A Be Midterm Spring Key

8/6/2019 A Be Midterm Spring Key

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Midterm Exam ABE425

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NAME:__________KEY_____________

1 (4) What is the Gibbs phenomenon, make a small drawing to illustrate it.

The Gibbs phenomenon shows that while approximating a function with a jump discontinuity (

for instance a square wave) by a Fourier Series, at the discontinuities the approximation forms

spikes for go to a finite limit.

2 (3) Explain the difference between a measurand (give an example) and a measurement

A measurand is the physical property that is measured (for instance temperature) and a

measurement is a numerical representation of the value of that measurand.

3 (6) Given is the Laplace transform of a sine function: 2 2sina

L at

s a

Also given is the Laplace transform of a derivative:

0df t

L sF s f dt

Use these two equations to calculate cos L at

2 2

2 2

sin cos

sin cos sin 0 cos

cos

cos

dat a at

dt

d L at L a at aL at

dt

aaL at s

s a

s L at

s a


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Midterm Exam ABE425

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4 (15)a) Explain what takes place when one calibrates an instrument.

Calibration means that an instrument of lesser accuracy is compared to an instrument of higheraccuracy and, if needed, adjusted so that they give the same results.

b) Explain the concept oftraceability in metrology (measurement science)Traceability shows that a measured value can be traced through calibration documentation to

through higher echelons to the highest authority in the land: in the USA this is the NIST.

Traceability requires the establishment of an unbroken chain of comparisons to stated

references each with a stated uncertainty

c) What does the acronym NIST stand for?National Institute for Standards and Technology.

d) The traditional definition for the Ampere was impossible to realize. Therefore, insteadtwo realizations for Volt and Resistance were developed that soon will become standardsin the metric system. Name one of the two Nobel laureates that laid the foundations for

one of these two standards.

Josephson (Josephson Junction for the Volt), Von Klitzing (Quantum Hall Effect for resistance).

e) Name two of the fundamental problems with the English Unit system.Firstly, the English unit system has many units for the same quantity (for instance length,

weight) which make is cumbersome, but the most fundamental problem is that it has nostandards.

,i i ie y f x


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Midterm Exam ABE425

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5 (16) Shown is a composite signal containing an AC square wave with a peak-peak value of20Volt, superimposed on a DC offset of 10Volt.

a) Calculate the RMS value of the AC componentThe mean value of the squared AC signal is 100 and the RMS is 100 10

b) Calculate the RMS value of the DC componentThe RMS value of a DC signal is the value of the DC signal itself, here 10 Volt.

c) Calculate the RMS value of the composite signal (as shown)The mean value of the squared composite signal is 200, and the RMS = 200 10 2

d) Explain why the sum of the first two a) and b) is not equal to the third c).You cannot add these directly since 2 2 2 210 10 10 2

Composite RMS RMS AC DC

e) Calculate the power that the AC component would produce in a 50 resistor2 210

250

ACUP Watt R

f) Calculate the power that the DC component would produce in a 50 resistor2 210

250

DCUP Watt R

g) Calculate the power that the composite signal (as shown) would produce in a50 resistor

2

2 10 24

50

ACUP Watt R

h) Explain why you CAN add the power produced by the components of the signals.

20V

0V

10V


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Midterm Exam ABE425

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The concept of the RMS value is based on equalized power for AC and DC signals. This means

that if theRMS AC DC value they will produce the same power in a resistor

6 (20) Given is an inverting OpAmp circuit where 1 21 , 10 R k R k and 1C nF. 1 2,Z Z are

impedances related to 1R and the combination 2 ,R C respectively. The transfer function of this

circuit is given as

2

1

O

i

V j Z

V j Z

a) First calculate the impedances:

1 1Z R

2Z

2

2 2 2

22

1

1/ /1 1

R

j C Z R C R j CR

Rj C

1Z


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Midterm Exam ABE425

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Write down the transfer function here:

2

22 2

1 1 1 2

1

1 1

* 1

OUT

IN

R j CRV Z R

V Z R R j CR

b) If the input of this circuit is 500mV DC, what is the output voltage?

For DC you need to set 0 and the output becomes

2

1

10,0000.5 5

1,000OUT IN

RV V V V

R

c) Calculate the corner frequency of this filter in Hz.The corner frequency ( in radian per second) is the reciprocal value of the time constant.

9 5

2 10,000*10 10 secR C

5

5

110

sec

10

15.912 2

C

C

C

rad

f kHz

d) Draw the Bode plot of this active filter


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Midterm Exam ABE425

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Midterm Exam ABE425

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7 (4) The foundation for the Ordinary Least Squares (OLS) approach is the following equation:

,i i i f x y e

a) Identify the model, the error and data vectors (draw arrows).

b) Explain what a parsimonious model is.A parsimonious model is a model that explains most of the data with the most compact model.

8 (6) In Fourier series, the OLS expression 1

T TA A A y

becomes much simpler:

2

2

2

T

N

A yN

N

. Here the Fourier coefficients are simply the inner products between

the basis functions and the data vector.

a) Explain the reason why the estimator equation becomes so simple in Fourier series.The model becomes so simple because the basis functions form an orthogonal function set.

The Fourier series on the interval 0,2 are given as follows:

0

1

2

0

0

2

0

2

0

cos sin

1

2

1cos

1 sin

n n

n

n

n

f t c a nt b nt

c f t dt

a f t ntdt

b f t ntdt

a) Write down the integrals that show how constant factors 12

and1

came into being.


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Midterm Exam ABE425

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2

0

2

0

2

0

2 1

cos *cos

sin *sin

dt

nt ntdt

nt ntdt

b) Explain in words what these integrals represent in terms of the basis functions.The factors are in fact the inner products of the basis functions with themselves.

9 (10) shown is a light emitting diode (LED).

The LED takes about 10 mA to light up properly. The

voltage drop at the top is 5V. The LED drops about 1.5 volt

in forward bias (this is when it lights up).

a) Calculate the value of the resistor in series with theLED in order to obtain the 10mA in the branch.

11 1 1 1 2

1

5 1.5350

10

RR

U VU i R R

i A

b) If 2 500R , calculate the total current i flowinginto the circuit.

22

2

1 2

510

500

10 10 20

RU V

i mAR

i i i mA mA mA

5V

2R 1R

1i 2i

i


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Midterm Exam ABE425

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10) (10)

Shown is a sensor on the left that drives an OpAmp. The OpAmp itself drives a load. Using this

drawing, explain the concept of input resistance INR and the output resistance OUTR and explain

in detail what the ideal values are in order to avoid loading errors.

IdealINR :

IdealOUTR :

Explanation (hint: write out the voltage divider equation)

Ideally you want to measure the value ofINU at U

. The resistances form a voltage

divider as follows:1

1

IN IN IN

OUT IN OUT

IN

RU U U

RR R

R

. To make the factor

1

1 OUT

IN

R

R

unity, either 0OUT

R orIN

R . An OpAmp attempts to achieve both.

INR

OUTR

OUTR

OUTR

INR

INR

U

U

INU


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Midterm Exam ABE425

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11) (4) Imagine that you have to drive a device of which you know that he input impedance is

low. What can you do in this case to avoid loading errors?

Place a buffer (voltage follower) in between the driver and the circuit being driven.

12) (4) a) Name the component shown on the right.

This is a variable resistor, or potentiometer (sometimes called rheostat)

b) What is the nominal value? 1k


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Midterm Exam ABE425

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A Be Midterm Spring Key

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