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Control Systems and Electromagnetics: A Comprehensive Guide with Exercises, Exercises of Electrical Engineering

University of the Philippines Open UniversityElectrical Engineering

A comprehensive overview of control systems and electromagnetics, covering fundamental concepts, types, and applications. It includes detailed explanations, examples, and exercises to enhance understanding. Suitable for students studying electrical engineering or related fields.

Typology: Exercises

2024/2025

Available from 04/05/2025

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CONTROL SYSTEMS
and
tUECTROHAGNETICS
This Free Quality Manual is
Found only at POWERLINE
E- MAI L: powerlinereviewcenter@yahoo.com
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' C O RN E R E SPAN A G. M. TOLENTINO STREETS
SAM P A L O C, MA N ILA
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TE L. NOS. (03 )261-2 2 44 Sc (03 2)2 61-8 4 52
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Download Control Systems and Electromagnetics: A Comprehensive Guide with Exercises and more Exercises Electrical Engineering in PDF only on Docsity!

CONTROL SYSTEMS

and

tUECTROHAGNETICS

This Free Quality Manual is

Found only at POWERLINE

E- MAI L: powerlinereviewcenter@yahoo.com

POWERLINE REVIEW CENTER

M EZ Z A N IN E FLO O R , D O N A A M P A R O BUILDING ' C O R N E R E S P A N A G. M. TOLEN TIN O S T R E E T S S A M P A L O C , M A N ILA TEL. NOS. 7 3 5 -7 3 -0 2 Sc 73 3-2 1 -1 8

TEL. NOS. (0 3 )2 6 1 -2 2 4 4 Sc (0 3 2 )2 6 1 -8 4 5 2

PART 1 : CONTROL SYSTEMS

Control System - is an arrangement of physical components connected or related in such a manner as to Commend, direct or regulate itself or another system. Purpose of control system The purpose of control system usually identifies or defines the “output” and “input”. If the output and input are given, it is possible to identify or define the nature of system's components.

Three Basic Types of Control Systems:

  1. Man-made control systems
  2. Natural, including biological control systems.
  3. Control systems whose.components are both man-made and natural.

Two General Categories of Control Systems:

  1. An Open-Ioop control systems is one in which the control action is independent of the output.
  2. A Closed-loop control system is one in which the control action is somehow dependent on the output. They are commonly called “ Feedback ” control systems.

The Control Systems Engineering Problem:

  1. Analysis - is the investigation of the properties of an existing system.
  2. Design - problem is the choice and arrangement of control system components to perform a specific task

The Representation of the Problem: The Model Three basic representations (models) of physical components and systems are extensively employed in the study of control systems.

  1. Differential equation^and other mathematical relations. 2 Block diagrams
  2. Signal flow graphs

The Solution of Linear Constant Coefficient Ordinary Differential Eouation: Consider the class of Differential Equation of the form: A ' _dy_* A d lx > a. - 4 - = > — P d tl to d tl Where: “ t ” is time, the coefficients a t & bt are constant,

X = x{t) is a known time function and y = y {t) is the unknown solution of the equation. n — is called the order of the differential equation. The solution of a differential equation of this class can be divided into two parts, a “ free responsed ’ and a “ force responsed. The sum of these two responses constitute the “ total respon sed or the solution y (t ) of the equation. Electrical Analogies: There are two electrical analogies of mechanical systems; 1 The Voltage-Force or Mass-lnductance analogy 2 The Current-Force or Mas$~Capacitance analogy The following table shows both the voltage-force & current-force analogies for mechanical systems:

Voltage-force analogy Current-force analogy Force, F Velocity, dx/dt Damping, b Mass, m Spring constant, k

Voltage, u Current, / Resistance, R Inductance, L Elastance, 1/C

Force, F Velocity, dx/dt Damping, b Mass, m Spring constant, k

Current, / Voltage, o Conductance, 6 Capacitance, C Reciprocal of L (ML)

In general, in laying out equivalent electric circuits for mechanical systems, the following rule is observed If the forces act in series in the mechanical system, the electrical elements representing these forces are put in parallel. Forces in parallel are represented by elements in series in electric circuits.

PART 2 : ELECTROMAGNETICS

Coulombs Law / Electrostatics: It states that the force between two very small objects separated in a vacuum or free space by a distance which is large compared to their size is proportional to the charge on each and inversely proportional to the square of the distance between them.

Mathematically. F - - — — ■- a R Newtons 4 kS 'R 2

Where: Q x, Q2 = respective charges in Coulomb, R = separation distance in meters £ = permittivity of the medium = S0 • Sr ( for air 8r = 1.0 )

£0 — permittivity constant = 8.854 xlO 12 F/m « ----- *10 9 F/m 36 k a R = unit vector in the direction of R

Electric Field Intensity: It is a vector force on a unit positive test charge. Electric field intensity must be measured by the unit- Newton per Coulomb or Volts per meter.

Mathematically^ mm^ : E = — - —-^1?^ Q^ a R Volts^ ----o r Newton ---

4 ns-R m Coul Standard Charge Conf/auartion: '

  1. Point charge - a field of a single point charge Q is given by:

„ Q Volts Newton ,. f ^ ■■ a r ------- o r -------- — 4 Trs-r m Coul

  1. Infinite line charge - if charge distribution with uniform density of p f (Coul/m) along an infinite straight line which will be chosen as the Z-axis, then the field intensity is given by:

.j^ = -----^ p P :—^ a r ------- Volts^ o r ------- — Newton^ Cylindrical coordinates> , 2ns * r m Coul

  1. Infinite sheet charge - if charge is distributed with a uniform density of p s (Coul/m2) over the plane, then the field intensity is given by:

P s Volts Newton k = — a R ------- o r ----------- 2s m Coul Where: a n = unit vector perpendicular (normal) to the plane Gauss's Law: The electric flux passing through any closed surface is equal to the total charge enclosed. For the r -» electric flux density D we have: y/ —Qcndoscd = f D« dS

Where: if/ — electric flux in Coulomb (a scalar)

D = electric flux density in Coulomb/m2 (a vector)

dS = differential surface in m2 (a vector) & j> = integral of closed surface

Note: D = £ 0£ rE in Coulomb/m

Gauss Divergence Theorem:

The surface integral of normal component D over the surface is equal to the volume integral of divergence D over the volume integral.

! » • dS ~ \ p vdV = f (V • D )dV in Coulomb J vl 1 V

But p v = div D = V • D from Maxwell’s equation

d d d

Where V = ”del” or “nabla” operator = — a v 4-— H-----^ a y

dx x dy y dz z

Electric Potential Between Two Points: The potential of point UA” with respect to point “B57 is defined as the work done in moving a unit positive charge <2u from “B” to “A”

Work f. Joule

^-us = “ — - - J E • <11 in volts or

Qu Coul

Relation ofV toE is E = — V V 9 where V V = gradient of V. Current and Current Density: The electric charges in motion constitute a current, Current density is a vector J.

I = j \ j • dS in Amp

Poisson’s Equation & Laplace's Equation:

From V • D = p , also D = S * E and E = —V V

Consider, V • 6'(—V F ) — p thus, V • V V = —— or V 2V = —— 8 £ Where: V • VV — divergence of divergence of V = Laplacian of V

Ampere's Law & Magnetic Field:

Biot-Savart Law - a differential field strength , results from differential current element I ♦ dt

. The field varies inversely with the distance squared, is independent of the surrounding medium, and

has a direction given by the cross-product of I * d l and a^.

■ I < # x a » H =

Amp

in --------

4 n r 2 meter

Current Density. J & Magnetic Field. H:

V x H = J = (curl H ) = crE Where: <7 “ conductivity of the material in mho/meter

Curl H =

a , a y (^) a* d d d dx (^) d> H y H z

dHz 8H y A dy dz

a v +

( d n y an dx a>’ +

'dHy dH, dx dy

F I j x Density & Stokes Theorem: (Ampere's Circuital Law)

B = j A * d( = £ b • dS ~ / i0j>J • dS

Where: A = magnetic potential vector. Electromagnetic Waves:

Wave Equations: V x H = (cr + jo)e)E also V x E = ~jcojuH V • E = 0 & V « H = 0 Propagatfon Constant : y - a + jj

Where: a = attenuation factor in neper/m = co

P O W E R U n e r e v i e w c e n t e r Inc.

The Ultimate E.E. Review Center

CONTROL SYSTEMS & ELECTROMAGNETICS

REVIEW PROBLEMS

  1. Any externally introduced input signal affecting the controlled output is known a s _____. A. signal C. feedback B. stimulus D. gain control
  2. A power-amplifying feedback control system where in the controlled variable is mechanical position or a time derivative of positiqn such as velocity or acceleration is... A. timer C; servomechanism B. regulator D. automation
  3. Is that part of the total response which does not approach zero as time approaches

infinity? TA.; steady state V1! ' C. signal B. transient D. force

  1. Is that part of the total response which approaches zero as time approaches infinity? A. steady state C. signal B. transient t ^ „ D. force
  2. Analysis of control systems by Laplace Transform technique is NOT applicable f o r ____. A. discrete-time system C. linear system B. time-invariant system D. unstable continuous-time systems
  3. Method of determining continuous system stability, a criterion that can be applied to an nth. Order characteristic equation. j V ) Routh criteria C. continued fraction criteria B. Hurwitzcriteria D. locus of point
  4. The laplace transform of a unit impulse function, 8 (t) is ______. A. 1/s C. 1/s . B. 1.0 D. 0
  5. Determine the initial value of the function f ( t ) whose Laplace is F(a) = - ------- s 4 + 353 + 52 + 5 A. 1 B. 2 , C. 0 D. 3
  6. Determine if the following characteristic equation represents a stable system:

' 53 + 4 s 2 +85 + 1 2 - 0 A. No, it is unstable C. indeterminate B. Yes, it is stable D. undefined

10. Consider the system with differential equation y'+2y - u'+u , find the transfer function.

a 5 + 1 r , ' S+ 2 „ s ^ s

A. * B. ------- C. ------- D.
• S+ 2 5 + 1 5 + 2 5 + 1
  1. Find the transfer function C (s )/ R{s ) for the system shown.
B.
G\G
1 - G 2H,

Gl + G 2

c. G |~ G 2

D.
\ - G 2H,
1 - G ,H \
X
  1. Determine the damping ratio C and undamped natural frequency <y„in rad/sec for the

d 2y , Ady dt_ A 0,5, 2 B. 0.5, 4 C. 0.25, 1 D. 2, 4

second order system: 2 — ~ + 4 — + 8>> = 8x

  1. Determine the time constant r in sec. and damped natural frequency o)d in rad/sec for

d 2y , dy the second order system:

A. 0.4, 0.

d t

B. 0.5, 1.
  • 5— + 9 y = 9x dt C. 0.4, 1.66 D. 0.25, 0.
  1. A servo system for a pen plotter is given by the following block diagram. Find the value of K required to get the fastest response without any overshoot is given by:

m

- o

\ w IP- (^) K (^) ■---------------^ 1 s (s + 2) c % )

A. 10 B. 1 C. 5 D. 100
  1. What is the flux crosses the closed surface which contains a charge distribution in the

form of a plane disk of radius 4m with a density p s

sin <j)

A. 2/71 Coul B. tc/2 Coul

2 r C. n Coul D. 2n Coul

Coul/m?

CONTROL SYSTEMS & ELECTROMAGNETICS

SUPPLEMENTARY PROBLEMS

  1. Is an algebraic or transcendental equality which involves more than one value of the dependent variable corresponding to more than one value of at least one of the independent variable(s). A) differential equation C. linear equation B difference equation D. quadratic equation
  2. Is a short-hand, pictorial representation of the cause-and-effect relationship between the input and output of a physical system. A. flow chart C block diagram B. bar chart D. signal flow graph
  3. Due to which of the following reason excessive bandwidth in control system should be avoided? A. noise is proportionalto band width C. it leads to a low relative stability B. it leads to slow speed response D. has no considerable effect
  4. Is another method for determining whether all the roots of the characteristic equation of a continuous system have negative real parts. A. Routh criteria C. continued fraction criteria B. Hurwitz criteria D. locus of point
  5. Find the initial value of the function / ( / ) whose Laplace is: F(s) = —------- t ------------ ■ 7 5 + 2s +9.y + 6 A. 1 C.) 0 B. 2 DT 1/a
  6. Determine if the characteristic equation given represents a stable or an unstable system:

s3 + 8 s 2 +145 + 24 = 0 A. stable system C. undefined Bf unstable system D. indeterminate

  1. What restrictions must be placed upon the parameter “K” in order to insure that the system is stable from the characteristic equation given:

s4 + 6s3 + l l s 2 +6s + K = 0

A. K>10, K<60 C. K>10, K<

B K< 10, K >Os D. K<60, K>

  1. What is the transfer function of a system whose input and output are related by the following differential equation: y"+3y'+2y = u'+u 5+1 „ 5 + 4 5 - 1 _ 5 + 3 A. —----------- B. —----------------------------C. ^ -------- D. —------------- 5 +35 + 2 5 +35 + 4 s~ — 3 5 - 2 5 " + 4 s + 5

9. Given y”+3y'+2y = u , with initial condition y(o) = 0, jv'(o) = 1, find the free response

A. e~‘ - e - 2' B. - e " + e '2'C. e~'+e~2' D. - e " - e ~ 2‘

  1. From the differential equation of prob. #9, find the force response if u(t) = 1 I 2

B. I ( l - e - ' + ^ ' ) D. H \ - 2 e - ‘ +2e-2‘)

A. 1—2e-' + e ~ 2‘ C. i(l-2(?w + ^ 2')

  1. Determine the damped natural frequency 0 )d, damping coefficient a respectively for the

following second order: - r f! > + 4 0 ^^ +^ s^ = 8 „ dt2 dt A. 2,1.5 C. 3,0. B. V2 , 1.25 D. V 3. 1

  1. Determine the damping ratio C, , undamped natural frequency a>n respectively for the following

d 2y , ,d y second order: + 5— + 9^ y^ = 9^ u dt2 dt A. 2/3, 2 rad/sec B. 4/5, 2.5 rad/sec C. 13 Find the transfer function C (.s)//?(s) for the system shown.

C. 5/6, 3 rad/sec D. 1/3, 3.5 rad/sec

A.
G,G
1 + G 2H , + G ,//,

g l + ^ \ - G xH , - G 2//,

C.
G, - G
1 - G , / / , + G,
G,G
I - G, 7/1 - C 2//,
  1. Find the output J ^ o f a system described by the differential equation: y"+3y’+2y = 1 + /

with initial conditions ,y(0) = o, y ( o ) = i

l& r' - l e ~ 2' + 2 t - \ \ C. 1/4[8ew + 3 e"2' - 2 / - l j

[8ew + 7e~2' +2/ + lJ D 1/4[8ew - 5 e - 2‘ - 2 t - \ \

  1. From the differential equation of prob. #11, find the transient response.

A. l / 4 [ 8 e " ' - 2 r - l j C. 1 / 4 ^ ^ + 7e"2( J

B. 1/4[2/ - 1] D. 1/4 [8e~' - 7e~2 1 j

  1. Given the following close-loop system: Find the value of K for the which the system becomes marginally stable and the frequency of oscillation respectively when K is set to make the system marginally stable. Hint : apply Routh-Hurwitz criterion

R(sy

K

------------- ^ 1

s (s + 2) (s 4 10)* f^ —^ ^ A -.

A. 10, 10 Hz B. 240, 0.71 Hz

C. 100, 1.2 Hz D. 60, 2 Hz

1 1 >3 / — (^) ~~ ✓ IVJLkLXu