Rr320201-analysis-of-linear-systems

Set No. 1

Code No: RR320201

III B.Tech Supplimentary Examinations, Aug/Sep 2008 ANALYSIS OF LINEAR SYSTEMS (Electrical & Electronic Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Distinguish between linear and non-linear systems with suitable examples. (b) Explain the D’Alembert’s Principle with the help of a mechanical rotational system. (c) For the mechanical system shown in figure 1c, draw the mechanical equivalent network. Hence develop the force-current analogous electric circuit and write the equations. [4+5+7]

Figure 1c 2. (a) Explain what is meant by state variable and Mention the advantages of state space approach. (b) Develop the state variable model equations of the network shown in figure 2b using equivalent source approach.

Figure 2b (c) Obtain the state-space representation of the series R-L-C circuit excited by e(t) and the response is i(t). [4+6+6] 3. (a) For the figure 3a shown, find the Laplace transforms of v(t).

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Set No. 1

Code No: RR320201

Figure 3a (b) Find the response of R-L series circuit with R=10Ω L=5H, fed by a shifted unit step function 5u(t-2) [8+8] 4. (a) Find the Laplace transforms function of the periodic waveform shown in figure 4a

Figure 4a (b) State and prove convolution theorem (c) Show that convolution of any function with unit impulse function is the functions itself [6+5+5] 5 11 5. (a) A source of ν(t) = sin πt + 10 sin 5πt− 100 sin 11πt is applied to a R-L series 1 circuit with R = 1Ω and L = π H. Find the average power and power factor of the circuit.

(b) Find the exponential Fourier series for the waveform shown in figure 5b.[8+8]

Figure 5b 6. (a) Find the Fourier transform of the signal F(t)=1.0 for −T1 < t < +T1 = 0 elsewhere (b) If f(t) = Ke−at u(t). Find the Fourier transform of the function F (jω). Compare this with Laplace transform of the given function. 2 of 3

Set No. 1

Code No: RR320201

(c) Find the function V(t) corresponding to the function V(f) shown in figure 6c using inverse Fourier transform. [6+6+4]

Figure 6c 7. (a) Check whether the following polynomial is Hurwitz or not? H(s) = s4 + 2s2 + 3s + 6 (b) Find the range of values of ‘a’ so that H(s) = s4 + s3 + as2 + s + 3 is Hurwitz. [7+9] 8. (a) Synthesize L-C Admittance function into first Cauer form network Z(s) = (2s4 + 20s2 + 18)/s(s2 + 4) (b) Synthesize the admittance function into second Cauer form network Z(s) = (s2 + 2)(s2 + 4)/s(s2 + 3)(s2 + 5) ⋆⋆⋆⋆⋆

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[8+8]

Set No. 2

Code No: RR320201

III B.Tech Supplimentary Examinations, Aug/Sep 2008 ANALYSIS OF LINEAR SYSTEMS (Electrical & Electronic Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) For the figure 1a shown below , draw the mechanical system. And hence write the equilibrium equations.

Figure 1a (b) Draw the electrical analogous circuits for the mechanical system shown in figure 1b. [8+8]

Figure 1b 2. (a) Develop the state equations of the following network: figure 2a.

Figure 2a ˙ (b) Derive the expression to find the solution of the state equations X(t) = A x(t) + B u(t) with x (0) = x0 using state Transition Matrix approach. [8+8] 3. (a) The voltage waveform of a non-periodic function is shown in figure 3a. Find the Laplace transform of the function.

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Set No. 2

Code No: RR320201

Figure 3a (b) Find the current in a series R-L circuit consisting of a resistor R=6Ω, L=3H when fed by shifted ramp functions 3r(t-4) [8+8] 4. (a) Assuming stair case function shown in figure 4a, is not repeated, and is applied to an R-L series circuit with R=1Ω, L=1H, find the current i(t).

Figure 4a s using convolution (b) Find inverse Laplace transform of F (s) = (s+1)(s+2)(s+3) theorem. [8+8] 5. (a) Derive the expression for RMS value of a complex (of voltage) wave which is expressed in terms of fourier series. (b) A complex voltage e(t) = 100 sin w t + 30 sin 3wt + 20 sin 5 wt where w = 100t. If this voltage is applied to a load of 10 ohms in series with 0.01H, find the current, average power and power factor of the circuit. [6+10] 6. (a) Find the Fourier transform of a gate function G(t) = 1 f or − T2 < t < T2 = 0 otherwise (b) Find the Fourier transform of the constant signal f(t) = A(−∞ < t < ∞) [8+8] 7. (a) Test whether the following polynomial is Hurwitz or not? H(s) = s6 + 5s5 + 13s4 + 21s3 + 20s2 +16s+8 (b) Check whether the following function is positive real or not F (s) = (s2 + 4s + 3)/(s2 + 2s)

[8+8]

8. (a) State and explain second Cauer form of synthesizing a network. (b) Synthesize the network in first Cauer form of the function: F(s) = 2 (s+1) (s+3) / s (s+2) 2 of 3

[7+9]

Set No. 2

Code No: RR320201 ⋆⋆⋆⋆⋆

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Set No. 3

Code No: RR320201

III B.Tech Supplimentary Examinations, Aug/Sep 2008 ANALYSIS OF LINEAR SYSTEMS (Electrical & Electronic Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. For the mechanical system shown in figure 1.

Figure 1 (a) Draw the mechanical network (b) Draw the force-current analogous electric circuit, and hence develop the state variable model. (c) Develop the force-voltage analogous circuit.

[3+7+6]

2. (a) Write the state equations using topological method for the network: shown in figure 2a.

Figure 2a (b) In the circuit shown in figure 2b, the inputs are e1 , e2 and outputs are υ1 , υ2 , and Vc. Obtain the state variable model for this system . [8+8]

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Set No. 3

Code No: RR320201

Figure 2b 3. (a) Find the voltage Vc (t) for the circuit shown in figure 3a with the input given in Figure. 3a

Figure 3a

Figure 3a (b) State and explain scaling theorem. (c) The Laplace transform equation for the current is I(s) = i. Find the current I(t). ii. Using scaling theorem, find the current i1 (t) if I1 (s) =

2 (s)(s+2) 2 [7+4+2+3] (3s)(3s+2)

4. (a) Find the Laplace transforms function of the periodic waveform shown in figure 4a

Figure 4a (b) State and prove convolution theorem (c) Show that convolution of any function with unit impulse function is the functions itself [6+5+5] 5. (a) Obtain the trigonometric Fourier series representation of voltage waveform shown in figure 5a 2 of 3

Set No. 3

Code No: RR320201

Figure 5a (b) Find the exponential form of the Fourier series for the following waveform shown in figure 5b [8+8]

Figure 5b 6. (a) Find the Fourier transform of the waveform shown in figure 6a.

Figure 6a (b) Find the Fourier transform of double sided exponential f (t) = Ae−a|t| for−α < t<α [8+8] 7. (a) Check whether the following polynomial is Hurwitz or not? P (s) = 2s4 + 5s3 + 6s2 + 2s + 1 (b) “ All driving point immittances of passive networks are positive real functions”. Substantiate the statement. (c) State the analytical tests to be considered for a polynomial to check whether it is a positive real function or not? [7+5+4] 8. (a) Explain how the removal of pole at infinity of an impedance Z(s) can realize an element in the network. (b) Realize the network with the following driving point impedance function using first Foster form. Z(s) = (s+2) / s(2s+5) [8+8] ⋆⋆⋆⋆⋆ 3 of 3

Set No. 4

Code No: RR320201

III B.Tech Supplimentary Examinations, Aug/Sep 2008 ANALYSIS OF LINEAR SYSTEMS (Electrical & Electronic Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) For the figure 1a shown below , draw the mechanical system. And hence write the equilibrium equations.

Figure 1a (b) Draw the electrical analogous circuits for the mechanical system shown in figure 1b. [8+8]

Figure 1b 2

2. The differential equation of a system is described by ddt2y + 4 dy + 3y = u(t).Obtain dt the complete response of the system to a unit step input with initial conditions: (0) = 1 [8+8] y(0) = 0 and dy dt 3. (a) Distinguish between unit impulse function and unit doublet function and hence develop the Laplace transform of these functions. (b) Find the expressions for the current i(t) in a series R-L-C circuit, with R=5Ω, [3+3+10] L=1H, C= 14 F, when it is fed by a ramp voltage of 12 r(t-2). 4. (a) State and Explain the graphical interpretation of convolution theorem. (b) Determine the convolution integral for the functions (e−2t ) (sin2t) (c) Given that impulse response of a systems is input of e−2t .

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s s+1

, find the response for an [4+6+6]

Set No. 4

Code No: RR320201

5. Find the Fourier series coefficients of the periodic waveforms υ(t) = sint for 0 ≤ t ≤ π with a period π. If this voltage is applied to a series R-L-C circuit with R = 1Ω, L = 1H, C =1F, find (a) RMS value of the applied voltage (b) RMS value of the current wave (c) Average power and power factor of the circuit

[5+5+6]

6. (a) State and explain Parseval’s theorem. (b) Derive the expression for Fourier transform of unit step function.

[7+9]

7. (a) Check whether the following polynomial is Hurwitz or not? H(s) = s4 + 2s2 + 3s + 6 (b) Find the range of values of ‘a’ so that H(s) = s4 + s3 + as2 + s + 3 is Hurwitz. [7+9] 8. (a) State and explain the properties of L-C immittance functions, deriving necessary expressions. (b) Explain the procedure to test whether a function can be represented as an L-C immittance form and hence explain how synthesizing the network is to be carried out in second Foster form. [8+8] ⋆⋆⋆⋆⋆

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