ELEC 300 LINEAR CIRCUITS IIjbornema/ELEC300/Elec300-LabManual-2013.pdfThe front page of a lab report...

LABORATORY MANUAL

ELEC 300 LINEAR CIRCUITS: II

Poman So and Adam Zielinski

Department of Electrical and Computer Engineering

University of Victoria

Revised in January 2013

Table of Contents

Laboratory and Manual Information ...................................................................................... V

I. Preparation ............................................................................................................... V

II. Equipment ............................................................................................................ V

III. Execution of Experiment ...................................................................................... V

III.1 Layout .......................................................................................................... V

III.2 Current limiting ........................................................................................... V

III.3 Troubleshooting ........................................................................................... V

IV. The Laboratory Report ........................................................................................ VI

IV.1 Appearance .............................................................................................. VIII

IV.2 Is the general appearance of your report pleasing to the reader? ............ VIII

IV.3 Presentation ............................................................................................. VIII

IV.4 Technical Contents................................................................................... VIII

IV.5 Punctuality ............................................................................................... VIII

IV.6 Remember ................................................................................................ VIII

V. Acknowledgement ............................................................................................... IX

Experiment 1 — Dependent Sources .................................................................................... 10

I. Objective ................................................................................................................. 10

II. Theory ................................................................................................................. 10

II.1 Voltage and Current Sources ....................................................................... 10

II.2 Operational Amplifier (op-amp) .................................................................. 10

II.3 Voltage-Controlled Voltage Source (VCVS) ............................................... 11

II.4 Voltage-Controlled Current Source (VCCS) ............................................... 12

II.5 Current-Controlled Voltage Source (CCVS) ............................................... 13

Introduction I

II.6 Current-Controlled Current Source (CCCS) ............................................... 14

III. Procedure ............................................................................................................. 14

III.1 Voltage-Controlled Voltage Source ............................................................ 14

III.2 Voltage-Controlled Current Source ............................................................ 15

IV. Devices Required ................................................................................................ 15

V. Preparation prior to the laboratory session .......................................................... 15

Experiment 2 — Frequency Response of Linear Systems .................................................... 20

I. Objective ................................................................................................................. 20

II. Theory ................................................................................................................. 20

III. Procedure ............................................................................................................. 23



Experiment 3 — Time-Domain Responses ........................................................................... 27

I. Objective ................................................................................................................. 27

II. Theory ................................................................................................................. 27

II.1 Active Second-Order Circuit ....................................................................... 27

II.2 Time-Domain Analysis Using the Laplace Transform ................................ 28

II.3 Impulse Response ........................................................................................ 29

II.4 Step Response .............................................................................................. 29

III. Procedure ............................................................................................................. 30



VI. Appendix — Principle of Operation of a Digital Oscilloscope .......................... 32

VI.1 Refresh Mode ............................................................................................. 32

VI.2 Role Mode .................................................................................................. 32

Introduction II

Experiment 4 — Analysis and Applications of Active Networks ......................................... 35

I. Objective ................................................................................................................. 35

II. Theory ................................................................................................................. 35

II.1 Inverting Voltage Amplifier ......................................................................... 37

II.2 Inverting Adder (Summer) ........................................................................... 37

II.3 Inverting Integrator ...................................................................................... 37

II.4 Summing Integrator ..................................................................................... 37

III. Procedure ............................................................................................................. 38



Experiment 5 — Two-Port Networks ................................................................................... 43

I. Objective ................................................................................................................. 43

II. Theory ................................................................................................................. 43

II.1 Motivation ................................................................................................... 43

II.2 Two-Port Networks ...................................................................................... 44

II.3 Impedance Parameters (z-parameters) ......................................................... 44

II.4 Admittance Parameters (y-parameters) ....................................................... 46

II.5 Transmission Parameters (T) ....................................................................... 46

II.6 Y–𝚫𝚫 and 𝚫𝚫–Y Transformations ................................................................. 47

III. Procedure ............................................................................................................. 48



Experiment 6 — Computer Simulation of Dynamic Systems .............................................. 54

I. Objective ................................................................................................................. 54

II. Theory ................................................................................................................. 54

Introduction III

II.1 Introduction ................................................................................................. 54

II.2 Analog Simulation ....................................................................................... 54

II.3 Digital Simulation ........................................................................................ 55

III. Procedure ............................................................................................................. 57

Introduction IV

LABORATORY AND MANUAL INFORMATION

I. Preparation The laboratory experiments are intended as a practical demonstration of various circuit

theorems introduced during regular classes. Due to the rigid timetable of the laboratory periods, it may happen on occasion that an experiment may precede the class material. This manual is intended as self-contained study material, which will allow a student to be prepared in advance for a laboratory experiment irrespective of material covered in class. All calculations and designs not requiring experimental results should be done prior to the laboratory period. To assess the degree of understanding of theory and procedures, a short oral or written test may be conducted by the instructor and used as a grading component.

II. Equipment The following equipment constitutes one laboratory station intended for a laboratory group

consisting of two students. ♦ Universal Breadboard ♦ Power Supply ♦ Function Generator ♦ Multi-meter ♦ Oscilloscope ♦ Multifunction Counter

Additional equipment and electronic devices required are listed at the end of each experiment. The students should familiarize themselves with the theory and operation of the laboratory equipment by consulting operating manuals and related material.

III. Execution of Experiment III.1 Layout

A lot of time can be saved and frustration avoided by a careful layout and firm connections of the circuit on the breadboard. In the layout, follow the original schematic drawings as closely as possible. Avoid unnecessary crossing of the connecting wires. Use the BNC posts to connect the input/output signals.

III.2 Current limiting

Before connecting a power supply to the circuit, estimate the maximum current that will be taken by the circuit. Set this value as the maximum current which can be delivered by the supply (refer to the operating manual for details). This will protect the circuit in case of wrong connections

III.3 Troubleshooting

Very often a prototype circuit will not work when first switched on. If so, check for proper

Introduction V

supply voltages; measure the supply voltage as close to the circuit as possible; in the case of integrated circuits, measure across the supply pins. Are supplies properly referred to the ground? If connections are messy, it is better to disconnect all components, rebuild the circuit and start afresh. Check the signal lines and component values. Sometimes the integrated circuit can be at fault. If so, replace it.

IV. The Laboratory Report A laboratory report is required from each group for each experiment performed and shall be

submitted within one week after the experiment. The front page of the report is shown in Fig 0-1 and should be reproduced for each laboratory report.

Introduction VI

Department of Electrical and Computer Engineering University of Victoria

ELEC 300 - Linear Circuits II

LABORATORY REPORT

Experiment No.: ____________________________________________________________

Title: ____________________________________________________________

Date of Experiment: ____________________________________________________________

(should be as scheduled)

Report Submitted on: ____________________________________________________________

(should be within one week from the time of experiment)

To: ____________________________________________________________

Laboratory Group #: ____________________________________________________________

Names: (please print) ____________________________________________________________

____________________________________________________________

____________________________________________________________

____________________________________________________________

Fig. 0-1. The front page of a lab report

Introduction VII

The report should be divided into the following parts: objective, introduction, result, discussion and conclusion. The following are some points you should consider while preparing your lab report:

IV.1 Appearance

IV.2 Is the general appearance of your report pleasing to the reader?

IV.3 Presentation ♦ Is your write-up easy to follow, logical and complete? ♦ Did you make an effort to write your report in precise and concise style? Get other

members of your group to read the manuscript before submission. ♦ Did you make efficient use of relevant figures and tables? ♦ Are the tables used clearly described? ♦ Are graphs properly labelled (titles, units, etc.)? ♦ Are proper headings used?

IV.4 Technical Contents ♦ Are all required points covered? ♦ Did you attempt to make some nontrivial observations and measurements on your

own? ♦ How well did you justify your conclusions and results? ♦ Did you use consistently proper units?

IV.5 Punctuality ♦ Do you adhere to the schedule for labs and submission of lab reports?

IV.6 Remember ♦ A report is communication with other persons. Keep in mind the reader's point of view

when writing your report

Introduction VIII

V. Acknowledgement The first edition of this manual was written 1985 by Dr. A. Zielinski; many revisions were

done in the past to keep the contents up-to-date. The efforts of Mr. Michael Shack, who did the initial proofreading of the manuscript and testing of the experiments, are acknowledged. Thanks are due to Dr. Andreas Antoniou and Dr. Jens Bornemann for their suggestions, corrections and improvements, and to Mrs. Marilyn Lacate for typing the original TeX version of the manual. The following sources were used in preparation of this manual:

♦ Bennett, A. W., Introduction to Computer Simulation, West Publishing Company, 1974.

♦ Dorf, R. G., Modern Control Systems, Addison-Wesley Publishing Co., 1967. ♦ Kaufman, M. and Wilson, J. A., Electronics Technology, McGraw-Hill Book Co.,

1982. ♦ Lancaster, D., Active-Filter Cookbook, Howard W. Sams & Co., 1975. ♦ Van Valkenburg, M. E., Network Analysis, Prentice-Hall, 1974. ♦ Experiment 1 was largely based on material provided by Dr. F. H. El Guibaly and Dr.

Antoniou.

To make the document compatible with the word processing software in use by the department, the manual was converted to the Microsoft Word format. The original drawings created by Demetris Pavlides and Mr. Roger Kelly were replaced with new one created with Microsoft Visio. Finally, Dr. Adam Zielinski's professional contribution to this course as well as his effort in proofreading this new edition is very much appreciated.

Poman So, December 2005

Introduction IX

EXPERIMENT 1 — DEPENDENT SOURCES

I. Objective 1. To introduce an ideal operational amplifier (op-amp) and methods of analyzing circuits

with op-amp. 2. To construct and test simple dependent sources using an op-amp.

II. Theory II.1 Voltage and Current Sources

We are familiar with a laboratory power supply which can function as a voltage source (VS) or a current source (CS). We can control the output of such sources by adjusting the appropriate knob. Often it is convenient or necessary to use an electric quantity such as voltage or current to control the output of a voltage or current source. Such a source is called a dependent source. Four types of dependent sources are possible:

3. Voltage-Controlled Voltage Source (VCVS) or voltage amplifier 4. Voltage-Controlled Current Source (VCCS) or transconductance amplifier 5. Current-Controlled Voltage Source (CCVS) or transresistance amplifier 6. Current-Controlled Current Source (CCCS) or current amplifier

The circuit symbols for these sources are shown in Fig. 1.1 (sometimes different symbols are used). Realistic sources have finite input and output resistances, but very often they can be neglected as suggested in Fig. 1.1, leading to idealized sources.

II.2 Operational Amplifier (op-amp)

The operational amplifier is a device that is very useful in many applications. To illustrate its usefulness, we will employ it in some simple circuits to construct the four types of dependent sources.

The operational amplifier has two input ports and one output port as shown in Fig. 1.2a. The word 'operational' comes from the original application of these devices to perform mathematical operations such as addition, multiplication, integration and differentiation. The device is called an amplifier because the output voltage is a magnified version of the voltage difference between the non-inverting and inverting inputs. We can write ( )−+ −= VVAVo where A is defined as the gain of the amplifier.

In practice, an operational amplifier has a very high gain in the range 510 to 610 , a low output resistance of the order of 10 to 200 Ω, and presents a very high input resistance between the two inputs and between each input and the ground (in excess of 2 MΩ). Hence it is convenient to define an ideal operational amplifier as a device which has an infinite gain, zero output resistance, and infinite input resistances. The circuit representation of an ideal operational amplifier is illustrated in Fig. 1.2b and, as can be seen, it operates as a voltage-controlled voltage source.

Experiment 2 10

The operational amplifier is said to be an active device because it supplies power to the circuit connected at its output. This power comes from the power supply connected to the operational amplifier.

Fig. 1.2c shows the pin diagram for a very common type of operational amplifier. In addition to the input and output pins, pins are also provided for the power supplies +

ccV and −

ccV .

The operational amplifier can be used as a component in the design of many types of devices such as negative impedance converters, precision rectifiers, limiters, etc. In this experiment we use the operational amplifier to design the four types of finite-gain controlled sources described above.

An approximate analysis of a circuit containing operational amplifiers can be readily carried out by the following procedure:

1. Replace each operational amplifier by its ideal model shown in Fig. 1.2b. 2. Apply Kirchhoff's voltage and current laws to obtain the circuit equations. 3. Solve the equations obtained in (2) for the desired dependent quantity. 4. Let the gain A become infinite and find the new expressions for the desired

dependent quantity.

An alternative simplified approach which yields identical results is based on the following observation. For proper operation, the output voltage of the op-amp. oV cannot exceed the supply voltages +

ccV or −ccV , which are typically in the 15± volts range. On the other hand,

( )−+ −= VVAVo , where A is a very large number. We must, therefore, conclude that ( )−+ −VV is extremely small (ideally zero). This observation leads to the following analysis, which will be applied in this experiment.

1. Without replacing the operational amplifier by its ideal model, assume that the input currents at the non-inverting and inverting terminals are equal to zero, and that the voltage difference between non-inverting and inverting input terminals is zero (this is sometimes referred to as "virtual ground").

2. Obtain the circuit equations by applying Kirchhoff's voltage and current laws. 3. Solve for the dependent quantity of interest.

II.3 Voltage-Controlled Voltage Source (VCVS)

A finite gain voltage-controlled voltage source can be designed as shown in Fig. 1.3. To find the relation between the output voltage and the input voltage, we use the simplified analysis described above. Referring to Fig. 1.3, we can write

iVVV == +− (1.1)

21 II = (1.2)

Experiment 2 11

The output voltage oV is given by

2211 RIRIVo += , (1.3)

and by using equation (1.1) and (1.2) we obtain

io VKV = , (1.4)

where 1

21RRK += .

Equation (1.4) states that the output voltage is proportional to the input voltage and is independent of the output current. In other words, we have constructed a dependent voltage source.

The relationship between iV and oV holds only if the operational amplifier operates as a linear device, that is, neither of the two voltages approaches the supply voltage value (in practice 0.5-2 volts less than the supply voltage). For linear operation, iV and oV must satisfy the inequality

icc VV <− and +< cco VV . (1.5)

If this inequality is violated, the relationship between oV and iV in equation (1.4) will not hold.

II.4 Voltage-Controlled Current Source (VCCS)

Figure 1.4 is the circuit diagram for a voltage-controlled current source. Notice that this circuit has the advantage that both the input and output of the dependent source are connected to ground. Assuming an ideal operational amplifier, we can write

oVVV == +− (1.6)

21 II = (1.7)

Using the above two equations, we can write

222 RIVV o += (1.8)

33RIVo += (1.9)

and consequently, for RRRRR ==== 4321 we have

Experiment 2 12

32 II = (1.10)

oVV 22 = (1.11) The output current is obtained as 3III io += (1.12)

( )

RV

RVV ooi +

−= (1.13)

and by using equations (1.10) and (1.11), we finally obtain

io VGI = (1.14)

where RG /1= .

Equation (1.14) is the required expression relating the output current to the input voltage. This expression is true as long as none of the voltages in the circuit exceeds the supply voltage. From equation (1.11) the limits on the output voltage are set as

22

+−

<< cco

cc VVV (1.15)

and from equation (1.14) and inequality (1.15) we can write the permissible limits of iV as

L

cci

L

cc

RRVV

RRV

⋅<<⋅+−

22 (1.16)

II.5 Current-Controlled Voltage Source (CCVS)

One method to construct a current controlled voltage source is to use the circuit of Fig. 1.3 after modifying its input as shown in Fig. 1.5. Equation (1.4) is still valid and we can write

oVVV == −+ , (1.17) where

31

21 RRRR ⋅

+= , (1.18)

3R

VI ii = . (1.19)

Equation (1.17) is the required expression relating the output voltage to the input current. Of course, the limits on the input voltage and the output voltage are the same as those in equation

Experiment 2 13

(1.5).

II.6 Current-Controlled Current Source (CCCS)

Fig. 1.6 shows the circuit diagram for the implementation of a current-controlled current source. A simple equation relating the input and output currents is obtained by using the simplified analysis described above. We have

oVVV == −+ (1.20)

iII =1 . (1.21)

On the other hand

2IIo = (1.22)

2211 RIRI = . (1.23)

Now by solving for oI in terms of iI using the above equations, we obtain

io IKI = , (1.24) where

2

1

RRK = (1.25)

Again, equation (1.24) is valid only if the circuit voltages do not exceed the supply voltage. This imposes limits on the input and output currents as follows:

( )Locc RRIV +<−2 or ( ) +<+ ccLi VRKRI 1 . (1.26)

III. Procedure

Set the 6030 (0-30 volt) supplies for VVcc 10<+ and VVcc 10−<+ with respect to ground by connecting the (–) terminal of one module to ground (for +

ccV ) and the (+) terminal of the second module to ground (for −

ccV ). Set the current limit control of both modules for the maximum current you expect your circuit to draw (say, ±100 mA, in this experiment). This will protect electronic components to a certain extent, because the supplies will not deliver more than 100 mA of current even if a short circuit occurs. Follow this procedure in all your future experiments. Refer to the supply manual for details.

III.1 Voltage-Controlled Voltage Source

1. Connect the circuit of Fig. 1.3 using Fig. 1.2c as a guide for correct connection of the

Experiment 2 14

op-amp. Set ∞=LR (open circuit) and 0=iV (short circuit). Connect the power supplies −

ccV and +ccV and turn them on. Recheck your circuit if supplies operate in

current limiting mode. This might indicate improper connections. 2. Use the function generator in the dc-offset mode to produce the input voltage iV ,

(select D.C. waveform and pull D.C. offset control to engage). This source has internal impedance of 50Ω or 600Ω. Switch it to 50Ω. Vary iV between –6 and +6 V and plot oV against iV . Observe when the relationship between oV and iV breaks down. Does the slope of the straight line agree with the value of K in equation (1.4)?

3. Adjust iV to obtain ppo VV 2= at frequency f = 1 kHz. Load the circuit with Ω= 150LR and observe how this affects oV . Based on this observation, estimate the

value of internal resistance of the source.

III.2 Voltage-Controlled Current Source

1. Connect the circuit as shown in Fig. 1.4. Connect a load resistance Ω= k1LR . Note that the source might have an inner resistance. If so, a similar resistance must be added (series-connected) to 1R in order to preserve symmetry.

2. Vary iV and plot oI versus iV for ±8 volts of iV .

3. When oI stops following iV , measure oV and compare with inequalities (1.15) and (1.16).

4. Adjust iV such that mA1=oI as measured by an ammeter with negligible internal

resistance. Change LR from 1 kΩ to 0 Ω and observe oI . Does oI change?

Estimate the output resistance of the source.

5. Measure the voltage difference between the inverting and non-inverting inputs of the op-amp to confirm the validity of the assumption made about the op-amp.

IV. Devices Required

• Op-amp: 741 × 1 • Resistors: 1 kΩ × 5, 150 Ω × 1

V. Preparation prior to the laboratory session 1. Use PSPICE to model the circuits to be built. 2. Justify the simulation results with theoretically with your own calculations.

Experiment 2 15

Fig. 1.1: Types of Dependent Sources

+

–

iv ∞=iR

0=oR+

–

VCVS

io kvv =

+

–

iv ∞=iR ∞=oR

+

–

VCCS

io Gvi =

ii ∞=iR ∞=oR

+

–

CCCS

io Kii =

∞=iR

0=oR+

–

CCVS

io Riv =ii

Experiment 2 16

(a) Circuit symbol for an operational amplifier

(b) Equivalent circuit for an operational amplifier

(c) Pin diagram for the 741 operational amplifier

Fig. 1.2: Operational Amplifier

ov

−v

+v

–

+

−v

+v

)( −+ −= vvAvo

1

2

3

4

8

7

6

5

−v

+v−ccv

+ccv

ov

–

+

Experiment 2 17

Fig. 1.3: Circuit diagram for a voltage-controlled voltage source (VCVS)

Fig. 1.4: Current diagram for a voltage controlled current source (VCCS)

ov

−v

+v +

–741

iv

1R

2RLR

+

–

oi

1i2i

Ω== k121 RR

ov

−v

+v +

–741iv

2R1R

LR+

–

oi

2i

1i

4R

4i

3R

3i

Ω==== k14321 RRRR

Experiment 2 18

Fig. 1.5: Circuit diagram for a current-controlled voltage source (CCVS)

Fig. 1.6: Circuit diagram for a current-controlled current source (CCCS)

ov

−v

+v +

–741

1R

2RLR

+

–

oi

1i2i

ii 3R iv+

–

ov

−v

+v +

–741ii

1R

LR+

–

oi

1i

2R

2i

Experiment 2 19

EXPERIMENT 2 — FREQUENCY RESPONSE OF LINEAR SYSTEMS

I. Objective

♦ To investigate the frequency response (amplitude and phase) of linear systems and its relationship with pole-zero diagram.

♦ To introduce the logarithmic representation of frequency plots (Bode plots), and their approximation.

♦ To design a simple network and investigate its properties in the frequency domain.

II. Theory A very important signal is the sinusoidal waveform with amplitude A , and radian

frequency ( )fπω 2= , that is

( ) ( ) tjAetAtx ωω Recos == (2.1)

If such a waveform is applied to a linear system for a sufficiently long time, then the steady-state response ( )ty of the system will also be a sinusoidal waveform of the same frequency as the input waveform, but with a different amplitude and phase, that is

( ) ( )φω += tBty cos (2.2)

Both amplitude B and phase φ depend on the frequency of the input signal. Their exact values can be found by phasor analysis of the circuit.

If the transfer function ( )sH of the circuit is given, then the so-called frequency response ( )ωjH , can be obtained by direct substitution of s by ωj , that is

( ) ( ) ωω jssHjH == (2.3)

( )ωjH is a complex function of ω . It allows us to calculate the output amplitude and phase at any particular input frequency oω namely

( ) ( )oo jHAB ωω = (2.4a)

and ( ) ( ) οωωφ jHo arg= (2.4b)

The ratio ( )ωjHAB = as function of frequency is called the amplitude response of a linear circuit, and function ( )ωφ is called the phase response of the circuit.

Experiment 2 20

As an illustration, let us consider the simple RC circuit shown in Fig. 2.1 (already transformed to s-domain). To find its transfer function, we proceed in the usual way by writing

( ) ( )C

RC

ZZZ

sXsY ⋅+

=

or

( ) ( )( ) o

o

ssXsYsH

ωω+

== ; RCo1

=ω (2.5)

The value of ps = for which ( ) ∞=sH is called a pole. The value of zs = for which ( ) 0=sH is called the zero of a transfer function. In this particular case, ( )sH has no finite

zeros and only one pole at ops ω−== . The frequency response of this circuit can be obtained using Eqns. (2.3) and (2.4). The amplitude response is

( )( )222 1

1

oo

o

o

o

o

o

jjjH

ωωωωω

ωωω

ωωωω

+=

+=

+=

+= (2.6)

Similarly, the phase response is given by

( ) ( )ooo jjH ωωωωωωθ 1tanargargarg −−=+−== (2.7)

The two plots are shown in Fig. 2.2.

We can see that this circuit passes low-frequency signals (including dc), but attenuates high-frequency signals. It is appropriate to call it a low-pass filter. The amplitude response (sometimes called gain) can vary over a wide range. To accommodate this range, a logarithmic scale is often used. The gain is then said to be measured in decibels, that is

( ) ( )ωω jHjHdB

log20= (2.8)

For the circuit of Fig. 2.1, Eqns. (2.6) and (2.8) give

( ) ( )21log10 odBjH ωωω +−= (2.9)

A useful approximation of (2.9) can be proposed for 1<<oωω and 1>>oωω , namely

Experiment 2 21

( ) ( ) ( )[ ]

>>−×<<

≈oo

odB

jHωωωωωω

ωfor loglog20for 0

(2.10)

This approximation becomes particularly simple if the frequency is also plotted on a logarithmic scale. Two straight lines are obtained which meet at point ( )oωlog , as illustrated in the upper part of Fig. 2.3. Frequency oω is called, appropriately, the corner frequency.

The exact value of ( )dB

jH ω is shown by a dashed line for the sake of comparison. The maximum error of approximation is 3 dB and occurs at the corner frequency. The unit for logarithmic frequency scale (base of ten) is the decade and represents a separation of two frequencies, one ten times the other.

We can see from Eqn. (2.10) that the slope of the approximating line for ( oωω loglog > ) is –20 dB/decade. Sometimes a logarithm of base two is used for the frequency scale and Eqn. (2.10) can be expressed as

( ) ( ) ( )[ ]

>>−×<<

≈oo

odB

jHωωωωωω

ωfor loglog6for 0

22

by noting that

xxx 22

210 log6

10loglog20log20 ==

Therefore, if ( )dB

jH ω is plotted against ω2log , a piecewise linear graph is obtained, as before. In this case, the unit of the frequency scale is called an octave, and represents a separation of two frequencies, one twice the other. The slope of the approximating line (for

oωω 22 loglog > ) is –6dB/octave.

A similar piecewise approximation is possible for the phase response given by Eqn. (2.7). The approximation is shown at the lower part of Fig. 2.3. As we can see, the presence of a pole affects the phase characteristic within two decades around the corner frequency. Similar approximations hold for a system with a transfer function containing a zero, z , such as

( ) zssH −= ; oz ω−= (2.11)

In this case, the slope of the amplitude response of piecewise approximation will be dec/dB20± starting from the corner frequency oω . The phase characteristic will have a slope of

dec/45± for 2 decades around the corner frequency to end up with 90± maximum phase shift contribution.

In general, a transfer function can have several poles and zeros. In such a case, the total frequency response will be equal to the sum of amplitude and phase contributions from each pole and zero.

Experiment 2 22

The location of poles (×) and zeros (。) in the s plane can be used directly to construct the frequency response. This is illustrated in detail in Fig 2.4 for the amplitude response.

III. Procedure 1. The network shown in Fig. 2.5a is called a phase lag network, and has a transfer

function

( )pszsKsH

−−

=

Find the relationships between K, z and p and circuit elements 1R , 2R and C .

2. Assume that secrad 100022 1 /fp ×== ππ and secrad 000,1022 2 /fz ×== ππ Plot the Bode approximation of the amplitude response of the circuit. The frequency should be expressed in cycles/sec (Hz) rather than radians/sec and should cover 3 decades starting from 100 Hz as suggested in Fig. 2.5b. The range on the vertical axis (gain expressed in dB) should be properly selected for clear display.

3. Calculate the exact values of ( )dB

jH ω for the corner frequencies 1f and 2f given in point 2, and plot them on the same coordinates as were used in point 2.

4. Assume F01.0 µ=C and calculate the required values for 1R and 2R to obtain the corner frequencies as given in point 2.

5. Connect the circuit of Fig. 2.5a, using the values for 1R and 2R calculated in point 4. Using the function generator, apply a sinusoidal waveform of varying frequency and plot the experimental amplitude response on the coordinates which were used in point 3.

6. Use two channels of the oscilloscope to investigate the phase response of the circuit. Justify the name of this circuit.


♦ Resistors: 1.6 kΩ × 1 10 kΩ × 1

♦ Potentiometer: 10 kΩ × 1 ♦ Capacitor: F01.0 µ × 1


Experiment 2 23

Fig 2.1: Circuit in s-domain

Fig 2.2: Frequency response of a first-order system

)(sX )(sYsC

ZC1

=

RZR =+

–

+

–

1

21 ( )ωjH

( )ωφ j45−

90−

oω ω

Experiment 2 24

Fig 2.3: The frequency response of a first-order system in logarithmic scale

Fig 2.4: Approximate frequency response from the pole-zero plot

dB0dB3−

dB20−

0

45−

90−

oω1.0 oω oω10

Slope = –20dB/dec

Corner frequency

log-scaleω

( )dB

jH ω

( )ωφ j

s-plane

Pole-zero plot

p2 p1z1

log|p1| log|z1| log|p2|log|ω|

( )dB

ωjH

Experiment 2 25

(a) Electrical circuit

(b) Amplitude response

Fig 2.5: The phase lag network

+

–

+

–

)(tx )(ty

1R

2R

C

310 410 510 610(Hz) f

Experiment 2 26

EXPERIMENT 3 — TIME-DOMAIN RESPONSES

I. Objective

♦ To familiarize students with an active realization of a second-order system. ♦ To study its time-domain response to various excitations. ♦ To introduce a digital oscilloscope as a convenient device to capture and display

aperiodic signals.

II. Theory II.1 Active Second-Order Circuit

We are already familiar with a second-order passive system consisting of an inductor, capacitor and resistor. It is possible, however, to construct a second-order system without the use of a bulky and expensive inductor. Such a circuit involves the use of an amplifier, and is referred to as an active realization of a system. Shown in Fig. 3.1a is an example of an active realization of a second-order circuit. We can recognize that the portion of circuit enclosed by the dotted line is a voltage controlled voltage source (VCVS) or simply a voltage amplifier with gain

ba /RRG += 1 . This leads us to the equivalent circuit shown in Fig. 3.1b

The input and output voltages )(tx and )(ty are linked through a linear differential equation. To find this equation, we can write a set of simpler equations involving intermediate variables (such as )(tv ) as governed by Kirchhoff's current and voltage laws. These intermediate variables can then be eliminated leading to a single equation involving only )(tx , )(ty and the circuit parameters. This procedure might be quite tedious and a better method of analysis will be introduced later. In this particular case, one can find the differential equation linking )(tx with

)(ty as:

xGyyy ooo222 ωωζω =++ (3.1a)

where

o

o RCRC ω

ω 1or1== (3.1b)

and ζζ 23or2/)3( −=−= GG (3.1c)

We recognize this equation as the differential equation of the well-known second-order system with a natural frequency oω and a damping ratio ζ . It is very fortunate that, in this particular case, we can control the parameter oω (by the choice of RC) and ζ (by the choice of G) independently. We say that the parameters are decoupled. In particular, for a given ζ , we see from Equation (3.1c) that aR and bR in Fig. 3.1a should satisfy:

( )ζ−= 12ba RR (3.2)

Experiment 3 27

II.2 Time-Domain Analysis Using the Laplace Transform

We will investigate the time-domain response )(ty of the circuit shown in Fig. 3.1a to various input excitations )(tx . The output signal )(ty can be obtained by solving the differential equation given as Equation (3.1a). This can be done by classical methods or by using the Laplace transform. By applying the transform to both sides of Equation (3.1a) and assuming zero initial condition, we obtain

2 22 o o oy y y L G xζω ω ω+ + = L

or

)()()(2)( 222 sXGsYssYsYs ooo ωωζω =++ (3.3)

or

)(2

)( 22

2

sXss

GsYoo

o

ωζωω

++= (3.4)

where

( ) ( ) and ( )Y s L y t X(s) x t= = L

The ratio

22

2

2)(

)()(

oo

o

ssGsH

sXsY

ωζωω

++== (3.5)

is known as the transfer function of the system.

For a given input signal )(tx the system response )(ty can be found using the inverse Laplace transform; that is

1( ) ( ) ( )y t H s X s−= L (3.6)

The required inverse transform can be found by representing )()( sXsH as the sum of simpler terms with known inverse Laplace transforms (by utilizing partial fraction expansion), or by using a suitable table of Laplace transforms.

Experiment 3 28

II.3 Impulse Response

An important input excitation is the ideal impulse, also called a delta function or Dirac pulse, that is

)()( ttx δ= (3.7)

Since ( ) 1tδ =L , then following Equation (3.6) we have

1( ) ( ) ( )y t H s h t−= =L (3.8)

The response )()( thty = is called the impulse response of a system. In our case, for 1<ζ

)sin()exp()( ttGth ooo βωζω

βω

−= (3.9a)

and for 1=ζ (so called critical damping case)

)exp()( 2 ttGth oo ωω −= (3.9b)

where 21 ζβ −=

II.4 Step Response

Another important input excitation is the step function

<≥

==0for00for1

)()(tt

tutx (3.10)

Proceeding as before, we can find the following transformed response (step response) of our system

)2(

)()( 22

2

oo

o

sssGsAsY

ωζωω

++== (3.11)

Taking the inverse Laplace transform, we obtain for 1=ζ

+−−= )sin()exp(11)( θβωζω

βttGta oo (3.12a)

Experiment 3 29

where ( )ζβθ /tan 1−= and 21 ζβ −= , and for 1=ζ (real roots)

[ ])exp()1(1)( ttGta oo ωω −+−= (3.12b)

Note the following interesting points:

(a) From Eqn. (3.12) we can see that for ∞=t , Ga =∞)( and the output of the system will stabilize, after initial transients, at a constant value G. We can confirm this result using the final value theorem.

GssAas

==∞→

)(lim)(0

We can also obtain this result by inspection of the circuit given in Fig. 3.1a. At the steady DC-state, the capacitors act as open circuits, and the whole circuit reduces to a voltage amplifier with a gain of G.

(b) Very often it is necessary to identify system parameters oω and ζ based on an available step response. The following relationships derived from Eqn. (3.12a) can be used for this purpose (the symbols used are explained in Fig. 3.2).

)/exp( βζπ−=vO (3.13a)

βω

π

opT = (3.13b)

Using Eqn. (3.13a), one can estimate ζ and then, using this estimate and Eqn. (3.13b), the natural frequency ωo can be found.

(c) Comparing Eqns. (3.8) and (3.12), we notice that

)()( tadtdth = (3.14)

The explanation of this property is based on system linearity and the fact that

)()( tudtdt =δ

Because of Eqn. (3.14), the impulse response can be deduced from the step response, which is easier to obtain.

III. Procedure 1. Connect the circuit of Fig. 3.1a using the component values indicated. Use a 100 kΩ

Experiment 3 30

potentiometer for aR and 15±=ccV V. 2. Apply a rectangular waveform at a suitable frequency to the input of the circuit and

observe the output using an oscilloscope. Confirm proper operation of the circuit by varying the value of aR and observing any change in the character of the response.

3. Calculate and adjust the value of aR to obtain 1.0=ζ . Using the digital oscilloscope (see Appendix 3.1 for a brief introduction to the instrument) and the digital plotter stations, obtain a hard copy of the step response of your circuit. The input step should have approximately 0.1 volt amplitude (and can be considered a unit).

4. Using the response obtained in point 3 and Eqn. (3.13), estimate the values of ζ and oω .

5. Using values for ζ and oω obtained in point 4 and Eqn. (3.12a), calculate a few points of the response and plot them for comparison with the experimental response obtained in point 3.


• Op-amp: 741 × 1 • Capacitors: 16 nF × 2 • Resistors: 10 kΩ × 2, 39 kΩ × 1 • Potentiometer: 100 kΩ × 1


Experiment 3 31

VI. Appendix — Principle of Operation of a Digital Oscilloscope A block diagram of a digital oscilloscope is shown in Fig. 3.3. An analog signal is

converted to a digital form by an analog-to-digital converter (ADC) and stored in the memory. The beginning of a conversion is initiated by the trigger circuit set up in a proper mode.

Data from the memory can be non-destructively and repetitively read at instants and rates determined by the control circuit. After being converted to an analog form by a digital-to-analog converter (DAC), the signal can be displayed on the cathode ray tube (CRT) in a conventional way. The sweep generator provides a sweeping deflection voltage as determined by the control circuit.

The principle of operation of a digital scope allows for some unique modes of operation which are not available on conventional analog oscilloscopes.

VI.1 Refresh Mode

The new signal replaces the old contained in the memory and the display updates from the left-hand edge of the screen to the right. This happens every time a trigger occurs which initiates the sweep. If a trigger is not received, the signal captured on the previous sweep will be retained on the display.

VI.2 Role Mode

♦ Continuous

The memory functions as a shift register and is displayed each time a new signal sample is received. This creates a display continuously moving from the right edge of the screen to the left, thereby imitating a paper chart recorder.

♦ Single Shot

In this mode, the signal is continuously acquired by the memory which functions as a shift register. If a trigger signal is received, this process is stopped and the memory content is displayed. This arrangement allows the scope to display the portion of the signal preceding the trigger (so called pre-trigger feature which can be set at different percentages of the total horizontal length of the display).

Experiment 3 32

Ω−=Ω==Ω===== k)1(78,k39,k10,nF16 2121 ζab RRRRRCCC

(a) Active realization of a second-order system

RRR == 21

(b) Equivalent circuit of (a)

Fig 3.1: Second-order system

)(ty)(tx

bR aR

+

–

b

a

RRG +=1

+

–7412R1R

1C

2C

+

–

)(ty)(tx

+

–

2R1R

1C

2C

+

–

Gvv+

–

Experiment 3 33

Fig 3.2: Normalized step response of a second-order system

Fig 3.3: Functional diagram for a digital oscilloscope

0

1

βωπτo

p =

τ

)()(

∞aa τ

Overshoot :vO

Input

ExtTrigger Control Sweep

ADC DACMemory

CRT

Experiment 3 34

EXPERIMENT 4 — ANALYSIS AND APPLICATIONS OF ACTIVE NETWORKS

I. Objective

♦ To introduce the s-domain network analysis and to illustrate it on several useful active circuits.

II. Theory As discussed in Experiment 2, the Laplace transform can be a useful tool for the solution of

a linear integro-differential equation describing an electrical network.

The simplest network will contain only one element: resistor (R), capacitor (C) or inductor (L). The relationship between current and voltage across a resistor is given by Ohm's law as

)()( tiRtv RR ⋅= (4.1)

Applying the Laplace transform ( L transform) to both sides of the above equation, we have

)()( sIRsV RR ⋅= (4.2)

Eqn. (4.2) can be thought of as Ohm's law in the s-domain. Similarly, for a capacitor C we have

∫∞−

=t

CC diC

tv ττ )(1)( (4.3)

or, after applying the L transform, and assuming zero initial condition, we have

)(1)( sIsC

sV CC = (4.4)

By taking the ratio of the transformed voltage to the transformed current, we can introduce a "generalized resistance"

sCsI

sVsZC

CC

1)()()( == (4.5)

This generalized resistance is called impedance. The relationship can be thought of as generalized Ohm's law. A similar development holds for an inductor L, leading to the impedance of an inductor given by

sLsIsVsZ

L

LL ==

)()()( (4.6)

Experiment 4 35

Analysis of a complex network can be simplified if the L transform is applied at the element level rather than on the full differential equation. This is done by replacing all simple elements by their impedances. Since the L transform is a linear operation, all previously introduced network theorems apply to a circuit represented in the s domain. As an example, let us consider the network shown in Fig. 4.1a. The s-domain of the same network is shown in Fig 4.1b. Since all the quantities are functions of s, the symbol s is redundant and can be omitted for convenience. Applying the generalized voltage law to the circuit, we have

CR VVV += (4.7)

or using the generalized Ohm's law

IZZZIIZIZV CRCR =+=+= )( (4.8)

Eqn. (4.8) leads us to the concept of equivalent impedance Z representing series connection of two impedances as illustrated in Fig 4.1c. Similar reduction can be performed on a parallel connection of impedances. In both cases, the rules for calculating the equivalent impedances are identical to those for simple resistors.

As an example, we will consider the circuit shown in Fig 4.2 which has already been transformed into the s-domain, as indicated by capital letters denoting transformed voltages and currents. The op. amp is assumed to be ideal, i.e. no current (or transformed current) flows into its inputs and no voltage difference (or its transform) exists between them.

Proceeding formally in the same manner as with a purely resistive network, we can write a sequence of equations to find the output oV as a function of the two input transformed voltages

1V and 2V (for simplicity, we will call them just voltages). We can write

IZVo −=

but, since

21 III +=

where

111 ZVI = and 222 ZVI =

we obtain

+−= 2

21

1

VZZV

ZZVo (4.9)

Many practical circuits can be derived as the special cases of Eqn. (4.9) and the circuit of Fig 4.2.

Experiment 4 36

II.1 Inverting Voltage Amplifier

For RZ = , 11 RZ = and ∞== 22 RZ , Eqn. (4.9) becomes

)()( 11

sVRRsVo −=

or in the time domain

)()( 11 tvKtvo −= (4.10)

where

11 RRK =

II.2 Inverting Adder (Summer)

For RZ = , 11 RZ = and 22 RZ = , Eqn. (4.9) in the time domain becomes

[ ])()()( 2211 tvKtvKtvo +−= (4.11)

where

11 RRK = and 22 RRK =

II.3 Inverting Integrator

Let us assume that a capacitor C is connected between output and inverting terminal of the

op. amp. so that sC

Z 1= , 11 RZ = and let ∞== 22 RZ , Eqn. (4.9) becomes

1111

11

111 Vs

GVsCR

VZZVo −=⋅−=−= (4.12)

where CR

G1

11

= . Taking the inverse L transform, we have in the time domain

∫−=t

o dvGtv0

11 )()( ττ (4.13)

II.4 Summing Integrator

Experiment 4 37

For sC

Z 1= , e.g. 11 RZ = and 22 RZ = , we have

[ ]22111 VGVGs

Vo +−= (4.14)

or

[ ]

+−= ∫

t

o dvGvGtv0

2211 )()()( τττ (4.15)

where CR

G1

11

= and CR

G2

21

= .

All these cases are summarized in Fig 4.3. The circuits described can be used as building blocks to construct more complex systems. We illustrate this point by considering the circuit shown in Fig. 4.4a. We can quickly recognize it as a summing integrator. An interesting feature of this circuit is that the second input to the integrator is the output signal (like a dog eating his own tail). Such a connection is called feedback. The symbolic representation of circuit 4.4a showing only the essential feature is shown in Fig 4.4b (in s-domain). Using this representation, we can write

( )

−⋅+=

sGVVV oo 1

or

sG

GsHVVo

+−== )(

1

(4.16)

where RC

G 1= . We recognize this as the transfer function of a first-order system shown in

Fig 4.4c.

III. Procedure 1. Connect the circuit of Fig. 4.3a and confirm its operation as a unity gain inverter. Use a

sinusoidal signal at a frequency of approximately 1 kHz as the )(1 tv input.

2. Connect the circuit of Fig. 4.3b and confirm its operation as an adder. For the input signal )(2 tv , use the low-voltage power supply (model 6007). Input )(1 tv remains as in point 1.

3. Connect the circuit of Fig. 4.3c and confirm its operation as an integrator. Proceed as

Experiment 4 38

follows:

♦ Apply a small dc voltage to its input ( 1.0−≈ volts). This will quickly drive the integrator to saturation (the output will "clamp" to the positive supply voltage).

♦ Reset the integrator by shorting the capacitor C.

♦ Remove the shorting wire and capture the increasing output voltage using the pre-trigger feature of the digital scope. Plot the captured signal using the plotter.

4. Connect the circuit of Fig. 4.4a. By investigating its time response (such as the response to a unit step), confirm that this is indeed a first-order system. Use the digital scope and plotter to document your findings.


• Op-amp: 741 × 1 • Capacitor: 10 nF × 1 • Resistors: 10 kΩ × 3


Experiment 4 39

(a) Time-domain representation

(b) s-domain representation

(c) Equivalent impedance

Fig 4.1: Network Representations

)(ti R

C

+

–

)(tv

)(tvR

)(tvC

+ –+

–

)(sI RZR =

sCZC

1=

+

–

)(sV

)(sVR

)(sVC

+ –+

–

)(sI

CR ZZZ +=

+

–

)(sV

+

–

Experiment 4 40

Fig 4.2: An active network in s-domain

Fig 4.3: Circuits with an op-amp

)(1 sZ

)(2 sZ)(2 sI

)(1 sI )(sZ )(sI

)(2 sV

)(1 sV

)(sVo

–

+741

)(tvo

1R

R

–

+741

++

– –

)(1 tv

Ω== k10:Inverter (a) 1RR

)(tvo

1R

R

–

+741

2R

++

+

– –

)(1 tv

)(2 tv

Ω=== k10:Adder (b) 21 RRR

)(tvo

1R

C

–

+741

++

– –

)(1 tv

Ω== k10nF,16:Intgerator (c) 1RC

)(tvo

1R–

+741

2R

++

+

– –

)(1 tv

)(2 tv

C

Ω=== k10nF,16:Integrator Summing (d) 21 RRC

Experiment 4 41

(a) A circuit with an integrator

(b) Equivalent representation of circuit (a)

(c) Equivalent representation of circuit (b)

Fig 4.4: Representation of a circuit with an integrator

)(tvo1R

–

+741

2R

++

– –

)(1 tv

C

Ω=== k10nF,16 21 RRC

+sG

−+

)(1 tv )(tvo

+

sGG+

−+

)(1 tv )(tvo

+

Experiment 4 42

EXPERIMENT 5 — TWO-PORT NETWORKS

I. Objective

♦ To introduce methods of description, analysis and measurement of two-port networks.

II. Theory II.1 Motivation

In order to analyze a complex circuit (system), it is convenient to subdivide it into several simpler circuits (subsystems) and mathematically describe their properties. The next step will involve analysis of the whole system using a description of its subsystems and taking into consideration the nature of their interconnections.

For example, consider a cascade connection of the two simple RC networks shown in Fig. 5.1a. Each of them can be described by an easily derivable transfer function, that is

( ) ( )sHVVsH

VV

23

41

1

2 and ==

These functions are derived assuming open-circuit output conditions. In other words, there is no current taken from the circuit (no loading).

Assume now that these two circuits are connected in series as shown in Fig. 5.1(b), and that we want to find the relationship between the input 1V and the output 4V . We note that an ideal amplifier with a unit gain is inserted between the circuits. The input impedance of such an amplifier is very high (infinity), and therefore no current 2I is drawn from the preceding circuit (no loading). Under these circumstances, the transfer functions derived for open output conditions will be valid, and we can write the following relationships:

324

112

VHVVHV

==

but since 32 VV =

( ) ( ) 1214 VsHsHV = (5.1)

Now consider the circuit shown in Fig. 5.1(c). We can see that the ( )sH 2 network is loading the ( )sH1 network ( )02 ≠I . For this reason, Eqn. (5.1) is no longer valid. In order to find a relationship between 4V and 1V , we have to write and solve a set of two simultaneous equations. Alternatively, one can utilize a methodology fully describing such a network including input and output currents.

Experiment 5 43

II.2 Two-Port Networks

Any complex linear network (with or without dependent sources) can be represented as a “black box” with four terminals — two input terminals (input port) and two output terminals (output port) as shown in Fig. 5.2. Four variables, 1V , 2V , 1I and 2I , can be measured at the input and output terminals, and can be used to describe the network. Any two networks which behave identically with respect to these variables can be considered equivalent. Many equivalent networks are possible and the selection of a particular one will depend upon the application.

II.3 Impedance Parameters (z-parameters)

Assuming linearity of the black box shown in Fig. 5.2, we can write

2221212

2121111

IzIzVIzIzV

+=+=

(5.2)

The ijz parameters describing a network can be obtained by setting 1I or 2I to zero (open circuits), and then applying voltages 1V or 2V and calculating or measuring the voltage-to-current ratios. In this way, we can obtain

02

2220

2

112

01

2210

1

111

11

22

,

,

==

==

==

==

II

II

IVz

IVz

IVz

IVz

(5.3)

Considering Eqn. (5.3), it is proper to call the ijz parameters as open circuit impedances. Once parameters ijz are found, a convenient equivalent circuit such as that shown in Fig. 5.3 can be used to simulate the black box. As we can see, the equivalent circuit involves the use of two current-controlled voltage sources.

Example l

We will use Eqn. (5.3) to derive the z-parameters for the T network shown in Fig. 5.4. Assuming open output terminals ( )02 =I and voltage 1V at the input, we can write

31

11 RR

VI+

=

or

01

13111 2 ==+= II

VRRz

Experiment 5 44

Similarly

312 RIV =

or

01

2321 2 === II

VRz

We now assume that the input terminals are opened, and voltage 2V is applied at the output terminals. Proceeding as before, we obtain

3123222 and RzRRz =+=

Note that 2112 zz = . The equality is valid for any passive network.

Eqn. (5.2) can be used to derive the input impedance of a two-port network. With open output terminals ( )02 =I , we obtain from Eqn. (5.2)

1111 IzV = or

111

1 zIVZ ocin == (5.4)

With shorted output terminals ( )02 =V , using Eqn. (5.2), yields

( )2222

22122211

1

1

zz

zzzzz

IVZ scin

∆=

−== (5.5)

In general, the input impedance inZ of a two-port network depends on the load LZ connected to its output.

A very important parameter of a two-port network is the so-called characteristic impedance (or image impedance) defined as a load oL ZZ = for which the input impedance oin ZZ = . A practical consequence of the characteristic impedance is that any number of two-port networks can be cascaded (with the last stage loaded with oL ZZ + ), without affecting the input impedance seen at the first port.

It can be shown that the characteristic impedance is given by

zzzZZZ scinocino ∆==

22

11 (5.6)

Experiment 5 45

II.4 Admittance Parameters (y-parameters)

Eqn. (5.2) can be solved with respect to currents I_1 and I_2 yielding an alternative description of a two-port network, namely

2221212

2121111

VyVyIVyVyI

+=+=

(5.6)

where

zzyzzyzzyzzy

∆=∆−=∆−=∆=

1122

2121

1212

2211

(5.7)

211222112221

1211det zzzzzzzz

z −==∆

The parameters ijy can also be obtained directly from Eqn. (5.6), that is

02

222

02

121

01

212

01

111

1

1

2

2

=

=

=

=

=

=

=

=

V

V

V

V

VIy

VIy

VIy

VIy

(5.8)

Setting the voltages 1V or 2V to zero involves shortening of the input or output terminals.

Parameters y are particularly useful in a parallel connection of two networks as shown in Fig. 5.5. It can be proved that the y parameters of an equivalent two-port network are equal to the sum of the y parameters of networks aN and bN , that is

bijaijij yyy __ += (5.9)

II.5 Transmission Parameters (T)

Experiment 5 46

If we solve Eqn. (5.2) with respect to 1V and 1I , we obtain a new form, involving the so-called transmission parameters A, B, C and D. We can write

221

221

DICVIBIAVV

−=−=

(5.10)

where

11

22

21

2121

11

,1

,

zzD

zC

zzB

zzA

==

∆==

(5.11)

The T parameters can be derived directly from Eqn. (5.10), that is

0

2

10

2

1

02

10

2

1

22

22

,

,

==

==

−==

−==

VI

VI

IID

VIC

IVB

VVA

(5.12)

The T parameters are particularly useful for the analysis of cascade connections of two-port networks as shown in Fig. 5.6. It can be proved that the T parameters of the equivalent network are given by the following matrix equation:

×

=

bb

bb

aa

aa

DCBA

DCBA

DCBA

II.6 Y–𝚫𝚫 and 𝚫𝚫–Y Transformations

Any two-port networks described by identical parameters will be considered completely equivalent from the point of view of their terminal behaviour. Shown in Fig. 5.7 are examples of frequently used networks. The T network is sometimes called the Y network and the \pi network is sometimes called the ∆ Network.

These two networks can be proved to be equivalent if the following conditions are satisfied:

123 ,, zzzzzzzzz TCTBTA === (5.14)

where 313221 zzzzzzzT ++= , or

π

π

π

zzzzzzzzzzzz

CB

CA

BA

===

3

2

1

(5.15)

Experiment 5 47

where CBA zzzz ++=π .

The above transformations are called Y–∆ and ∆–Y transformations.

III. Procedure 1. For the twin-T network shown in Fig. 5.8, calculate both open and short circuit input

impedances, ocinz and scinz . Using Eqn. (5.6), find the characteristic impedance oz .

2. Find the y parameters for the T network shown in Fig. 5.4 and the twin-T network shown in Fig. 5.8 by recognizing that it is a parallel connection of two T networks.

3. Connect the twin-T network of Fig. 5.8 and experimentally verify the values of ocinz and scinz obtained in point 1. Since this is a resistive network, some

measurements in this experiment can be done with an ohmeter only. 4. Measure the z parameters of the network of Fig. 5.8 and verify Eqns. (5.4) and (5.5).

5. Connect the characteristic impedance oz calculated in point 1 at the output of the network of Fig. 5.8, and measure the input impedance to confirm that oin zz = .

6. Measure the y parameters of the above network and compare them with the values obtained in point 2.


♦ Resistors: 1 kΩ × 6


Experiment 5 48

(a)

(b)

(c)

Fig 5.1: Series connection of two-port networks

2V1V

( )sH1

1R

+

–

+

–

1C

4V3V

( )sH 2

2R

+

–

+

–

2C

–

+( )sH 2( )sH11V

+

–

4V

+

–

2V 3V

+ +

– –

02 =I

1V 1R

+

–

1C

4V2R

+

–

2C02 ≠I

Experiment 5 49

Fig 5.2: Two-port network

Fig 5.3: Equivalent representation of a two-port network.

Fig 5.4: T-network

1V

+

–

2V

+

–

Black BoxPort 1 Port 2

1I 2I

1V

+

–

2V

+

–

Port 1 Port 2

1I 2I11z 22z

212Iz 121Iz

1V

+

–

2V

+

–

1I 2I1R 2R

3R

Experiment 5 50

Fig. 5.5: Parallel connection of two-port networks with a common ground

Fig 5.6: Cascade connection of two-port networks

bN

aN

1V

+

–

2V

+

–

1I 2I

1V

+

–

2V

+

–

1I 2I

aN bN

Experiment 5 51

(a) T-network (Y-network)

(b) π-network (∆-network)

Fig 5.7: Two-port networks

1V

+

–

2V

+

–

1I 2I1Z 2Z

3Z

1V

+

–

2V

+

–

1I 2I

BZ CZ

AZ

Experiment 5 52

(a) Twin-T

(b) Equivalent representation of circuit (a)

Fig 5.8: Equivalent two-port networks

1V

+

–

2V

+

–

1I 2I1R 2R

3R

aR bR

cR

1V

+

–

1I

2V

+

–

2IaR

bR

cR

1R

2R

3R

Experiment 5 53

EXPERIMENT 6 — COMPUTER SIMULATION OF DYNAMIC SYSTEMS

I. Objective

♦ To introduce some principles of analog and digital simulation of dynamic systems.

II. Theory II.1 Introduction

Before an engineering system or process is implemented, an extensive computer simulation is often carried out to ensure that the system will function as predicted. This approach is much more cost effective than building and testing of a prototype. In order to simulate a complex system, several simpler subsystems and their connections are first identified. Each subsystem is then described by appropriate equations (mathematical model).

Once reduced to mathematical relationships, the system simulation can be carried out in a variety of ways. For example, the behaviour of a mechanical system can be simulated by an electronic network described by the same equations (for instance, a resonant circuit is an electrical equivalent of a mechanical spring-mass-damping system). In the past, such simulations were carried out by a collection of versatile electrical networks called collectively an analog computer.

Presently, most of the simulation is being done by digital techniques, although in some cases, specialized analog computers are still being used. We will discuss analog simulation techniques as a convenient way to introduce digital simulation.

II.2 Analog Simulation

The basic building blocks of an analog computer are the summer-inverter and summer-integrator-inverter as introduced in Experiment 3 (circuits in Fig. 3.3b and d), and shown symbolically in Fig. 6.1.

II.2.1 First-Order System

Let us construct a circuit to simulate a first-order system described by the transfer function

oo

o

sK

sG

sXsYsH

ωωω

+=

+==

)()()( (6.1)

where oo GKG ωω =,, are constants.

Cross-multiplying in Eqn. (6.1), we obtain

XKYYs o =+ ω

Applying the inverse L transform and regrouping terms, we obtain a mathematical form

Experiment 6 54

suitable for simulation given by

Kxyy o =+ ω

( )yKxyKxy oo ωω −−=+−=− (6.2a)

∫=− dtyy (6.2b)

Eqn. (6.2) can readily be implemented using building blocks from Fig. 6.1 to obtain the simulator shown in Fig. 6.2a. By combining functions of the summer and integrator, the simulator can be simplified as shown in Fig. 6.2b. Actual implementation of the simulator leads to the circuit of Fig. 3.4 discussed in Experiment 3.

II.2.2 Second-Order System

The transfer function of a second-order system is given by Eqn. (2.5), that is

bass

KsXsYsH

++== 2)(

)()( (6.3)

where 2oGK ω= , oa ζω2= , 2

ob ω=

Coefficients G, oω and ζ have the physical interpretations discussed in Experiment 2. Proceeding in a similar way as for the first-order system, we obtain

[ ]Kxbyyay −+−= (6.4)

Eqn. (6.4) can be implemented by a summer followed by two cascaded integrators as shown in Fig. 6.3a. Combining, as before, summing functions with integration, a simplified simulator can be obtained, as shown in Fig. 6.3b.

II.3 Digital Simulation

The simulation of continuous systems can be carried out by digital means. The basic difference between analog and digital simulation (and source of error) is that digital processing can only be performed on sampled signals ( ) nxnTx = where T is the sampling period. The shorter the sampling period or the more frequent the samples, the better the representation of an analog signal by its sampled version. On the other hand, the cost of processing increases with the density of sampling.

The basic simulation building block is the integrator. An analog signal can be integrated using its samples by several possible algorithms. The simplest are algorithms based on the well known rectangular and trapezoidal integration.

Experiment 6 55

The output ny of a digital integrator using rectangular integration is described by the simple recursive formula

11 −− += nnn Txyy (6.5)

where 1−ny designates a previous (delayed by T) sample. The implementation of (6.5) is illustrated in Fig. 6.4.

We can simulate analog systems using analog simulator diagrams by replacing analog integrators with corresponding digital versions. For example, the first-order system of Fig. 6.2b can be modified for digital simulation as shown in Fig. 6.5. The recursive formula describing this simulator can be obtained from Eqn. (6.5) by replacing nx by

nonn yKxz ω−=

that is

[ ] ( ) 11111 1 −−−−− +−=−+= nnononnn TKxyTyKxTyy ωω (6.6)

There is a technique which allows us to derive a recursive formula directly from the transfer function of an analog system to be simulated by digital means. This technique is based on the so-called Z transform which is a linear transform and has the following property

( ) kn zzKXKxZ −

− =1 (6.7)

The Z transform has a unique inverse given by

( ) knk KxzzKXZ −

−− =1 (6.8)

Symbol kz− represents a k-unit-delay operator. Applying the Z transform to both sides of Eqn. (6.5), we obtain

11 −− += TXzYzY

or

11 1

1

−=

−= −

−

zT

zTz

XY (6.9)

Eqn. (6.9) is a different form of recursive formula simulating an analog integrator described by the transfer function 1/s.

Experiment 6 56

A recursive formula for any analog system described by transfer function H(s) can be obtained by the following steps:

1. Replace s in the transfer function by ( ) Tz /1− . 2. Perform all required algebraic manipulations to simplify the expression. 3. Apply the inverse Z transform as given by Eqn. (6.8) to obtain the desired recursive

formula.

We shall illustrate these steps by considering a first-order system with transfer function given by Eqn. (6.1). We can write

os

KXYsH

ω+==)(

or XKYYs o =+ ω

and on replacing s by ( ) Tz /1− , we have

XKYT

zY o =+− ω1

or ( ) XKTYTYz o =−+ 1ω

Multiplying both sides by 1−z , we obtain

( ) 111 −− =−+ XKTzYzTY oω

Performing the inverse Z transfer, we obtain the recursive formula

( ) 111 −− +−= nnon KTxyTy ω

which is identical to that in Eqn. (6.6). Similar developments hold for the second-order system described by Eqn. (6.3)

III. Procedure

1. Select values for oω , T and K and program an available computer to implement the recursive formula (6.6) simulating a first-order system. Apply a sampled unit step as input sequence xn and observe output yn. Vary T and observe its effect on the output. Compare yn with the response of the analog system.

2. Derive a recursive relationship to simulate a second-order system with transfer function given by Eqn. (6.3).

3. Repeat point 1 for the recursive formula derived in point 2.

Experiment 6 57

(a) Summer-Inverter

(b) Summer-Integrator

Fig 6.1: Building blocks of an analog computer

(a) Simulation of a first-order system

(b) Simplified simulator

Fig 6.2: Simulation of Kxyy o +−= ω

( )21 bxaxy +−=1x

2x

ya

b

1x

2x

( ) ( )

+−= ∫∫

tt

dxbdxay0

20

1 ττττ

y

a

b

x K

oωy− 1 y− y−

1

x

y−oω

K

Experiment 6 58

(a) Simulator for a second-order system

(b) Simplified simulator

Fig 6.3: Simulation of [ ]Kxbyyay −+−=

Kb y y− y

x

a

x−1

1

1 1

y

1y−b

a

K

y

1y−

x

Experiment 6 59

Fig 6.4: Digital Integrator

Fig 6.5: Digital simulation of the first-order system

× +T Delaynx

T

1−nx

T Delay

1−nTxny

1−ny

×

+

×

Digital Integratornx

K

oω−

nynzno yω−

Experiment 6 60

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