Operating Systems Chapter 2 Homework Show That And The Transfer Function From

2020 CHAPTER 2. DYNAMIC MODELS

Figure 2.48: Circuit for Problem 12.

(a)

Vin V

Rin

=VVout

Rf

12. Show that the op amp connection shown in Fig. 2.48 results in Vo=Vin

if the op amp is ideal. Give the transfer function if the op amp has the

non-ideal transfer function of Problem 2.11.

Solution:

Ideal case:

13. A common connection for a motor power amplifier is shown in Fig. 2.49.

The idea is to have the motor current follow the input voltage and the

connection is called a current amplifier. Assume that the sense resistor,

Rsis very small compared with the feedback resistor, Rand find the

transfer function from Vin to Ia:Also show the transfer function when

Rf=1:

Node A

Node B

Figure 2.49: Op Amp circuit for Problem 13 with nodes marked.

Solution:

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The dynamics of the motor is modeled with negligible inductance as

Substituting this into Eq.93

14. An op amp connection with feedback to both the negative and the positive

terminals is shown in Fig 2.50. If the op amp has the non-ideal transfer

function given in Problem 11, give the maximum value possible for the

positive feedback ratio, P=r

r+Rin terms of the negative feedback

ratio,N=Rin

Rin +Rf

for the circuit to remain stable.

Solution:

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Figure 2.50: Op Amp circuit for Problem 14.

15. Write the dynamic equations and find the transfer functions for the circuits

shown in Fig. 2.51.

(a) passive lead circuit

(b) active lead circuit

(c) active lag circuit.

(d) passive notch circuit

Solution:

(a) Passive lead circuit

With the node at y+, summing currents into that node, we get

(b) Active lead circuit

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Laplace Transform

(c) active lag circuit

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(d) notch circuit

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16. The very ‡exible circuit shown in Fig. 2.52 is called a biquad because

its transfer function can be made to be the ratio of two second-order or

quadratic polynomials. By selecting di¤erent values for Ra; Rb; Rc;and

Rdthe circuit can realise a low-pass, band-pass, high-pass, or band-reject

(notch) filter.

(a) Show that if Ra=R; and Rb=Rc=Rd=1;the transfer function

from Vin to Vout can be written as the low-pass filter

Vout

Vin

=A

s2

!2

n

+ 2s

!n

+ 1

(99)

where

A=R

R1

!n=1

RC

=R

2R2

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Figure 2.52: Op-amp biquad

(b) Using the MATLAB comand step compute and plot on the same

graph the step responses for the biquad of Fig. 2.52 for A= 2;

!n= 2;and = 0:1;0:5;and 1:0:

Solution:

Before going in to the specific problem, let’s find the general form of the

transfer function for the circuit.

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Plug in V1,V2and V3to the fourth equation.

Finally,

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(a) If Ra=R; and Rb=Rc=Rd=1;

(b) Step response using MatLab

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Step responses

17. Find the equations and transfer function for the biquad circuit of Fig. 2.52

if Ra=R; Rd=R1and Rb=Rc=1:

Solution:

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Problems and Solutions for Section 2.3

18. The torque constant of a motor is the ratio of torque to current and is

often given in ounce-inches per ampere. (ounce-inches have dimension

force-distance where an ounce is 1=16 of a pound.) The electric constant

of a motor is the ratio of back emf to speed and is often given in volts per

1000 rpm. In consistent units the two constants are the same for a given

motor.

(a) Show that the units ounce-inches per ampere are proportional to

volts per 1000 rpm by reducing both to MKS (SI) units.

(b) A certain motor has a back emf of 25 V at 1000 rpm. What is its

torque constant in ounce-inches per ampere?

(c) What is the torque constant of the motor of part (b) in newton-meters

per ampere?

Solution:

Before going into the problem, let’s review the units.

Some remarks on non SI units.

Relation between SI units

–Voltage and Current

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(a) Relation between torque constant and electric constant.

Torque constant:

19. The electromechanical system shown in Fig. 2.53 represents a simplified

model of a capacitor microphone. The system consists in part of a parallel

plate capacitor connected into an electric circuit. Capacitor plate ais

rigidly fastened to the microphone frame. Sound waves pass through the

mouthpiece and exert a force fs(t)on plate b, which has mass Mand is

connected to the frame by a set of springs and dampers. The capacitance

Cis a function of the distance xbetween the plates, as follows:

C(x) = “A

x;

where

“= dielectric constant of the material between the plates;

A= surface area of the plates:

The charge qand the voltage eacross the plates are related by

q=C(x)e:

The electric field in turn produces the following force feon the movable

plate that opposes its motion:

fe=q2

2“A