UNIVERSITY COLLEGE DUBLIN

SCHOOL OF ELECTRICAL, ELECTRONIC & COMMUNICATION ENG.

EEEN30150 MODELLING AND SIMULATION

Experiment MS1

SOLUTION OF EQUATIONS BY ITERATION

1. Objective:

To employ the Regula-Falsi or the Newton-Raphson algorithm as appropriate in the solution of

equations.

2. Background Information:

See module notes and handouts for MS0_1 and MS0_2. In general you may solve problems using

just the command line, using loops, using inline functions, using scripts or using MATLAB function

m-files, save where the latter are explicitly called for, which it should be noted is common. All of

your MATLAB function M-files must include help files, version information and appropriate error

detection/correction. You will also, in any event, have to avoid bad practice such as choosing

meaningless names for variables and/or functions, embedding universal numbers into the code,

writing excessively inefficient code or failing to include comments. In printing your report you must

acknowledge that MATLAB employs colour coding, accordingly presentation of MATLAB

function m-files will need to be in colour. Samples of acceptable MATLAB function m-files are

presented in lectures and in Sample Report available from module Brightspace page.

The possible workload for this laboratory comprises: (i) two hour preparation such as downloading

and reading this laboratory description, revising or reviewing relevant module notes, textbooks

(such as Erwin Kreyzig, Advanced Engineering Mathematics) or websites, preliminary thoughts on

choice of second problem which you will solve, gaining familiarity with tools required to complete

report to satisfactory standard, (ii) two hours of this laboratory session where you should be aiming

at the very least to complete the first problem, (iii) eight hours after this laboratory session where

you will need to solve, or complete the solution of the second, more difficult problem and (iv) two

hours preparing a report on your work for submission.

The Teaching Assistants will grade this laboratory. Each TA will need to grade approximately 40

reports, which they should be able to do reasonably quickly. Accordingly, you should receive

feedback for this laboratory. As they will not even be due until very near the end of or after the

teaching trimester, you will not receive feedback for the team minor project MP2 (or the alternative

laboratory MS3). They are treated as examinations and you will learn your grade with the normal

issuing of the examination results. The module coordinator will grade the first solo minor project.

No grades are released until all grades are available.

Some comments on the style and substance of the report are called for. Mostly these comments

apply to all of the reporting for this module and are not specific to this laboratory, so take note. A

similar section regarding reporting will not appear in later handouts. This laboratory requires that

you solve two problems. Your report should therefore consist of two sections, one section for each

problem. The sample report issued would amount to one of those sections, although I have chosen to

divide it into subsections. Your report does not need to and therefore should not contain any

additional sections. Accordingly, it is neither required nor desired that an abstract, introduction or

conclusion be included, very much in contradiction of what you have been previously taught. I do

not repeat sections of the laboratory description. Rather this subsection essentially comprises the

“analysis”, that is to say the conversion of the problem to specific equations (Equations (1) and (2))

which need to be solved. Each problem can be solved by deciding upon or developing an algorithm

and implementing this using a number of commands. Accordingly, one of the most important

components in each section will be a record of the commands employed with good comments

explaining the objective and effect of each command. Remember that this module is primarily about

numerical algorithms not one random piece of software, so these comments are very important. Of

course a second important component in each section will be the presentation of the results. In

general in this module you will have choices as to how you present your results. If you are after the

highest grades which indicate excellence, then you must indeed excel and presentation is one area in

which you can do this. The report must be typed. Scanned hand-written material is unacceptable and

this year, from the outset, will not be graded. This comment is your one and only warning, so take

note. Electronic submission of your report must be as pdf attachment to e-mail sent to

paul.curran@ucd.ie for the purposes of identifying plagiarism and record. Production quality is your

responsibility and so you must take responsibility. Of course you must include your name and

student number to clearly identify yourself and it would be more secure to include this information

on each page either as a header or as a footer. Your report may well include equations. You should

employ a proper equation editor and you should employ proper mathematical notation. Visualisation

and presentation are important parts of this module and this extends to the reports.

It is not the job of either the module coordinator or the teaching assistants to edit your report.

Students can at times make de facto editing demands of their assessors by adopting the policy of

including everything on the off chance that it might be important. Appropriate concision is a very

important skill to develop. You will impress almost nobody by being unnecessarily long-winded and

you will be comprehensible to almost nobody by being unacceptably brief. Word counts and page

counts are deemed irrelevant here. You should have as many or as few words and pages as it takes

to effectively communicate all of the necessary information, but you should have no more and no

less. It is clear that in order to write a report to conform to these rather vague guidelines you will

need to determine what in fact is the necessary information. That is exactly as it should be and is in

fact the primary purpose in making these demands upon your editorial skills. It is noted elsewhere

that the most egregious example of poor editing in this module, which results in significant loss of

grades, has traditionally been omission. Many students undertake an analysis but do not report that

analysis or do so only in the form of code. Note yet again: present all of the important equations,

assumptions and arguments of your analysis. Moreover do so in the main body of the report, do

not push them into appendices (of which you should generally have none) or code (which amounts

to reporting through code, an entirely unacceptable practice). Need it even be said that another part

of the editor’s job is to ensure that all spelling is correct? With the widespread availability of spell

checkers incorrect spelling is now both unacceptable and inexcusable. American and English

spelling are both accepted. Extremely long lists of numbers running over many pages, which some

students appear to consider reasonable material and presentation, are indicative in fact of a missing

table. Tabulation or graphing are the appropriate means of presenting such data.

The issue of what code to present and what to withhold is fraught. Unfortunately it amounts, at least

initially to a case of guessing what the module coordinator requires. It is to be hoped however that

the majority can exercise a degree of common sense. So for example if you plot a graph then just

include the plot, do not bother to include the code whose sole purpose was to generate that plot, with

the exception discussed below. If you develop a MATLAB function m-file then the entire file

should be presented in the report (in colour). This file is employed in much the same way as one

would employ a graph. It amounts to your evidence. You would not present a graph without any text

commenting upon it and observing for the reader some of the more important information contained

within it. The same is true of the MATLAB function m-files which you present. You must actually

write a report. Just presenting code indicates your belief that you may report through code. You may

not. In spite of this clear stipulation, a large minority report through code nonetheless, probably

because they do not understand the phrase “report through code”. Accordingly, let us state it again

in a different manner. If all or almost all of your report is comprised of code then you can expect to

earn almost no grades.

Two additional points regarding code should be made: Firstly it is important in code development

that when there is a standard you write to that standard. Whereas in open source code such as

Python there is no universally accepted standard, it being to some degree a matter of taste, in

MATLAB code there is a standard since the code is owned by a single entity, Mathworks.

Accordingly, the standard for how a MATLAB function m-file should be written is the standard

used by Mathworks. Secondly a method for writing millions of lines of bug-free code is to not write

millions of lines of code. One way to achieve this is to modularise the code. Individual modules,

which essentially equate to MATLAB function m-files in MATLAB, should be simple but flexible.

Simplicity usually means that individual modules of code do not contain many lines of code and can

be fully debugged. Simplicity generally requires that a module of code should do just one job. An

error is to write MATLAB function m-files which do too many jobs, where in fact you should be

writing several MATLAB function m-files. Flexibility means that whereas the module does just one

job it can be used, usually through changing input arguments, to solve a variety of related problems.

When done correctly your core modules of code will be executed many times during the solution of

a problem. I refer to this as “reusing not rewriting”, although the more common phrase in Computer

Science is DRY (don’t repeat yourself). When a MATLAB function m-file is being used perhaps

hundreds of times in your solution of a problem, that m-file is code that I will wish to see. This is

the exception mentioned above. If a core module is repeatedly used to produce a series of plots or a

video, then I will wish to see that code even though I will also wish to see the main outcome of that

code, namely the plots or the video. Emphasis is on the word repeatedly.

Another example of inappropriate reporting occurs if you execute an identical line of code forty

times and subsequently print it forty times. All you need do is to print it one time and to report that

it was executed forty times. A “thought experiment” which you can attempt to perform to stop

yourself from doing something rather silly is “exaggeration”. Suppose that the amount of data which

you had was a thousand times what you do in fact have. Would you list thousands of pages of

numbers or print thousands of repetitions of an identical piece of code? At what point would it occur

to you to call a halt and realise that this has become entirely self-defeating and that some other

means of presentation ought to be sought? The purpose of the exaggeration thought-experiment is to

realise the need for this before much work on inappropriate presentation has been undertaken.

Within the rather wide parameters set by these guidelines you may make your own decisions

concerning the report. Some will wish, as other modules will have demanded of them, to have

introductory and conclusory sections, i.e. sections both titled “Introduction” and “Conclusion” and

containing the kind of material normally present in such sections. As noted previously their

inclusion will be accepted, although the preference is that they be excluded since the majority of

conclusions are pointless and irrelevant and the majority of introductions amount to wholesale

repetition of the laboratory description and as such border upon plagiarism.

You must include with the report (at the front) a (digitally) signed cover sheet declaring that you

have read and understood the university policy on plagiarism. Failure to do so will result in an

automatic deduction of one grade step. Actually many of you will not have read this policy and

many more will not have understood it. The principle infringements likely in this module are

copying (mostly from other students) without citing. Downloading of large sections of code written

by others, copying large pieces of text from websites or from each other, these things do not

comprise plagiarism. Plagiarism only arises when you do these things without clearly stating in the

report that you have done them and giving sources for the material so that an assessor can determine

for themselves how much of your work is original and how much is just copied from others.

Obviously if almost all of your work is copied then it is not in fact your work and you have not

earned a good grade. It is a widely adopted practice to present copied text in italic and in quotes for

example, since indeed a quote is nothing more than copying, although it is not plagiarising.

Very many of you will effectively work in groups. This is acceptable and indeed encouraged to

some extent, although obviously not to excess. For those who struggle with the material a

reasonable level of help from a colleague will be most beneficial. For those who take to the material

well the exercise of attempting to explain it to others will greatly enhance both your understanding

and your reporting. When it comes to writing the report however the work should become your own.

Best practice is to say farewell to those with whom you have collaborated up to this point and

undertake the last task of reporting entirely on your own. In the event of obvious copying wholesale

of sections of report all students involved will suffer a loss of grades. No attempt will be made to

identify the content-creator. Allowing your report to be copied is equal to copying and suffers equal

penalty. If copying reaches extreme levels then the matter will be referred to the plagiarism

committee of the School of Electrical and Electronic Engineering for their adjudication and, where

appropriate, determination of penalty.

To finish on a positive note I reiterate point made in introductory slides. If you have questions

please ask. To those who are making an effort I promise that I will gladly support you in that effort.

Laboratories will be primarily run by the Teaching Assistants this year. Part of the reason for this is

to free up significant time on my part to engage via e-mail or small group zoom sessions with any

questions that you may have.

3. Experiment:

3.1. Required problem:

Problem: Clapeyron’s ideal gas law, as the name suggests, holds for ideal gases only where

there is essentially no interaction between atoms and molecules. The Redlich-Kwong equation

comprises a more accurate relationship for real gases. It is commonly written in one of two

equivalent forms (and it must be said that notation is not consistent across various accounts). We

have either:

and where the largest root of the cubic equation is taken. The parameters Tc and Pc are the

temperature and pressure at the critical point. Values are widely available for a variety of

substances. Find the critical temperature and pressure for Carbon Dioxide gas (CO2). In these

equations R is the ideal gas constant, P is the pressure, T is the temperature and v is the molar

volume, i.e. the volume occupied by 1 mole of the gas at the given pressure and temperature. Show

that the two forms of the equation given are indeed equivalent. By implementing either the RegulaFalsi

or the Newton-Raphson algorithms develop a MATLAB function m-file to find the molar

volume (in L/mol) of Carbon Dioxide (CO2) at a temperature of 293 K for pressures of 1 atm, 1.5

atm, 2 atm, 2.5 atm, 3 atm, 5 atm, 10 atm, 15, atm, 25 atm, 50 atm and 100 atm. Compare these

predictions with those of the ideal gas law:

Pv = RT

. You may find a numerical value for the ideal

gas constant R on the internet, but be careful with units. You should make this comparison using

good visualisation. For example plot the ideal molar volume and the calculated more realistic molar

volume on the same axes using the hold on command. You should also undertake to avoid the

inclusion of obvious machine artefact. Consulting online sources, for the given temperature

determine for what range of pressures is the Redlich-Kwong equation of state deemed to be

accurate.

(9 grade steps)

3.2. Modelling and solving equations: Applications

Solve one of the following problems or alternatively a problem of your own choosing (provided it

has been approved by the module co-ordinator). Note carefully that core code in any case must take

the form of a MATLAB function m-file. A correctly reported problem solution will generally have

three components: (i) mathematical analysis of problem (either setting up of equations or

confirming equations given); (ii) numerical analysis of problem (implementing algorithm as

MATLAB function m-file to find approximate numerical solutions to equations; (iii) presentation of

results (commonly in the form of graphs or tables). Please take note. This is a numerical methods

module. You are to solve problems numerically using methods introduced in module, not

analytically, not using algorithms found from other sources (cited or otherwise) and not using builtin

MATLAB functions.

(12 grade steps)

Problem 3.2.1:A horizontal handrail is securely fixed at both ends. Make your own decision as to

what might comprise a reasonable material for the construction and thus find suitable values for the

Young’s modulus of elasticity and the density. The rail has an annular cross section giving a second

moment of area about either axis within the plane of the cross section of ( )

where r1

and r2 are respectively the inner and outer radius. By considering some standard sizes of handrails

determine appropriates values for the length L (between fixed points), the inner radius r1 and the

outer radius r2 of the rail. Obviously there is a significant level of approximation here as typical rails

are fixed (or pinned) at multiple points (as shown in Figure 1) and treating a single section between

fixed points as if it was unconnected to the rest of the rail is inappropriate. Nevertheless that is what

you may do on this occasion. We will model the rail (or section of rail) using the Euler-Bernoulli

beam equation. As the rail is fairly light we will ignore loading. Accordingly the free vibrations of

the rail are very approximately described by the partial differential equation:

where y is the transverse deflection in the vertical direction (as shown in Figure 1), m is the mass per

unit length, E is the Young’s modulus of elasticity and I is the second moment of area about an axis

perpendicular to the direction of deflection, being the same for every such axis as above. The fact

that the rail is fixed at both ends, i.e. we are just modelling the section of the rail between adjacent

supports, the following boundary conditions are applied:

It is possible to solve this partial differential equation using the method of separation of variables.

This method offers the following oscillatory solution:

y(x,t) (c sin(? x) c cos(? x) c sinh(? x) c cosh(? x))(c sin(?t) c cos(?t)) = 1 + 2 + 3 + 4 5 + 6

where c1, … , c6 and ? are constants of integration which are determined by appropriate boundary

and initial conditions.

y(x,t)

x x = 0 x = L

Figure 1: handrail

By substituting this formula back into the partial differential equation confirm that it is indeed a

solution, provided the parameters ? and ? are related by:

m

EI 2 ? = ? .

Interpret the formula for y(x,t) by showing that the frequency of vibration of the corresponding

deflection of the beam is ? in rad/sec.

By using the boundary conditions derive the constraints on the constants of integration c1, c2 , c3 , c4

and ? ? These constraints comprise a system of four simultaneous linear equations in the four

unknowns c1, c2 , c3 , c4 . Clearly the would-be solution offered by the method of separation of

variables is only non-zero if at least one of these unknowns c1, c2 , c3 , c4 is non-zero. By employing

the theory of determinants show that this occurs if and only if

cos(? L)cosh(? L)?1= 0.

Implement an algorithm of your choosing (Regula Falsi or Newton-Raphson) to determine the

smaller solutions ? of this equation, presenting an account of the code used. Find the lowest nine

frequencies of oscillation (free vibration) of the handrail in Hz (cycles per sec).

You do not have to use it, but there is an obvious scaling which can be employed here.

Problem 3.2.2:When water flows through a pipe there is friction between the water and the inner

surface of the pipe. One method of modelling the effect of this friction is to compute the resulting

head loss. The Darcy-Weisbach equation expresses the head loss due to friction:

where L is the pipe length, Dh is the hydraulic diameter, V is the average velocity of the fluid flow

being equal to the volumetric flow rate per unit cross-sectional wetted area, g is the local

acceleration due to gravity and fD is a dimensionless coefficient called the Darcy friction factor. One

must be careful not to confuse the Darcy friction factor with the Fanning friction factor f. The two

are related by the rather trivial relation:

f f D

= 4 .

Rather strangely the hydraulic diameter is defined to be four times (not twice) the hydraulic radius.

This is due to the curious nature of the definition of the hydraulic radius. The hydraulic radius is

defined as:

P

A

Rh =

where A is the cross-sectional area of the flow normal to the direction of the flow and P is the wetted

perimeter, i.e. the length of the perimeter in contact with the water. The cross-sectional wetted area

is the cross-sectional area of the pipe which is wetted, i.e. in contact with water. This is equal to A.

For a pipe of circular cross section of internal diameter d where the flow completely fills the pipe

the hydraulic radius is given by:

So the hydraulic radius comes out as one quarter of the internal diameter in this case. The hydraulic

diameter is defined so that it is equal to the actual internal diameter in this case, i.e. as four times the

hydraulic radius. In this respect it is the definition of the hydraulic radius which is strange, being

essentially half what it arguably could be.

If rather than head loss one requires the pressure loss due to friction then the standard formula

?p = ? ghf

may be employed, where ? is the density of the water.

To use the Darcy-Weisbach equation one must know the Darcy friction factor. There are two

methods for determining this. One is graphical. A Moody chart is an experimentally derived plot of

Darcy friction factor vs Reynolds’ number (Re) for a variety of relative roughness values (? /D) and

flow regimes assuming fully developed flow in a circular pipe. Alternatively for laminar flow (i.e.

very low Reynolds’ number) Poiseuille gives the formula:

Re

64 f D

=

whereas for turbulent flow (i.e. high Reynolds’ number) Colebrook gives the more difficult formula:

For relative roughness of 0.001, 0.002, 0.003, 0.004, 0.005, 0.007, 0.01, 0.02 and 0.05 employ the

Colebrook equation to determine the Darcy friction factor for a range of Reynolds’ numbers from

400 to 500,000,000, presenting your results in the manner of a Moody chart.

A circular pipe has a radius 1.2 m, length 40 m and a relative roughness 0.007. Water flows through

the pipe. The flow is turbulent, with a Reynolds’ number of 1,800,000, but it is uniform, meaning

that the depth of the water is constant along the pipe. The depth of the flow is 175 cm and the

volumetric flow rate is 12.5 m3

/sec. Use the standard value g = 9.81 m/s2

.

Calculate the Darcy friction factor for the given Reynolds’ number and relative roughness.

The depth of the flow is larger than the radius of the pipe. The situation is as depicted in Figure 2.

q r

y

h

Figure 2: water in circular pipe of depth y exceeding radius

Show that. Hence show that the wetted perimeter

is the cross sectional area of the water normal

to the flow. Hence determine the value of the hydraulic diameter and the average velocity of the

flow. You may now employ the Darcy-Weisbach equation to determine the head loss. What is the

associated loss of pressure?

Problem 3.2.3:An electrical wire is 1 m long and has a circular cross section with cross-sectional

area of 70 mm2

. The wire is made of copper which has a resistivity ? = 16.78 n?·m at 20oC and a

thermal conductivity kwire = 401 W/m·K. The wire is enclosed in an electrically insulating sheath

made of flexible PVC. Flexible PVC has a thermal conductivity of about kPVC = 0.15 W/m·K and

begins to decompose at about 140oC. The insulated wire is surrounded by air at an ambient

temperature of 20oC. The dominant mechanism of heat transfer in the air is convection with a heat

transfer coefficient of h = 25 W/m2K. Making the unrealistic assumption that all physical

parameters remain constant over the range of temperatures involved and assuming a DC current of

615A is flowing in the wire, what is the minimum acceptable thickness of the PVC sheath such that

the plastic does not begin to decompose?

Assuming steady-state has been achieved with the surface temperature of the wire at T1, ignoring

edge effects occurring at the ends of the wire and assuming the dominant mechanism of heat

transfer within the material to be conduction the temperature distribution may be described by the

steady-state heat equation with a heat generation term. In polar co-ordinates this is:

with the heat generation term: ???????? ? =????2????W/m3

where R is the resistance of the wire, l is the

length of the wire, A is the cross-sectional area of the wire and i is the current flowing through the

wire.

This equation may be solved giving the general solution: ??(??) = ??1????(??) ????????????????????(??24) + ??2 for

arbitrary constants of integration C1 and C2.

By substituting this formula into the ordinary differential equation show that it is indeed a solution.

We have one explicit boundary condition: ??(??????????) = ??1 but we have a second implicit boundary

condition: ??(0) finite. Show that these boundary conditions lead to the solution:

The outward heat flow at the surface of the wire is: ??? = ?????????????????????

????

????

at ?? = ?????????? where Awire

is the surface area of the wire. This heat flow equals the heat flow into the PVC insulating sheath

through the inner surface. Show that:

??? = ????

2

.

Again assuming steady-state has been achieved the surface temperature of the wire at T1 ,the outer

surface temperature of the PVC insulating sheath at T2, ignoring edge effects occurring at the ends

of the wire and assuming the dominant mechanism of heat transfer within the material to be

conduction the temperature distribution in the PVC may again be described by the steady-state heat

equation. In polar co-ordinates this is:

1

??

??

???? (??????????

????

????) = 0 , ??(??????????) = ??1

To reflect the heat transfer across the inner boundary we also have a second boundary condition

??? = ???????????????????

????

????

at ?? = ?????????? where ??? is the heat flow into the PVC as previously calculated.

The general solution of this equation is: ??(??) = ??1

????(??) + ??2

where C1 and C2 are constants of

integration.

By substituting this formula into the ordinary differential equation show that it is indeed a solution.

Show that the boundary conditions lead to the solution:

??(??) = ??1 ?

???

????????2????

???? (

??

??????????

) .

Show that the heat flow out across the outer boundary of the cable as given by ?????????????????????

????

????

at

?? = ???????????? is equal to ??? the heat flow in, where Acable is the outer surface area of the cable (i.e. the

wire plus insulating sheath).

Again assuming steady-state has been achieved with ,the outer surface temperature of the PVC

insulating sheath at T2 , ignoring edge effects occurring at the ends of the wire, assuming the

surrounding air at a suitable distance from the sheath to be at temperature ??∞ = 20oC , this being

the ambient temperature and assuming the dominant mechanism of heat transfer within the air at the

surface of the sheath to be convection the heat transfer away from the sheath is given by Newton’s

law of cooling:

??? = ?????????????(??2 ? ??∞).

We seek the value of rcable such that T1 =140oC. Show that in this event the following equations

hold:

??2 = ??1 ?

???

????????2???? ln (

????????????

??????????

)

??? = ?2????????????????(??2 ? ??∞)

??? = ????

2

Hence find rcable. The insulation thickness which is actually sought equals rcable – rwire .

Problem 3.2.4: The npn BJT has been modelled by Ebers and Moll as a pair of ideal diodes,

back-to-back.

n+ p nEmitter

Base Collector IC

IB

IE

Figure 3: npn bipolar junction transistor idealised structure

There are two pn junctions (base-emitter and base-collector). Under the conditions where the

voltages VBE and VCE are positive the base-emitter pn junction is forward-biased whereas the basecollector

pn junction is reverse-biased. Normally the current flowing through a reverse-biased pn

junction should be small. But normally the p-doped base region should have relatively few mobile

electrons. With the base-emitter pn junction forward-biased there can be a massive influx of mobile

electrons from the emitter (where they are in relatively great abundance). The base region, being ptype

notwithstanding, actually has a very large number of mobile electrons and the result is that the

current flowing through the reverse–biased base-collector pn junction is far from small (relatively

speaking). A minor extension of the analysis of Ebers and Moll leads to the following equations for

the collector and base currents:

Ebers and Moll in their original paper establish that

Let this quantity equal IS hence

where

Kirchoff’s voltage law yields an additional equation concerning the voltages:

VCE = ?VBC +VBE

In these equations k = 1.3806503 x 10-23 m2kg s-2 K-1

is Boltzmann’s constant, q = 1.60217646 x

10-19 C is the magnitude of the electron charge, T is the temperature in K. The quantity VT = kT/q is

called the thermal voltage at temperature T. As is reasonably common in electronics we will

perform the analysis for a temperature of 300 K. Model parameter IS is the reverse saturation

current. Model parameter VA is called the Early voltage. It models the Early effect whereby as the

collector-base voltage becomes more positive the depletion region of the base-collector pn junction

becomes wider. The result is that the base region effectively becomes narrower. As a result fewer of

the mobile electrons injected from the emitter recombine in the base region, so that there are more

of them to cross into the collector giving rise to an increase in the collector current. Model

parameters ?F and ?R are the forward and reverse common-emitter current gain respectively.

The npn BJT is employed in the very basic amplifier circuit shown in Figure 4. The collector supply

voltage VCC = 15 V. The base bias resistor RB = 22 k?. The load resistor RL = 1.2 k?. The base

bias voltage E = 1.5 V. The input signal e is sinusoidal being Asin(?t). The amplitude of the input

signal is A V. The frequency of the input signal is ? rad/sec. Assume the input signal frequency to

be 1.2 kHz. Model the npn BJT by the modified Ebers-Moll equations above with the model

parameters IS = 0.85 pA, VA = 120 V, ?F = 250 and ?R = 8. Assume the amplitude, A, of input signal

e is 20 mV. Write down the equations which describe the circuit and hence by employing any

numerical algorithm from the module notes you choose (Regula Falsi or Newton-Raphson)

determine the output voltage Vo appearing across the load resistor RL (as indicated in Figure 4) for

several (at least three) cycles of the input signal. To facilitate comparison, plot the output signal Vo

and the input signal e(t) vs time t for several cycles on the same axes. Hence deduce whether

amplification has indeed occurred and determine the gain. For the purposes of this example we will

define the gain as:

Peak - to - peak variation of input signal

Peak - to - peak variation of output signal Gain = .

RB

VBE

IB IC VCE

RL

VCC

E e

Vo

Figure 4: basic amplifier circuit

Problem 3.2.5:A model of the space shuttle is shown in Figure 5. Thrust is provided by the two

solid rocket boosters and the three liquid-fuelled orbiter engines. The centre of mass of the fully

loaded craft (external tank, boosters and orbiter) is shown in Figure 5 and marked as G. At lift off

the orbiter engine thrust is directed at an angle q. NASA offers the following information:

External tank: 78,000 lb (empty), 143,000 US gallons of liquid oxygen, 383,000 US gallons of

liquid hydrogen. Find the total mass (in kg) of the external tank at lift off.

Solid rocket booster: 185,000 lb (empty), 1,100,000 lb fuel. Find the total mass (in kg) of the two

solid rocket boosters at lift off.

Orbiter: 165,000 lb (empty), payload 65,000 lb (including fuel). Find the total mass (in kg) of

the orbiter at lift off.

The combined thrust from the solid rocket boosters is

23.6 10 N

6

TB

= ?

. The combined thrust from

the three orbiter engines is

5.03 10 N

6

TS = ?

. Clearly any residual moment about the centre of mass

of the craft will cause it to rotate about G to catastrophic effect.

Set up an equation for the net moment about G at lift off as a function of q. What is the required

angle q at lift off such that the net moment about G is zero?

Research fuel usage and thereby determine the mass of the external tank when each of the solid

rocket boosters has used up 25% of its fuel. Making the incorrect assumption that the attitude of the

shuttle is unchanged (i.e. ignore roll and similar manoeuvres and assume a simple vertical flight

path) determine the new location of the centre of mass and the new required angle q such that the

net moment about this new centre of mass remains zero. You may assume that the loading and use

of fuel is such that that the centre of mass does not shift in the vertical (i.e. longitudinal) direction.

Solid rocket booster External tank Orbiter G

q

Ws WB

TB TS

17.31''

28 ft

38 ft

Figure 5: Model of Space Shuttle

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