

Io

VmIo R0.63(VmIoR)

Figure 20: RL Circuit and Transient Responses. Left (a), Upper (b), Lower (c)

7.2.3 RL and RC Circuit Transients

A series RL circuit with a step input voltage is shown in Figure 20(a). For an initial current iL(0) = Io, which maybe positive or negative, the inductor current and voltage transient responses for t 0 are given by

iL(t) =VmR

VmR

Io

et/ (5)

andvL(t) = (Vm IoR) et/ (6)

where = L/R (7)

is the circuit time constant. Figures 20(b) and 20(c) depict the responses given by equations (5) and (6) withVm > 0 and Io < 0.

A basic feature of the exponential function having the general form

y(t) = yf [yf yi]et/ (8)where yf is the final value of y and yi is its initial value, is that can be calculated using any two points, y1 and

y2, corresponding to t1 and t2, respectively, viz,

=t2 t1

ln(yf y1) ln(yf y2) (9)

It is noted that y y(t 5).Derive equation (9) and include the derivation in your report.For the special case where (t2 t1) = , equation (9) yields:

y2 y1 = (1 e1

)(yf y1) = 0.632(yf y1) (10)That is, about 63% of the change from y1 to yf occurs in one time constant. Likewise, one can show that 99.3%

of this change occurs in five time constants.


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24

Readjust the time scale to display vL(t) over a time span of 3 to 4 time constants on a full screen. Measure vL(t)at t = 2 and 3. Compare the results with theoretical calculations.

Exchange the positions of R and L in the circuit to enable the display of vR using the common ground betweenthe oscilloscope and the FG. Since R is known, this measurement yields iL, viz, iL = vR/R.In your report, draw [vL(t) + vR(t)] from your accurate sketches of vL(t) and vR(t). Compare this with the inputvoltage.

7.4 RC-Circuit Transient Tests

These tests are similar to those performed on the RL circuit.

For the RC circuit shown in Figure 21(a), replace the step input with the same square-wave used on the RLcircuit. Select R = 100k and C = 10nF. Measure the actual value of R with an ohmmeter and calculate atheoretical value of the time constant = RC.

Make the measurements needed to determine as you did in the RL circuit but use only the vC(t) transient. Measure vC(t) at t = 2 and 3 as you did before on vL(t). Make an accurate sketch of vC(t) during the positive half period only. Record several values of time and the

corresponding voltage levels.

Exchange the positions of R and C, and make an accurate sketch of vR(t) = RiC(t) as you did for vC(t).In your report, analyze the data for the RC circuit as you do for the RL circuit. Again, draw [vC(t) + vR(t)] fromyour accurate sketches, and compare the result with the input voltage.


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8 Transients in RLC Circuits

8.1 Objective

Measurement verification of transient parameters, in RLC circuits, viz, damping factor, and natural frequency.

8.2 RLC Circuit Transients

A series RLC circuit is shown in Figure 22(a) with a step input voltage. This circuit exhibits three types of transientresponses. These are determined by the roots of the characteristic equation:

s2 +R

Ls +

1

LC= 0 (14)

viz,s1,2 =

2 2o (15)

The damping factor, , and the resonant frequency, o, are given by

=R

2L (16)

and

o =1LC

(17)

If o > , the two roots s1 and s2 given by equation (15) are complex conjugate, and the response is an exponentiallydecaying sinusoidal oscillation. Such a response is said to be underdamped. Figure 22(b) shows the current responsewith initial conditions vC(0) = vC0 and i(0) = 0.

If o = , the two roots are real and equal, and the current response in this case is an exponential pulse as shownin Figure 22(c). This response is said to be critically damped, and settles toward its final value considerably fasterthan the underdamped response.

If o < , the two roots are real and unequal, and the current response is also an exponential pulse as shown inFigure 22(d). However, it settles toward its final value more slowly than the critically damped response, and is said

to be overdamped.These three cases will be discussed separately, and a test procedure will be specified for each. The values of L and

C willbe held constant, and the three cases will be generated by adjusting the value of R.

8.2.1 The Underdamped Case

With assumed initial conditions vC(0) = VC0 and i(0) = 0, the current in Figure 22(b) is given by

i(t) = Ieet sin(dt) (18)

whered =

2o 2 (19)

is the natural frequency of the system, and

Ie = (Vm VC0)/(dL) (20)is the amplitude of the exponential envelope, Ieet, shown by the dotted lines, at t = 0, and Vm is the input

square-wave peak. It is clear form equation (18) that the zero crossings of i(t) occur at multiples of T /2, whereT = 2/d is the period of oscillation. Thus, d, may be found from a measurement of the period T, i.e.,

d =2

T(21)

For small damping, i.e.,


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+ Vmu(t)

R

+ VR

L

C+

vCi

(a)

0 1 2 3 4 55

0

5x 10

4

t in multiples of T

i(t)

Underdamped

0 1 2 3 4 50

1

2x 10

4

t in multiples of T

icd(t)

Critically damped

0 1 2 3 4 50

0.5

1

x 104

t in multiples of T

iover(t)

Overdamped

Figure 22: RLC Circuit (a) and Transient Responses: (b) Underdamped, (c) Critically-Damped, (d) Overdamped

Construct the circuit of Figure 22(a) using R = 1500, L = 500mH, and C = 10nF. Instead of the step input,use a 4-V p-p, 100 Hz, square wave from the FG to create a repeating transient for viewing on the oscilloscope.

Measure the dc resistance of the inductor used, and remember to account for the 50 FG internal resistance. Display about two periods of oscillation of the voltage vR(t), which is proportional to the desired current i(t), as

shown in Figure 22(b). Adjust the vertical sensitivity to obtain the largest possible display. Make an accuratesketch of it, and record the amplitude and time values indicated in the figure as accurately as possible.

From the measured data, calculate d and using equations (21) and (22). Then use equation (19) to calculate o.Compare these experimental values with their theoretical counterparts calculated using equations (16), (17), and (19).

8.2.2 The Critically-Damped Case

Assuming the initial conditions are again vC(0) = VC0 and i(0) = 0, the exponential current pulse in this case is given

by:

i(t) = [(Vm VC0)/L]tet (23)The maximum value of this current is

Im = [2(Vm VC0)e1]/R (24)and occurs at

tm =1

(25)

The student is required to prove all of these relations, and also show that an alternative formula to equation (25)for calculating from experimental data is:

= [ln(t2/t1)]/(t2 t1) (26)where t2 and t1 are any two points with i(t2) = i(t1) = I12, as indicated in Figure 22(c).

A measurement procedure for this case is as follows:


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Use the same circuit used in the previous case with a decade box for R. Display vR(t) on the oscilloscope.Increase R gradually, and record its value where the oscillation just disappears. This is the critical-dampingresistance R. For this purpose, use the highest possible vertical sensitivity on the oscilloscope.

Read a sufficient number of amplitude and time values to make an accurate sketch of i(t) as illustrated in Figure22(c). To obtain accurate readings, position the trace on the screen and spread it vertically and horizontally asneeded. In your data, include Im, tm, and two values of time, t1 and t2, at some convenient current level, I12,near Im/2.

Interchange the physical positions of R and C in the test circuit. Display vC(t) on the oscilloscope and measureits initial value VC0.

From the measured data, determine a using equations (25) and (26). Compare the two results with each otherand with the theoretical value obtained using equation (16). Also, compare the values of Im measured directlywith the value calculated using equation (24).

8.2.3 The Overdamped Case

Once more assuming the initial conditions vC(0) = VC0 and i(0) = 0, the exponential current pulse in this case isgiven by:

i(t) = A[e1t e2t] (27)where

A = (Vm VC0)/[(2 1)L] (28)1 =

2 2o (29)

and2 = +

2 2o (30)

The maximum value, Im, of this current occurs at

tm = [ln(2/1)]/(2 1) (31)

In general, two distinct measurements of current, I1 = i(t1) and I2 = i(t2) are sufficient to determine 1 and 2.This, however, requires the numerical solution of two simultaneous transcendental equations of the form (21).

With any appreciable overdamping, 1 and 2 become widely separated, i.e., 1 2tm, the current may be wellapproximated by

i(t) Ae1t; t > 2tm (32)Now, 1 is simply expressed in terms of two experimental measurements as illustrated in Figure 22(c), viz,

1 = [ln(I1/I2]/(t2

t1) (33)

Substituting this and the previously measured value of o into equation (29) yields:

(21 + 2o)/21 (34)

Finally, using equations (29) and (30),

2 = 2 1 (35)The student is required to prove all of these relations.The following test procedure specifies a value for R that is large enough to justify the stated approximations.

Use the same test circuit used in the previous case with R = 25k. Display vR(t) on the oscilloscope, and adjustthe time base so that a full screen covers approximately from 0 to 10 tm.

Read a sufficient number of amplitude and time values for an accurate sketch of i(t). In your data, include thepoints t = tm and t = 8tm.


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Interchange the physical positions of R and C in the test circuit. Display vC(t) on the oscilloscope and measureits initial value VC0.

From your measurements, determine 1 using equation (33). Then determine using equation (34), and comparewith the theoretical value calculated from equation (16). Next determine 2 using equation (35). Calculate tmfrom equation (31) and compare with the measured value. Finally, use equations (27) and (28) to calculateIm = i(tm), and compare with the measured value.


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9 Sinusoidal AC Circuit Measurements

9.1 Objectives

1. Learn phase-angle measurement techniques using the oscilloscope, and verify the sinusoidal average power formulausing the power factor.

2. Measure voltage and current phasors in a series-parallel RLC circuit to verify Kirchhoffs Current and VoltageLaws, KCL and KVL.

3. Determine the Thevenin equivalent of an RLC circuit by open- circuit voltage and short-circuit current mea-surements.

4. Verify the Maximum Power Transfer Theorem for ac circuits.

9.2 Phase-Angle Measurements and Average Power

There are two popular methods for measuring the phase angle between two sinusoidal functions using the oscilloscope.

These are now discussed and applied to measure the phase angle between current and voltage in RL and RC seriescircuits.

9.2.1 Time-Difference Method

Here, the two sinusoids are displayed on the oscilloscope together using a common trigger signal. Figure 23 illustratesthis for the two voltages:

v1(t) = Vm1cost (36)

andv2(t) = Vm2cos(t ) (37)

0 1 2 3 4 52

1.5

1

0.5

0

0.5

1

1.5

2

Figure 1: TimeDifference Method for PhaseAngle Measurement

v1=vm1 cos(omega t)

v2=vm2 cos(omega t2pi/3)

v1(t)

v2(t)

Vm2

Vm1

t1 t2

t=t2t1

Figure 23: Time-Difference Method for Phase-Angle Measurement


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Two adjacent zeros on v1(t) and v2(t) are shown at t1 and t2 as well as the common period T = 2/. Clearly,from equations (1) and (2), t1 = (2n + 1)/2 = (t2 ), which gives:

=t2 t1

T 2 radians (38)

=t2 t1

T 360 degrees (39)

Thus, the phase angle is determined from the time difference t = (t2 t1) and the period T.

9.2.2 Ellipse Method

In this method, one of the sinusoidal functions is used to provide the oscilloscope horizontal deflection (x-axis), whilethe other is used to provide the vertical deflection (y-axis). For example,

x = Vm1 cos t (40)

y = Vm2 cos(t ) (41)

When the time parameter, t, is eliminated, it can be shown that these two equations generate the ellipse equation:x

Vm1

2+

y

Vm2

2+ 2

x

Vm1

y

Vm2

cos = sin2 (42)

For = /2, the ellipse is rotated as shown in Figure 2. For = /2, the ellipse axes are along the x and y axes.Since x = 0 when t = (2n + 1)/2, and y = 0 when t = (2n + 1)/2 + , equations (5) and (6) give the maximum

and intercept values on the x and y axes, viz,

xm = Vm1; xi = Vm1 sin (43)

andym = Vm2; yi = Vm2 sin (44)

Thus, the phase angle may be calculated from these measurements as:

= sin1(xi/xm) or = sin1(yi/ym) (45)

The following figure demonstrates the shape of the ellipse for different phase (angle) differences.These two methods are now applied to measure the phase angle between current and voltage in an RL and an RC

circuit using the following procedure:

Construct the circuit shown in Figure 3 using the nominal values R1 = R2 = 1200, L = 1H, and C = 0.5F.Measure the actual values of R1 and R2 and the dc resistance of the inductor with an ohmmeter. For vs(t) usea 400-Hz, 16-V p-p sinusoidal voltage from the function generator.

Apply the input voltage vs(t) to oscilloscope Ch1. Apply the voltage v1(t) which represents inductor currenti1(t) to Ch2. With vs(t) as reference, measure the phase angle of v1(t) using the time-difference method. Expandthe time scale as much as possible to obtain good accuracy in measuring t = (t2 t1).

Place the oscilloscope in the X-Y mode to display an ellipse. Measure maximum and intercept values along bothaxes to determine the phase angle. Make the ellipse as large as a full screen; it is not necessary to have theoscilloscope channels in the calibrated mode since a ratio of two measurements will be used.

Apply v2(t) which represents capacitor current i2 to Ch2, and repeat the two measurements.Calculate the phase angles using the measurement data from both methods. Compare the results with each other andwith the theoretical values obtained from the familiar phase-angle formula = tan1(X/R).

Calculate the average power for the RL and the RC circuits using the two equations:

Pavg =1

2VsmIm cos (46)

and

Pavg =

1

2 I2

mR cos (47)where Vsm is the amplitude of vs(t), is the power-factor angle, Im = Vm1/R1 and R = R1 for the RL circuit, and

Im = Vm2/R2 and R = R2 for the RC circuit. Compare the two results.


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31

1

1

2

1 2123

ymyi

xmxi

Figure 24: Ellipse Method for Phase-Angle Measurement

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.22

1.5

1

0.5

0

0.5

1

1.5

2

PhaseAngle Measurement, =0,30,60,90,120,180,225, and 270 degrees

0

30

60

90

120

180

225

270

Figure 25: Different Shapes of the ellipse for different phase differences


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32

+

vg

+

Rg +

vs

L

i1

R1

+

v1

C

i2

R2

+

v2

exp6fig3.m4

CH1

CH2

Figure 26: Circuit for Measuring Phase Angle between Voltage Across and Currents through RL and RC Impedances

vg

+

Rg +

vs

+ v1

R1i1 a

R2

+

v2

L

+

vL

i2

R3

+

v3

C+

vCi3

exp6fig4.m4

g2

g1

Figure 27: Phasor Measurement-1

9.3 Current and Voltage Phasor Measurements

A circuit containing R,L, and C elements in series and parallel combinations will be used to verify Kirchhoffs Currentand Voltage Laws experimentally. Amplitude and phase-angle measurements will be made to determine the phasorsneeded using the following procedure:

Construct the circuit of Figure 27 with the function generator ground connected to point g2 as shown. Thenominal component values are: R1 = 1200, R2 = 470, R3 = 3300, L = 0.4H, and C = 0.2F. Use anohmmeter to measure the actual values of R1, R2, and R3, and the dc resistance of the inductor. For vs(t) use

a 400-Hz. 16-V p-p sinusoidal voltage from the function generator. With vs(t) used as a reference, measure the amplitude and phase angle of vab(t) using the time-difference method.

Similarly, measure the amplitude and phase angle of vL(t) and of vC(t).

Exchange the positions of L and R2 and of C and R3, and measure the amplitude and phase angle of v2(t) andv3(t).

Turn the function-generator connections around so that its ground is connected to point g2 , then measure v1(t).Express the phasors corresponding to the measured voltages in rectangular form, then check voltage sums and

KVL. For example, Vab = VL + V2 = VC + V3, and V1 + V2 + VL Vs = 0.Calculate the current phasors I1 = V1/R1, I2 = V2/R2, and I3 = V3/R3. Express them in rectangular form and

check KCL, i.e., I1 I2 I3 = 0?Use graph paper and draw to scale one phasor diagram for all voltages and another diagram for all currents.Check power balance by comparing |Vs||I1| cos 1 with |I1|2R1 + |I2|2R2 + |I3|2R3.


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34

Vs = Vsm0

o

+C

+VC

L+ VL

RL

R

+

VR

Is

seriesandparallelresonance.m4

Figure 29: RLC Circuit for Series Resonance Measurements

10 Series and Parallel Resonance

10.1 Objectives

1. Measure the input impedance and current response of an RLC series circuit and the input admittance and

voltage response of an RLC parallel circuit as functions of frequency.2. Determine the resonant frequency o, the half-power frequencies 1 and 2, the bandwidth B, and the quality factor Qo

from the measured responses.

3. Measure the effect of L and C on o, and the effect of R on B and Qo.

10.2 Series Resonance

A series RLC circuit driven by a variable-frequency source is shown in Figure 29, where RL represents the dc resistanceof the practical inductor used.

The total series circuit resistance isRs = R + RL (48)

and the current transfer function is given by

Is(s)

Vs(s)=

(1/L)s

s2 + 2s + 2(49)

where

=Rs2L

(50)

is the damping coefficient and

o = 2fo =1LC

(51)

is the resonant frequency.Another damping constant, , commonly used in control systems, is related to by

= /o = (Rs/2)

C/L (52)

and is called the damping ratio.As a function of s = j , the input impedance is

Zs(j ) = Rs +j(L 1/C) (53)

In terms of the quality factor Qo, equation (6) may be put in the form:

Zs(j ) = Rs

1 +jQo[

o o

]

(54)

whereQo = oL/Rs = 1/oRsC (55)

Thus, the magnitude and phase angle of Zs(j) are given by:


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Zs = Rs

1 + Q2o(

o o

)

(56)

and

Z = tan1[Qo(

o o

)] (57)

Clearly, Zs becomes a pure resistance at = o, and its magnitude assumes the minimum value Rs. Simultaneously,the current assumes its maximum value Im = Vsm/Rs, and becomes in phase with the input voltage Vs. Also, the twovoltages VL and VC, which are 180 out of phase, become equal and assume their maximum magnitude at o. Thatis, using equation (8), oLIm = (1/oC)Im = QoVsm, which may be considerably larger than the source voltage ifQo >> 1.

Equation (7) gives two frequencies, 1 and 2 , where Z increases by

2 of its minimum value Rs, andIs decreases by

2 of its maximum value Vsm/Rs which occur at o. These two frequencies are called the

lower and upper half-power frequencies, respectively, because the power input to the circuit drops to one half of its

maximum value V2sm/Rs at o. By equating the imaginary part in equation (7) to 1, it is easily shown that

1,2 = o

1 + [

1

2Qo]2 1

2Qo

(58)

Clearly, the phase angle of Zs is 45 at 1 and 45o at 2, and the reverse is true for the current Is. Equation (11)yields the following exact relations for the resonant frequency o and the half-power bandwidth B:

o =

12 (59)

and, with equations (3) and (8),B = 2 1 = o/Qo = 2 = Rs/L (60)

i.e., o is the geometric mean of 1 and 2, and B varies inversely with Qo.For large values of Qo, equation (11) also gives the approximate

1,2 o(1 12Qo

), Qo >> 1 (61)

o (2 + 1)/2, Qo >> 1 (62)That is, o can be approximated by the arithmetic mean of 1 and 2 when Qo is large.These basic concepts and results are illustrated by the typical responses shown in Figures 2 and 3 using a logarithmic

scale for .To determine the frequency response of the RLC series circuit, measurements will be made according to the

following procedure.

Construct the circuit shown in Figure 1, with C = 0.1F and L = 0.1H. Measure the dc resistance RL

of the inductor. Use a decade substitution box for R, and set it so that R + RL = 200. For Vs, use thesinusoidal output of the function generator (FG) and set it to 5 V rms with a digital multimeter (DMM).Make sure this value is maintained when the test frequency is varied.

Apply Vs to oscilloscope Ch 1 and VR to Ch 2. Use Vs as a source of trigger and as a reference in measuringthe phase angle of VR = RIs. Use the time-difference method in phase measurements as was done in a previousexperiment. Measure the rms values of VR, VL, and VC with a DMM.

Starting at about 100 Hz, increase the FG frequency while observing the phase angle of VR. When this anglebecomes exactly zero, i.e., at resonance, measure the period of the input voltage. Note the frequency-dial settingon the FG. Ensure that Vs is still 5 V rms, and measure the other voltages with the DMM.

Increase the frequency until |VR| decreases to about 90% of its maximum value at resonance. Measure the phaseangle ofVR, the period ofVs, and the rms voltage values as before. Repeat for |VR| = 80, 70.7, 50, 30, and 10%of its value at resonance.

Repeat these measurements for frequencies below resonance.


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Vs = Vsm0o

+

RL

RL

ILC

+Vp

IC

RLCparallelresonance.m4

Figure 30: RLC Circuit for Parallel Resonance Measurements

Change the value of R so that R + RL = 1k. Set Vs to 10 V rms and repeat the above measurements.From these measurement data, determine o, 1, 2, B , Qo, , and . Compare with theoretical calculationsusing circuit-element values. Determine |VL/Vs| and |VC/Vs| at resonance and compare with Qo. Plot |Is| =|VR|/R, |Zs| = |Vs/Is|, and the phase angle of VR vs. using a logarithmic scale for .

Set R to 250 and, keeping L at 0.1H, measure the resonant frequency for C = .05, 0.2, and 0.4F. Repeat forC fixed at 0.2 F but L = 0.05, 0.2, and 0.4H.Compare these last results with theoretical predictions.

10.3 Parallel Resonance

A parallel RLC circuit with a current source is the dual of a series RLC circuit with a voltage source. Therefore, allthe formulas given previously apply to the parallel circuit provided we replace R with 1/R, L with C, and C with L.

For laboratory measurements, we take advantage of source transformation and use a voltage source in series withR as an equivalent to a current source in parallel with R. This circuit is illustrated in Figure 4.

Here, the equivalent current source is Vs/R; the voltage Vp across L and C in parallel is the dual of the current Isthrough L and C in series; the parallel admittance Yp = (Vs/R)/Vp is the dual of the series impedance Zs = Vs/Is;

and the parallel currents IL and IC are the duals of the series voltages VC and VL, respectively.It can be shown that, for the circuit of Figure 4,

o =1LC

1 R

2C

L

(63)

However, it is easy to select laboratory components such that R2LC/L < 0.001. Then RL is safely ignored ando 1LC as before. For this parallel circuit, the following test procedure will be limited to a single combination ofelement values.

Construct the circuit of Figure 4 with R = 2k, L = 80mH, and C = 80nF. Set Vs with a DMM to 10 V rms,and make sure it remains constant during all measurements. With Vs on oscilloscope Ch 1 and Vp on Ch 2,observe the phase difference between them while you increase the source frequency from 100 Hz. Resonance isreached when the phase difference becomes exactly zero. At this point measure the period of the source voltageand the rms values of Vp, IL, and IC.

Increase the frequency above resonance, and repeat the phase and current measurements where |Vp| decreasesto 90, 80, 70.7, 50, 30, and 10% of its maximum value at resonance. Decrease the frequency below resonanceand repeat.

From these measurement data, determine o, 1, 2, B , Qo, , and . Compare with the theoretical values.Also, determine |IL/(Vs/R)| and |IC/(Vs/R)| at resonance, and compare with Qo. Is Qo large enough to justifyapproximate calculations?

Plot |Vp|, |Yp| = |(Vs/R)/Vp|, and the phase angle of Vp vs. using a logarithmic scale for .


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102

101

100

101

0

1

2

3

4

5

6

, rad/sec

|H()|

= 0.1

= 1.0

102

101

100

101

200

150

100

50

0

, rad/sec

Phase

angle,

in

degrees

o=1, Normalized plots

Figure 31: Lowpass-Filter Bode Diagram

w=wr*wo;

wval(row)=wr;

Hval(row,col)=abs(Hjw(k,zeta,w,wo));

Pval(row,col)=angle(Hjw(k,zeta,w,wo))*180/pi;

end; %for wr

end %for zeta

subplot(2,1,1);semilogx(waxis,Hval); xlabel(\omega, rad/sec); ylabel(|H(\omega)|);

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