APPLICATION OF

1
CHAPTER 5

• APPLICATION OF
• LAPLACE TRANSFORMS

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Chapter 5LAPLACE TRANSFORMS - DEFINATION

• WHAT IS LAPLACE TRANSFORMS ?
• It is a powerful techniques to model the analog
• systems such as an electrical systems and used
• to solve linear differential and integral
• equations in s-domain. It can be used in
• network stability analysis and network
• synthesis.
• The Laplaces domain ? s
• The Fouriers domain ? j?
• v(t) gt V(s), i(t) gt I(s)
• dv(t)/dt gt sV(s), di(t)/dt gt sI(s)

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Chapter 5 LAPLACE TRANSFORMS - DEFINATION

• Steps in applying Laplace Transform
• 1. Transform circuit from time domain, t to
• s domain.
• 2. Solve the circuit problem either using
• nodal analysis, mesh analysis,
• superposition, source transformation
• 3. Take the inverse transform of the
• solution and obtain the solution in the
• time domain.

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Chapter 5 LAPLACE TRANSFORM CIRCUIT MODELS

• For the basic circuit consists of power supply
  and resistor, the circuit model in time domain is
  described by using Ohms Law as stated below
• v(t) Ri(t)
• Taking the Laplace Transform,
• V(s) RI(s)
• Voltage for Inductor in time domain is
• vL(t) Ldi(t)/dt
• Taking the Laplace Transform,
• V(s) LsI(s) i(0-)
• I(s) 1/sL V(s) i(0-)/s

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Chapter 5LAPLACE TRANSFORM CIRCUIT MODELS

• Laplace transform of inductor from time domain to
  s-domain is as shown in figure (a), (b) (c).

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Chapter 5 LAPLACE TRANSFORM CIRCUIT MODELS

• Laplace transform of Capacitor from time domain
  to s-domain is as shown in figure (a), (b) (c).

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Chapter 5 LAPLACE TRANSFORM CIRCUIT MODELS

• Current for Capacitor is defined in time domain
  is
• i(t) C dv(t)/d(t)
• Taking the Laplace Transform,
• I(s) C sV(s) - v(0-)
• V(s) 1/sC I(s) v(0-)/s
• The Laplace Transform can be used to solve first
  order and second order circuits.
• Initial conditions are part of the
  transformation.
• By using Phasor analysis, the Impedance and
  circuit reactance can be described in s domain as
  
• 1. Z(s) gt V(s) / I(s)
• 2. Resistor gt R
• 3. Inductor gt sL 4. Capacitor gt
  1/sC

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Chapter 5 LAPLACE TRANSFORM CIRCUIT MODELS

• EXAMPLE 1
• Find vo (t) for circuit network in Figure
  16.4 (Page 719) assuming zero initial conditions.

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Chapter 5 LAPLACE TRANSFORM CIRCUIT MODELS

• SOLUTION
• 1. Convert the circuit elements into
  s-domain.
• Voltage source, u(t) ? 1/s
• Inductor ? sL s(1H) s
• Capacitor ? 1/sC 1/s(1/3 F) 3/s
• 2. Perform Mesh Analysis for Mesh 1,
• 1/s (1 3/s)I1 (3/s)I2
• 3. Perform Mesh Analysis for Mesh 2,
• 3/s I2 I1 5I2 sI2 0
• I1 1/3 (s2 5s 3)I2

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Chapter 5LAPLACE TRANSFORM CIRCUIT MODELS

• Continued from EXAMPLE 1
• 4. Substitute I1 into the first equation to
  obtain
• 1/s (1 3/s)1/3 (s2 5s 3)I2
  (3/s)I2
• 5. Multiply the equation by 3s,
• I2 3 / (s3 8s2 18s)
• 6. Calculate Vo(s),
• Vo(s) sI2 3 / (s2 8s 18)
• 7. Now, perform inverse Laplace Transform to
  obtain
• vo(t),
• vo(t) 3/ e-4t sin t V, t
  ? 0

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Chapter 5 LAPLACE TRANSFORM CIRCUIT MODELS

• EXERCISE 1 (Practice Problem 16.2, page 721)
• Find vo (t) for the circuit shown in Figure
  16.9

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Chapter 5 LAPLACE TRANSFORM CIRCUIT MODELS

• SOLUTION
• 1. Convert the circuit elements to s-domain.
• Capacitor ? 1/s(1/4 F) 4/s
• Inductor ? s(1 H) s
• Current Source ? 2 u(t) ? 2/s
• 2. Perform a source transformation to obtain
• voltage source, V(s).
• V(s) 2 (4/s) 8/s
• 3. Now, use voltage division to obtain
  output voltage,
• Vo (s) 4 / (4 s 4/s)(8/s)
• 32 / s(s2)2
• 4. Perform Inverse Laplace Transform and use
  PFE
• method to obtain
• Vo(t) 8 (1 e-2t - 2te-2t ) u(t) V.

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Chapter 5 LAPLACE TRANSFORM CIRCUIT MODELS

• EXAMPLE 2
• Find the vo(t) in the circuit below with
  initial condition, vo(0) 5V.

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Chapter 5 LAPLACE TRANSFORM CIRCUIT MODELS

• SOLUTION
• 1. Convert the circuit elements to s-domain.
• 2. Find the initial condition for current
• source of Capacitor
• Cvo(0) (0.1F)(5V) 0.5A
• 3. Now, perform the nodal analysis
• 10(s1) Vo/10 2 0.5 Vo /10 Vo
  / (10/s)
• Vo (25s 35) / (s1)(s2)
• 4. Perform Inverse Laplace Transform an by
  using PFE
• method to obtain vo(t)
• Vo 10/(s1) 15/(s2)
• vo(t) (10e-t 15e-2t) u(t) V.

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Chapter 5 LAPLACE TRANSFORM CIRCUIT ANALYSIS

• The s-domain will make ease our circuit analysis.
• Just need to convert the circuit network from
  time domain to s-domain and then perform circuit
  analysis as usual. Next, use inverse Laplace
  transform to obtain back the circuit in time
  domain.
• The equivalent circuit with Capacitor Inductor
  only exits in the s-domain, cannot be transformed
  back into the time domain.

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Chapter 5 LAPLACE TRANSFORM CIRCUIT ANALYSIS

• EXAMPLE 3
• Consider the network shown. Find the voltage
  across capacitor by assuming vs(t) 10 u(t) V
  and at t0, -1A current flow through inductor and
  5V across the capacitor.

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Chapter 5 LAPLACE TRANSFORM CIRCUIT ANALYSIS

• SOLUTION
• 1. Perform Nodal Analysis
• (V1 Vs)/10/3 (V1 0)/5s i(0)/s
• (V1 v(0)/s)/1/(0.1s) 0
• 2. V1 (40 5s)/(s1)(s2)
• 35/(s1) 30/(s2)
• 3. Perform Inverse Laplace transform
• vo(t) (35e-t 30e-2t ) u(t), V

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Chapter 5 LAPLACE TRANSFORM CIRCUIT ANALYSIS

• EXERCISE 2 (Exercise Problem 16.52, page 750)
• The switch in network below has been closed
  for a long time. It is opened at t 0. Determine
  vo(t) for t gt 0 using Laplace transform.

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Chapter 5 LAPLACE TRANSFORM CIRCUIT ANALYSIS

• SOLUTION
• 1. Find the vo initial condition at t 0-.
• 3. The output voltage is

2. Applying KCL gives
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Chapter 5 LAPLACE TRANSFORM CIRCUIT ANALYSIS

• EXERCISE 3 (Exercise Problem 16.14, page 750)
• Determine io(t) in the network shown below.

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Chapter 5 LAPLACE TRANSFORM CIRCUIT ANALYSIS

• SOLUTION FOR EXERCISE 2
• 1. Find the initial condition of the circuit
• 2. Perform Nodal analysis

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Chapter 5 LAPLACE TRANSFORM CIRCUIT ANALYSIS

• Continued from Exercise 3
• At node o,

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Chapter 5 LAPLACE TRANSFORM CIRCUIT ANALYSIS

• Continued from Exercise 2

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Chapter 5 LAPLACE TRANSFORM CIRCUIT ANALYSIS

• Continued from Exercise 3

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Chapter 5 LAPLACE TRANSFORM TRANSFER FUNCTION

• Transfer function is key element in the signal
  processing. It can describe the behavior of the
  signals.
• The Transfer function is defined as the ratio of
  the output response Y(s) to the input X(s)
  assuming initial conditions are zero.
• H(s) Y(s) / X(s)
• There are 4 possible Transfer Function
• 1. H(s) Vo(s) / Vi (s) ? Voltage Gain
• 2. H(s) Io(s) / Ii (s) ? Current Gain
• 3. H(s) V(s) / I(s) ? Impedance, Z
• 4. H(s) I(s) / V(s) ? Admittance, 1/Z

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Chapter 5 LAPLACE TRANSFORM TRANSFER FUNCTION

• The inverse Laplace transform of the Transfer
  Function, H(s) will create the unit impulse
  response, h(t).
• Once we know the impulse response h(t) of the
  network, we can obtain the response of the
  network to any input signal in the s-domain, Y(s)
  H(s)X(s) or by using the convolution integral.

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Chapter 5 LAPLACE TRANSFORM TRANSFER FUNCTION

• EXAMPLE 4
• Given the Transfer function of a linear
  system
• H(s) 2s/(s6)
• Find the output, y(t) if the input to the
  system, x(t) e-3t u(t).

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Chapter 5 LAPLACE TRANSFORM TRANSFER FUNCTION

• SOLUTION TO EXAMPLE 4
• 1. Perform the Laplace transform to the input
• signal
• X(s) L e-3t 1/(s3)
• 2. Now, obtain the output
• Y(s) X(s)H(s) 1/(s3)2s/(s6)
  2s/(s3)(s6)
• 3. Next, perform inverse Laplace transform
  by using
• PFE method
• Y(s) -2/(s3) 4/(s6)
• y(t) -2e-3t 4e-6t

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Chapter 5 LAPLACE TRANSFORM TRANSFER FUNCTION

• EXERCISE 4 (Exercise Problem 16.34, page 753)
• Find the transfer function, H(s)
  Vo(s)/Vs(s) for the circuit shown below

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Chapter 5LAPLACE TRANSFORM TRANSFER FUNCTION

• SOLUTION TO EXERCISE 4
• 1. Convert the circuit element into s-domain
• as shown below
• 2. Perform Nodal Analysis

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Chapter 5LAPLACE TRANSFORM TRANSFER FUNCTION

• SOLUTION TO EXERCISE 4 (Continued)
• The transfer function is
• H(s) Vo(s) / Vs (s)

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Chapter 5LAPLACE TRANSFORM TRANSFER FUNCTION

• A transfer function, H(s) can be plot by using
  BODE Plot.
• It is used to find the frequency response of the
  system. It is a semilog plots of magnitude (in
  dB) and phase (in degree) of a transfer function
  over frequency.
• The transfer function can be written in the form
  of polar such as
• H(j?) H?? Hej?

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Chapter 5LAPLACE TRANSFORM TRANSFER FUNCTION

• For plotting the magnitude of Bode Plot,
• HdB 20log10 H
• Read chapter 14.4 for procedure to make a Bode
  plot.

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Chapter 5LAPLACE TRANSFORM TRANSFER FUNCTION

• EXAMPLE 5
• Construct the Bode plot of the following
  transfer function
• H(s) 200s / (s2 12s 20)
• Solution for Example 5
• 1. replace s by j? factor out the
• denominator
• H(j?) 200j? / (j? 2)(j? 10)
• 2. Dividing out the poles and zeros
• H(j?) 10j? / (1 j?/2)(1 j?/10)

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Chapter 5LAPLACE TRANSFORM TRANSFER FUNCTION

• Continued from example 5
• 3. calculate the magnitude phase
• 4. 10j?/1j?/21j?/10 ?90o tan-1?/2
• tan-1 / 10
• 5. Convert into decibels
• HdB 20log10(10) 20log10(j?)
  20log10
• (1j?/2) 20log10
  (1j?/10)
• ? 90o tan-1 ?/2 tan-1 ?/10
• 6. For the plot, refer to figure 14.13 (page
  14.13)
• 7. next, do practice problem 14.3

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Chapter 5LAPLACE TRANSFORM TRANSFER FUNCTION

• In MATLAB, the Bode Plot can be done as below
• 1. num 200 0
• 2. den 1 12 20
• 3. bode (num, den)

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Chapter 5LAPLACE TRANSFORM TRANSFER FUNCTION

• EXAMPLE 6
• Find the system output response, y(t) where
  the input to the system is x(t) 10e-t u(t) and
  the transfer function of the system is defined
  below
• H(s) s 4 / (s3 2s2 5s 10)
• Solution for Example 6
• 1. Use MATLAB to compute the output
  response
• gtgt t00.025 specify time
  interval with
• increment
  of 0.02.
• gtgt x10exp(-t) Specify the input
  signal
• gtgt num 1 4 Specify the
  numerator of H(s)
• gtgt den 1 2 5 10 Specify the den.
  Of H(s)
• gtgt y lsim(num, den, x, t) calculate
  the time response.
• gtgt plot(t,x,t,y) Plot the input and
  output response

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Chapter 5LAPLACE TRANSFORM TRANSFER FUNCTION

• EXERCISE 5 (Exercise Problem 16.35, page 753)
• Find the transfer function, H(s)
  Vo(s)/Vs(s) for the circuit shown below

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Chapter 5LAPLACE TRANSFORM TRANSFER FUNCTION

• SOLUTION TO EXERCISE 5
• 1. Convert the circuit element into s-domain
• as shown below
• 2. Perform Nodal Analysis
• where

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Chapter 5LAPLACE TRANSFORM TRANSFER FUNCTION

• SOLUTION TO EXERCISE 5 (CONTINUED)
• 3. Perform Nodal Analysis and replace I
• 4.