ECE 20200 : Linear Circuit Analysis II Summer 2012ee202/pastexams/M2.pdfPurdue University School of Electrical and Computer Engineering ECE 20200 : Linear Circuit Analysis II Summer

Purdue UniversitySchool of Electrical and Computer Engineering

ECE 20200 : Linear Circuit Analysis IISummer 2012

Instructor: Aung Kyi San

Midterm Examination IIJuly 19, 2012

Instructions:

1. Wait for the “BEGIN” signal before opening this booklet.

2. Enter your name, student ID number, e-mail address and your full signature in the spaceprovided on this page.

3. You have one hour to complete all 5 questions contained in this exam. When the endof the exam is announced, you must stop writing immediately. Anyone caughtwriting after the exam is over will get a grade of zero.

4. Read the questions carefully. Unless otherwise stated, you must fully justify your answers.You may use any method you want unless you are asked to use a specific method.

5. This booklet contains 16 pages including the Tables. Since only this booklet will be graded,make sure you have all your answers written in this booklet.

6. Notes, books, calculators, cell phones, pagers and any other electronic communicationdevice are strictly forbidden.

Name:

Student ID:

Email:

Signature:

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ECE20200, Summer 2012 EXAM II (Aung Kyi San)

(Total 20 pts) 1.

The pole-zero plot of the transfer function H(s) is given in the figure below.

(a) (4 pts) Find H(s) if the dc gain is −1.

(b) (6 pts) Compute the step response s(t). Identify the transient and steady-state response.

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(c) (6 pts) If the input x(t) is e−atu(t), find the positive number a such that the response y(t)does not have a term of the form Ke−atu(t). Find the zero-state response under this condition.

(d) (4 pts) Is the system described by the transfer function H(s) BIBO stable? Give reason.

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(Total 20 pts) 2.

(a) (5 pts) Sketch the Bode magnitude plot of H(s) =(s + 10)

(s + 1)(s + 100). Label the vertical axis

carefully.

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(b) (5 pts) The circuit given below is to be frequency and magnitude scaled so that the value ofH(s) at s = j2 becomes the value of Hnew(s) at s = j100 and the final value of the capacitor isto be 1 mF. Find the new final values of the resistor, Rnew (in Ω) and the indcutor, Lnew (in H).

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(c) (6 pts) Suppose H(s) =2(s + 2)(s− 2)

(s + 4 + 2j)(s + 4− 2j)=

2(s2 − 4)

(s + 4)2 + 4.

Find the sinusoidal input vin(t) that gives rise to a steady-state output vout(t) = −cos(2t)V.

(Hint: −cos(2t) = cos(2t + 180))

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(d) (4 pts) The following is the magnitude frequency response of a transfer function H(s).

Which of the following is the best candidate for H(s)? Justify your answer.

(i)(s2 + 2)

(s + 1)2

(ii)(s2 − 1)2[

(s + 1)2 + 1]

(iii)(s2 + 1)2[

(s + 0.1)2 + 1]2

(iv)(s2 + 10)2[

(s− 0.1)2 − 1]2

(v)(s + 1)4[

(s + 0.1)2 + 1]2

(vi)(s2 − 1)2[

(s + 0.1)2 + 1]

(vii)(s2 − 1)2[

(s + 0.1)2 + 1]2

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(Total 20 pts) 3.

(a) (10 pts) Find the convolution of x(t) = e−5tu(t− 1) and h(t) = 5u(t).

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(b) (10 pts)

If y(t) = f(t) ∗ f(t), then find

(i) y(4.5)

(ii) y(7)

(iii) y(-2.25)

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(Total 20 pts) 4. The following circuit is more or less a band pass circuit whose form is toocomplex for easy analysis because it contains lossy capacitor and lossy inductor. L = 0.1 H,C = 0.1 F, R1 = 0.05 Ω, R2 = 0.1 Ω, R3 = 10 Ω.

(a) (4 pts) Find the inductor-Q, QL and capacitor-Q, QC at ω0 =1√LC

.

(b) (4 pts) Next, showing your work, construct an APPROXIMATELY equivalent series RLCcircuit as shown below. Specify Rnew, Lnew and Cnew.

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(c) (12 pts) Showing your work, by obtaining the transfer function of the circuit in part (b),compute the approximate values for ωm, Bω, Qcir, the half power frequencies ω1 and ω2, andHm.

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(Total 20 pts) 5.

The transfer function of the Sallen and Key circuit below is Hcir(s) =2

s2 +1

Qs + 1

. This circuit

is to realize HLP (s) =48

s2 + 2s + 144.

(a) (6 pts) Compute the values of ωp and then Q which is needed for the first phase of the designrealization.

(b) (2 pts) Given your (correct) value of Q, specify the values of R1 and R2.

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(c) (4 pts) Compute the appropriate frequency scaled values of C1 and C2, denoted as C1N andC2N respectively.

(d) (8 pts) Use input attenuation to achieve the desired dc gain.

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Original Circuit Exact

Equivalent Circuit at

ω0

Approximate Equivalent circuit, for high Q,

(QL > 6 and QC > 6) and

ω within (1 ± 0.05)ω0


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Table 12.1 LAPLACE TRANSFORM PAIRS

Item Number f(t) LLLL[f(t)] = F(s)

1 Kδ(t) K

2 Ku(t) or K K s

3 r(t) 1 s2

4 tnu(t) n!

sn+1

5 e–atu(t) 1 (s + a)

6 te–atu(t) 1 (s + a)2

7 tne–atu(t) n!

(s + a)n+1

8 sin(ωt)u(t) ω

s2 + ω2

9 cos(ωt)u(t) s

s2 + ω2

10 e–atsin(ωt)u(t) ω

(s + a)2 + ω2

11 e–atcos(ωt)u(t) (s + a)

(s + a)2 + ω2

12 tsin(ωt)u(t) 2ωs

(s2 + ω2)2

13 tcos(ωt)u(t) s2 − ω2

(s2 + ω2)2

14 sin(ωt + φ)u(t) ssin(φ) + ω cos(φ)

s2 + ω2

15 cos(ωt + φ)u(t) scos(φ) − ω sin(φ)

s2 + ω2

16 e–at[sin(ωt) – ωtcos(ωt)]u(t) 2ω3

[(s + a)2 + ω2 ]2


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17 te–atsin(ωt)u(t) 2ωs + a

[(s + a)2 + ω2 ]2

18 e

−atC1 cos(ωt) +

C2 − C1aω

sin(ωt)

u(t)

C1s + C2s + a( )2 + ω2

Table 12.2 LAPLACE TRANSFORM PROPERTIES

Property Transform Pair

Linearity L[a1f1(t) + a2f2(t)] = a1F1(s) + a2F2(s)

Time Shift L[f(t – T)u(t – T)] = e–sTF(s), T > 0

Multiplication by t L[tf(t)u(t)] = –

d

dsF(s)

Multiplication by tn

L[tn f (t)] = (−1)n

dnF(s)

dsn

Frequency Shift L[e–atf(t)] = F(s + a)

Time Differentiation Ld

dtf (t)

= sF(s) – f(0

–)

Second-Order

Differentiation

Ld2 f (t)

dt2

= s2F(s) − sf (0− ) − f (1)(0− )

nth-Order

Differentiation

Ldn f (t)

dtn

= snF(s) − sn−1 f (0− ) − sn−2 f (1)(0− )

−K− f (n−1)(0− )

Time Integration

(i) L f (q)dq

−∞

t

∫

=F(s)

s+

f (q)dq−∞

0−

∫s

(ii) L f (q)dq

0−

t

∫

=F(s)

s

Time/Frequency Scaling L[f(at)] =

1

aFs

a


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ECE 20200 : Linear Circuit Analysis II Summer 2012ee202/pastexams/M2.pdfPurdue University School of Electrical and Computer Engineering ECE 20200 : Linear Circuit Analysis II Summer