T W O

Modeling in the

Frequency Domain

SOLUTIONS TO CASE STUDIES CHALLENGES

Antenna Control: Transfer Functions

Finding each transfer function:

Pot: θ

i

i

V (s)

(s) =

π

10 ;

Pre-Amp: p

i

V (s)

V (s) = K;

Power Amp:

a

p

E (s)

V (s) =



150

s 150

Motor: Jm = 0.05 + 5( 50 250

) 2

= 0.25

Dm =0.01 + 3( 50 250

) 2

= 0.13

t

a

K

R =

1

5

t b

a

K K

R =

1

5

Therefore: θm

a

(s)

E (s) =

 

t

a m

t b m

m a

K

R J

K K1 s(s (D ))

J R

= 0.8

s(s 1.32)

And:

θO

a

(s)

E (s) =

θm

a

(s)1

5 E (s) =



0.16

s(s 1.32)

Transfer Function of a Nonlinear Electrical Network

Writing the differential equation, 

 

   20 0 d(i i)

2(i i) 5 v(t) dt

. Linearizing i 2 about i0,

δ δ δ δ δ 

       0

2 2 2 2

0 0 0 0 0 0 i i

(i i) i 2i | i 2i i. Thus, (i i) i 2i i.

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2-2 Chapter 2: Modeling in the Frequency Domain

Substituting into the differential equation yields, δd i

dt + 2i0

2 + 4i0i - 5 = v(t). But, the

resistor voltage equals the battery voltage at equilibrium when the supply voltage is zero since

the voltage across the inductor is zero at dc. Hence, 2i0 2 = 5, or i0 = 1.58. Substituting into the linearized

differential equation, δd i

dt + 6.32i = v(t). Converting to a transfer function,

δi(s)

V(s) =

1

s 6.32 . Using

the linearized i about i0, and the fact that vr(t) is 5 volts at equilibrium, the linearized vr(t) is vr(t) = 2i 2 =

2(i0+i) 2 = 2(i0

2+2i0i) = 5+6.32i. For excursions away from equilibrium, vr(t) - 5 = 6.32i = vr(t).

Therefore, multiplying the transfer function by 6.32, yields, rV (s)

V(s)

 =

6.32

s 6.32 as the transfer function

about v(t) = 0.

ANSWERS TO REVIEW QUESTIONS

1. Transfer function

2. Linear time-invariant

3. Laplace

4. G(s) = C(s)/R(s), where c(t) is the output and r(t) is the input.

5. Initial conditions are zero

6. Equations of motion

7. Free body diagram

8. There are direct analogies between the electrical variables and components and the mechanical variables

and components.

9. Mechanical advantage for rotating systems

10. Armature inertia, armature damping, load inertia, load damping

11. Multiply the transfer function by the gear ratio relating armature position to load position.

12. (1) Recognize the nonlinear component, (2) Write the nonlinear differential equation, (3) Select the

equilibrium solution, (4) Linearize the nonlinear differential equation, (5) Take the Laplace transform of

the linearized differential equation, (6) Find the transfer function.

SOLUTIONS TO PROBLEMS

1.

a.

00

1 1 ( ) st stF s e dt e

s s



    

b. 2 20

00

( 1) ( ) ( 1)

st st

st

e st F s te dt st

s s e

        

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Solutions to Problems 2-3

Using L'Hopital's Rule

3 2

1 ( ) 0. Therefore, ( ) .

stt t

s F s F s

s e s 

   

c. 2 2 2 2

0 0

( ) sin ( sin cos ) st

st eF s t e dt s t t s s

    

 

      

 

d. 2 2 2 2

0 0

( ) cos ( cos sin ) st

st e sF s t e dt s t t s s

     

      

 

2.

a. Using the frequency shift theorem and the Laplace transform of sin t, F(s) = ω

ω2 2(s+a) + .

b. Using the frequency shift theorem and the Laplace transform of cos t, F(s) = ω2 2

(s+a)

(s+a) + .

c. Using the integration theorem, and successively integrating u(t) three times, dt = t; tdt = 2t

2 ;

2t 2

dt = 3t

6 , the Laplace transform of t3u(t), F(s) =

4

6

s .

3.

a. Taking the sum of the voltages around the loop and assuming zero initial conditions yields:

0

( ) 1 ( ) ( ) ( )

t di t

Ri t L i d v t dt C

   

b. Applying Laplace transform and solving for I(s)/V(s) gives:

( ) 1 1

1 1( ) ( )

I s

RV s Ls R L s

Cs L LCs

 

   

Substituting the values of R, L, and LC, we have:

2

( ) 2 2

16( ) 2 16 ( 2 )

I s s

V s s s s

s

   

 

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2-4 Chapter 2: Modeling in the Frequency Domain

Solving for I(s) and noting that V(s) = 1/s, we get:

2

2 ( )

2 16 I s

s s 

 

Observing that the denominator has complex roots, we re-write the above equation as:

2 2

2 ( )

( 1) ( 15) I s

s 

 

Applying the frequency shift theorem to the Laplace transform of sin t u(t), we find that the

transform for ( ) sin( )atf t e t is 2 2

( ) ( )

F s s a



 

  .

Comparing F(s) to I(s), we conclude that in the latter: a = 1 and  15 . Thus, the current, i(t),

may be given by:

2 ( ) 15 sin( 15 )

15

ti t e t

c.

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Solutions to Problems 2-5

4.

a. The Laplace transform of the differential equation, assuming zero initial conditions,

is, (s+7)X(s) = 2 2 5s

s 2 . Solving for X(s) and expanding by partial fractions,

   

  2 2 5 35 1 5 7 4

53 7 53( 7)( 4) 4

s s

ss s s

Or,

   

  2 2 5 35 1 5 7 2 4

53 7 53( 7)( 4) 4

s s

ss s s

Taking the inverse Laplace transform, x(t) = - 35

53 e-7t + (

35

53 cos 2t +

10

53 sin 2t).

b. The Laplace transform of the differential equation, assuming zero initial conditions, is,

(s2+6s+8)X(s) = 2

15

s 9 .

Solving for X(s)

   2 2

15 X(s)

(s 9)(s 6s 8)

and expanding by partial fractions,



     2

1 6s 9

3 3 1 15 19 X(s)

65 10 s 4 26 s 2s 9

Taking the inverse Laplace transform,

4t 2t18 1 3 15x(t) cos(3t) sin(3t) e e 65 65 10 26

     

c. The Laplace transform of the differential equation is, assuming zero initial conditions,

(s2+8s+25)x(s) = 10

s . Solving for X(s)

2

10 X(s)

s(s 8 s 25) 

 

and expanding by partial fractions,

2

4 1(s 4) 9

2 1 2 9 X(s) -

5 5 s 4 9s

 

  

Taking the inverse Laplace transform,

42 8 2( ) sin(3 ) cos(3 ) 5 15 5

tx t e t t  

     

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2-6 Chapter 2: Modeling in the Frequency Domain

5.

a. Taking the Laplace transform with initial conditions, s2X(s)-4s+4+2sX(s)-8+2X(s) = 2 2

2

s 2 .

Solving for X(s),

X(s) =

3 2

2 2

4 4 16 18

( 4)( 2 2)

s s s

s s s

  

   .

Expanding by partial fractions

2 2 2

1 s 2

1 1 21(s 1) 22X(s) 5 s 2 5 (s 1) 1

   