i. flow graphs...f o r e x a m a s c 1 i. flow graphs theory 1. state the definition of the graph...

F O R E X A M A S C 1

I. FLOW GRAPHS THEORY 1. State the definition of the graph (regardless of its type). A graph is a collection of points in a plane – called nodes – connected by arcs – called branches. 2. Define the source-node in a flow graph. It is a node where all the incident branches are diverging. 3. Define the sink-node in a flow graph. It is a node where all the incident branches are converging. 4. Define the intermediary-node in a flow graph. It is a node where the incident branches can be diverging or converging. 5. Define the path in a flow graph. It is a sequence of branches having the same orientation without loops. 6. Define the loop in a flow graph. It is a closed curve that connects a node to itself. 7. Define the irreducible flow graph. What are the transmittances of this graph? It is a graph that is equivalent to the given graph, having only source-nodes and sink-nodes. The transmittances of this graph are the total transmittances of the graph. 8. Define the transmittance of a branch in a flow graph. It is a factor that, if multiplied by the variable of the start-node, gives its contribution to the variable of the end-node. 9. Define the total transmittance between two nodes of a flow graph. It is the gain that takes into account all the paths that the variable of the start-node influences the variable of the end-node. 10. Write the Mason’s rule and define each term.

∑=k

kkT1T ΔΔ

where: T is the total transmittance between two nodes, Δ is the determinant of the graph, Tk is the

path k from the start-node to the end-node and Δk is the determinant of the complementary graph. 11. Define the parallel branches in a flow graph. They are branches with the same origin and the same end. 12. How can one eliminate an intermediary-node in a flow graph and when can’t this procedure be done? It is done by rebuilding all the paths that passed through that node. If the node has a self-loop, it can be eliminated. 13. How can one eliminate a branch in a flow graph? It is done by partially applying the procedure of elimination of the origin node, in the sense that all the paths of two branches that were passing through that node will be rebuilt. 14. How can one eliminate a self-loop in a flow graph? When isn’t this procedure possible? It is done by dividing the transmittances of the converging branches to b1 t− , where bt is the gain of the loop. If the gain of the self-loop is unity, then it can be eliminate. MCQ

1. With a flow graph one can represent:

A. the topology of a circuit. B. a differential equation system. C. an algebraic equation system. D. the Kirchhoff’s theorems and the Ohm’s law.

Solution: C, D 2. With a linear oriented graph one can represent:

2 F O R E X A M A S C

A. the topology of a circuit. B. a differential equation system. C. an algebraic equation system. D. the Kirchhoff’s theorems and the Ohm’s law.

Solution: A 3. When we inverse the orientation of a branch, one of the effects is:

A. in a linear oriented graph: the sign of the voltage and current changes. B. in a flow graph: the sign of the voltage and current changes. C. in a linear oriented graph: the branch gain inverses. D. in a flow graph: the branch gain inverses.

Solution: A, D PROBLEMS 1. Draw the FG that represents the equation system =Ax By , where:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−

=

1000

01

01

B;

2200012001120001

A

2. Draw FG that represents the equation system =Ax By , where:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

=

21

00

0021

B;

1300013001100022

A


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−=

2010

0002

B;

20002210

01310001

A


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

3000

0001

B;

2200031001200011

A

5. Write the equation system represented by the FG shown in the figure.

Solution:

1 1

1 2 3

2 3 1

3 4 2

x y2x x x 02x x y

2x 2x y

= −⎧⎪ − + =⎪⎨ − = −⎪⎪ + = −⎩


Solution:

⎪⎪⎩

⎪⎪⎨

⎧

−=+−=−

=−−=+

243

232

132

121

y2xx3yxx3y2xx

yx2x2

-2 -1

-1

-2 -2

1

-1

2

-1

x4

x1

x3

x2

y2

y1 2

2

-2

-3 -2

2 -3

-1

-1

-1

1

x4

x1

x3

x2

y2

y1

2

-2 -1

-1

-2 -2

1

-1

2

-1

x4

x1

x3

x2

y2

y1 2

2

-2

-3 -2

2 -3

-1

-1

-1

1

x4

x1

x3

x2

y2

y1

2

1

2 2

-1

-1

-1

2

-1

x4

x1

x3

x2

y2

y1 -2

-1

2 3

-1

-1

-1

1

1

x4

x1

x3

x2

y2

y1 -1

-2



Solution:

⎪⎪⎩

⎪⎪⎨

⎧

==−+

=++=

24

432

2321

11

y2x20x2x2x

yxx3xy2x


Solution:

⎪⎪⎩

⎪⎪⎨

⎧

−=−=+=−

=+

243

32

32

121

y3x2x20x3x0xx2

yxx

9. For the FG shown in the figure, find the determinant and the solution x1. Solution: 11 , x 0Δ = − = 10. For the FG shown in the figure, find the determinant and the solution x2. Solution: 2 1 21 , x 2y yΔ = − = − − 11. For the FG shown in the figure, find the determinant and the solution x1. Solution: 1 1 21 , x y yΔ = − = + 12. For the FG shown in the figure, find the determinant and the solution x2. Solution: 2 1 21 , x 3y 3yΔ = − = − − 13. For the FG shown in the figure, find the determinant and the solution x1. Solution: 1 1 22 , x y yΔ = − = + 14. For the FG shown in the figure, find the determinant and the solution x2. Solution: 2 1 22 , x y 0.5yΔ = − = − − 15. For the FG shown in the figure, find the determinant and the solution x3. Solution: 3 1 22 , x 3y 0.5 yΔ = − = − −

32 2

2

3

1

1

2

1 2

x4

x1

x3

x2

y2

y1

32 2

2

3

1

1

2

1 2

x4

x1

x3

x2

y2

y1

32 2

2

3

1

1

2

1 2

x4

x1

x3

x2

y2

y1

1 2

2

2

2

2

1 1

x2

x1

y2

y1

1 2

2

2

2

2

1 1

x2

x1

y2

y1

1 2

2

2

1

2

1 2

x2

x1

y2

y1

1 2

2

2

1

2

1 2

x2

x1

y2

y1

1

2 2

-1

-1

-1

2

-1

x4

x1

x3

x2

y2

y1 -2

-1

2 3

-1

-1

-1

1

1

x4

x1

x3

x2

y2

y1 -1

-2


16. For the FG shown in the figure, find the determinant and the solution x4. Solution: 4 1 22 , x 9 y 0.5yΔ = − = − 17. For the FG shown in the figure, find the determinant and the solution x1. Solution: 1 12 , x yΔ = − = − 18. For the FG shown in the figure, find the determinant and the solution x2. Solution: 2 1 22 , x y yΔ = − = − 19. For the FG shown in the figure, find the determinant and the solution x3. Solution: 3 1 22 , x 3y 2yΔ = − = − 20. For the FG shown in the figure, find the determinant and the solution x4. Solution: 4 1 22 , x 3y 1.5yΔ = − = − +

32 2

2

3

1

1

2

1 2

x4

x1

x3

x2

y2

y1

21 3

2

2

1

1

2

1 1

x4

x1

x3

x2

y2

y1

2 1 3

2

2

1

1

2

1 1

x4

x1

x3

x2

y2

y1

2 1 3

2

2

1

1

2

1 1

x4

x1

x3

x2

y2

y1

2 1 3

2

2

1

1

2

1 1

x4

x1

x3

x2

y2

10


II. Stability

THEORY 1. What is the necessary condition for a LTI system to study its stability? The system must be active and with reaction. 2. What is the necessary condition for the impulse response of an asymptotically stable system?

tlim h( t ) 0→∞

=

3. What is the necessary condition for the transfer function of an asymptotically stable system? All its poles must be in the left-half of the s-plane. 4. What is the necessary condition for the transfer function of system that is at the limit of stability? It has poles in the left-half of the s-plane and simple poles on the imaginary axis. 5. State the Mihailov criterion. A system is stable if, when ω varies from 0 to ∞, the hodograph of ( ) ( ) s jQ j Q s ωω == makes a counterclockwise

rotation of n2π

, where n is the order of the polynomial.

6. What are the algebraic stability criteria applied to? They are applied to the denominator polynomial of the global transfer function. 7. What is the Mihailov criterion applied to? It is applied to the denominator polynomial of the global transfer function. 8. What is the Nyquist criterion applied to? It is applied to the transfer function of the open-loop system. 9. Define the roots locus. It is the geometric locus of the roots of the characteristic equation (of the roots of the denominator of the transfer function; of the poles of the transfer function). MCQ 1. The stability of a LTI system depends on:

A. the position of the poles and zeros of the transfer function in the s-plane. B. the structure and the parameters of the system. C. the position of the poles of the transfer function in the s-plane. D. the structure of the system and the amplitude of the excitation.

Solution: B, C 2. Which of the following statements are true?

A. A system is invariant if its structure and parameters are time invariable. B. A system is asymptotically stable if all its poles are in the left-half of the s-plane. C. A system is non-asymptotically stable if its impulse response tends to 0, to a constant or to an

oscillation. D. A system is stable if its zeros are in the right-half of the s-plane.

Solution: A, B, C 3. The algebraic stability criteria are applied to:

A. the transfer function. B. the numerator polynomial of the transfer function. C. the impulse response. D. the denominator polynomial of the transfer function.

Solution: D


4. The system with the pole-zero configuration from the figure is: A. Asymptotically stable. B. Non-asymptotically stable. C. Unstable. D. Its stability cannot be determined.

Solution: A


Solution: B


Solution: C


Solution: C

8. Which of the following pole-zero configurations correspond to stable systems (not necessarily asymptotically stable)?

Solution: A, B 9. Which of the following impulse responses correspond to stable systems (not necessarily asymptotically stable)?

Solution: B, C, D

10. Which of the following impulse responses correspond to asymptotically stable systems?

Solution: C

11. Which of the following statements are true?

A. All the stability criteria are applied to the global transfer function. B. The algebraic criteria are applied to the denominator polynomial of the global transfer function. C. The Nyquist criterion is applied to the global transfer function. D. The Nyquist criterion is applied of the transfer function of the system’s loop.

Solution: B, D

A. B. C. D.

A. B. C. D.

A. B. C. D. (2)

(2)

(2)

(2)


12. The global transfer function of the system shown in the figure is:

A. [ ])s(F)s(F)s(R1)s(F)s(F

)s(H21

21

+⋅++

=

B. )s(R)s(F1

)s(F)s(R)s(F1

)s(F)s(H

2

2

1

1

⋅++

⋅+=

C. )s(R)s(F)s(F1

)s(F)s(F)s(H

21

21

⋅⋅+⋅

=

D. [ ])s(F)s(F)s(R1)s(F)s(F

)s(H21

21

+⋅−+

=

Solution: A 13. The system described by the impulse response: h( t ) t sin( t )= ⋅ is:

A. Unstable. B. Asymptotically stable.

C. At the limit of stability. D. Non-asymptotically stable.

Solution: A 14. The system described by the impulse response: th( t ) sin( t ) e−= + is:



Solution: C 15. The system described by the impulse response: h( t ) 5 / t , t 3= > ?



Solution: B 16. The system described by the impulse response: th( t ) 4 e−= ⋅ ?



Solution: B 17. The system having the impulse response shown in the figure is:

A. Asymptotically stable. B. Non-asymptotically stable. C. Unstable. D. Its stability cannot be determined.

Solution: C 18. The system having the impulse response shown in the figure is:


Solution: B 19. The system having the impulse response shown in the figure is:


Solution: A 20. The system having the impulse response shown in the figure is:


Solution: B

F1(s)

F2(s)

R(s)


III. STATE VARIABLES THEORY 1. Define the state variables. The state variables are a set of variables that uniquely determine the future state of a dynamic system, if the present

values and the excitations are known. 2. Define the “state” of a system. The state of a system is represented by the values of the state variables at a certain time. The state of the system evolves according as the state variables modify their values. 3. What is a “state trajectory”? The state trajectory is a curve in state space, representing the evolution of the state of a system. The state space is n-dimensional and on its axis one represents the values of the state variables. 4. Define the loop of capacitors (C-loop). It is a loop that contains only capacitors and independent voltage sources. 5. Define the junction of inductors (L-junction). It is a junction that has only inductors and independent current sources.. 6. What is the order of a passive circuit equal to (in the context of the state space)? It is equal to the number of state variables. It is equal to the number of reactive elements minus the number of C-loops minus the number of L-junctions. 7. Define the normal tree of a linear oriented graph. It is the tree that contains: all the independent voltage sources, no independent current sources, the maximum number of capacitors and the minimum number of inductors. 8. Write the normal form of the state evolution equation and explain the meaning of each term.

( ) ( ) ( )d t

t tdt

= ⋅ + ⋅x

A x B w , where: x is the state vector, w is the excitation vector, A(nxn) is the incidence matrix

(the system matrix), B(nxm) is the input matrix, n is the order of the system and m is the number of excitations.

9. Write the degenerated form of the state evolution equation and explain the meaning of each term. ( ) ( ) ( ) ( )

d t d tt t

dt dt= ⋅ + ⋅ + ⋅1 2

x wA x B e B , where: x is the state vector, w is the excitation vector, A(nxn) is the

incidence matrix (the system matrix), B1(nxm) şi B2(nxm) are input matrices, n is the order of the system and m is

the number of excitations.

10. Write the normal form of the output equation (in the state space) and explain the meaning of each term. ( ) ( ) ( )t t t= ⋅ + ⋅y C x D w , where: y is the output vector, x is the state vector, w is the excitation vector, C(kxn) is the

output matrix, D(kxm) is the direct transmission matrix, k is the number of outputs, n is the order of the system and

m is the number of excitations.

11. Write the normal form of the solution of the state evolution equation and explain the meaning of each term.

( ) ( ) ( )t

0

t dτ τ= ⋅ + ⋅ ⋅∫ A t -τAt 0x e x e B w , where: x is the state vector, w is the excitation vector, A(nxn) is the


incidence matrix (the system matrix), B(nxm) is the input matrix, n is the order of the system, m is the number of

excitations and x0 is the initial state of the system.

MCQ 1. Which of the following statements are true?

A. The order of a non-degenerated circuit is equal to the number of reactive elements of the circuit. B. The order of a circuit is equal to the number of reactive elements of the circuit. C. The order of a circuit is equal to the number of initial conditions that can be imposed. D. The order of a circuit is equal to the order of the characteristic equation.

Solution: A, C, D 2. Which of the following statements are true?

A. The order of a circuit is equal to the number of independent sources. B. The order of a circuit is equal to the number of the poles of the transfer function. C. The order of a circuit is equal to the number of zeros of the transfer function. D. The order of a circuit is equal to the number of L-junctions plus the number of C-loops.

Solution: B 3. Which of the following variables are the nodes of a flow graph with mixed formulation?

A. the chord voltages B. the chord currents C. the branch voltages D. the branch currents

Solution: B, C 4. In order to have a stable system, it is necessary that the eigenvalues of the system matrix are:

A. complex-conjugated. B. all with a negative real part. C. all with a positive real part. D. the stability of the system cannot be specified in terms of the eigenvalues.

Solution: B 5. In the state space, the stability of a system can be studied in terms of:

A. the system matrix. B. the input matrix. C. the eigenvalues of the system matrix. D. the system matrix and the input matrix.

Solution: A, C 6. The eigenvalues of the system matrix are:

A. the diagonal elements of the matrix. B. the roots of the characteristic equation of the system. C. the values for which the determinant of the matrix is zero. D. the poles of the transfer function of the system.

Solution: B, D

7. What can one say about the system having the system matrix: ⎥⎦

⎤⎢⎣

⎡−=

3002

A

A. Its state variables are independent (does not influence each other). B. The system is stable. C. The system contains only two reactive elements. D. The system is unstable.

Solution: A, D


⎤⎢⎣

⎡−

−=

9004

A


Solution: A, B


⎤⎢⎣

⎡ −−=

1234

A



Solution: B


⎤⎢⎣

⎡−−

=41

21A

E. Its state variables depend on each other. F. The system is stable. G. The system contains only two reactive elements. H. The system is unstable.

Solution: A, D 11. In order to write the state equation, the tree of the linear oriented graph must contain:

A. the minimum number of inductors and the maximum number of capacitors. B. the maximum number of inductors and the minimum number of capacitors. C. all the independent voltage sources and no independent current sources. D. all the independent current sources and no independent voltage sources.

Solution: A, C 12. If a circuit has 4 capacitors, 3 inductors and a C-loop, then the order of the circuit is:

A. 7 B. 8 C. 6 D. 5 Solution: C

13. For a given circuit, the number of the state variables is: A. at most equal to the number of C-loops and L-junctions. B. at most equal to the number of reactive elements C. equal to the number of C-loops. D. equal to the number of L-junctions.

Solution: B


IV. TWO-PORTS MCQ

1. The two-port with the impedance matrix ⎥⎦

⎤⎢⎣

⎡=

1121

1211zzzz

Z is :

A. symmetric and reciprocal B. symmetric and nonreciprocal C. asymmetric and reciprocal D. asymmetric and nonreciprocal

Solution: B

2. The two-port with the impedance matrix ⎥⎦

⎤⎢⎣

⎡=

1112

1211zzzz

Z is:

A. symmetric and reciprocal B. symmetric and nonreciprocal C. asymmetric and reciprocal D. asymmetric and nonreciprocal

Solution: A 3. If the longitudinal and transversal impedances of a symmetric two-port increase k times, its characteristic impedance Zc becomes:

A. ck Z⋅ B. c1 Zk⋅ C. c

1 Zk⋅ D. ck Z⋅

Solution: D 4. Between the image transfer constants θI1 (input-output) and θI2 (output-input) there is the relation:

A. 2I2I1I1I ZZ θθ = B. 12I1I =θθ C. 2I

1I

2I

1IZZ

=θθ D. 2I1I θθ =

Solution: D 5. Two symmetric two-ports, having the same characteristic impedance, are connected in chain. The equivalent transfer constant is:

A. 21 θθθ += B. 21 θθθ ⋅= C. 21

21θθθθ

θ+⋅

= D. 21

212θθθθ

θ+⋅

=

Solution: A

6. If the transfer constant is ( )ln 10 j3πθ = + , then the voltage and the current are:

A. 10 times amplified and the phase difference is π/3. B. 10 times amplified and the phase difference is – π/3. C. 10 times attenuated and the phase difference is π/3. D. 10 times attenuated and the phase difference is – π/3.

Solution: D 7. If the longitudinal and transversal impedances of a symmetric two-port increase k times, its transfer constant (θ) becomes:

A. θ B. θk C. kθθ = D. kθ

Solution: A THEORY 1. A two-port chain is matched if….. …if in any section of the chain the downstream impedance is equal to the upstream one. 2. Define the characteristic impedance of a symmetric two-port. It is the impedance Zc which, if connected at one of the ports, the input impedance at the other port is Zc. 3. Define the transfer constant for the symmetric two-ports.


It is the constant θ defined by: 2

1

2

1II

lnUU

ln ==θ , where U1, U2, I1 and I2 are the voltages and the currents at the

ports of the symmetric two-port when the two-port is matched.

4. Define the image impedances of an asymmetric two-port. If the image impedance ZI1 is connected at the input, the output impedance is ZI2. Similarly, if the image impedance ZI2 is connected at the output, the input impedance is ZI1.

5. Define the transfer constant for the asymmetric two-ports..

It is the constant θI defined by 22

11I IU

IUln=θ , where U1, U2, I1 and I2 are the voltages and the currents at the ports

of the symmetric two-port when the two-port is matched. 6. Define the iterative impedances for an asymmetric two-port. If the image impedance ZI1 is connected at the input, the output impedance is ZI1. Similarly, if the image impedance ZI2 is connected at the output, the input impedance is ZI2.

7. A real source (Eg , Zg) is connected to a load (Zs). Write the expressions of the voltage and current incident

waves.

g

gi

gi Z2

EI;

2

EU ==

8. Write the characteristic impedance in terms of the short and open impedances.

SC0c ZZZ = 9. Write the transfer constant for a symmetric two-port in terms of the de short and open impedances.

( )0

SCZ

Zth =θ

10. Write the image impedances in terms of the short and open impedances.

2SC021I1SC011I ZZZ;ZZZ == 11. Write the transfer constant for an asymmetric two-port in terms of the de short and open impedances.

( )02

2SC

01

1SCI Z

ZZ

Zth ==θ

PROBLEMS

1. The figure bellow represents a two-port with cZ 2= Ω and the transfer constant ( )ln 2θ = . If the two-port is

matched, fill in the empty boxes with the corresponding values, with their units.

2. The figure bellow represents a two-port with Zc = 75 Ω and the transfer constant θ = ln(3). If the two-port is

matched, fill in the empty boxes with the corresponding values, with their units.

10 V 2 Ω 5 V

2 Ω 2.5 A 2 Ω

5 V 2.5 V 2 Ω

1.25 A

NOTE: at problems 13 – 20, the values from te grey boxes are given, the other boxes contain the solutions.


3. If the two-port is matched, fill in the empty boxes with the corresponding values, with their units.






600mV

1 mA 300 Ω

300 mV 300 Ω 75 mV

250 μA

Zc = 300 Ω

θ = ln(4)

40 V

10 A 2 Ω

20 V 2 Ω 10 V

5 A

Zc = 2 Ω

θ = ln(2)

900 mV

6 mA 75 Ω

450 mV 75 Ω 150 mV

2 mA

Zc = 75 Ω

θ = ln(3)

900 mV 75 Ω 450 mV

75 Ω 6 mA 75 Ω

300 mV 150 mV 75 Ω

2 mA

8 V

2 A 2 Ω

4 V 2 Ω 4 V

2 A

Zc = 2 Ω

θ = 0

600 V

4 A 75 Ω

300 V 75 Ω 150 V

2 A

Zc = 75 Ω

θ = ln(2)

36 V

6 A 3 Ω

18 V 3 Ω 9 V

3 A

Zc = 3 Ω

θ = ln(2)


9. Three symmetric two-ports are matched with the load impedance sZ 2= Ω and their transfer constants are:

( )I ln 0.5 j 3πθ = + , ( )2 ln 2θ = and I j 3

πθ = − respectively. Find the voltage at the output of the two-port if

the source generates gE 12 V= ? Solution: 2U 6 V= 10. Three symmetric two-ports are matched with the load impedance sZ 75= Ω and their transfer constants are:

( )I ln 0.5 j 2πθ = − , ( )2 ln 2θ = and I j 3

πθ = − respectively. Find the source’s voltage if the voltage at the

output of the two-port is 2U 5 V= ? Solution: gE 10 V=

11. The two-port from the figure is asymmetric and matched. Fill in the empty boxes with the corresponding

values, with their units. 12.

13.

14.

15.

16.

80 V

aV = 4 θ =ln(2 )

aI = 1

4 Ω

40 V

10 A 10 A

10 V 1 Ω

18 V

aV = 1/9 θ =ln(1/3)

aI = 1

1 Ω

9 V

9 A 9 A

81 V 9 Ω

4 V

aV = 0.25 θ =ln(0.5)

aI = 1

1 Ω

2 V

1 A 1 A

4 V 4 Ω

36 V

aV = 1 θ =ln(3)

aI = 9

1 Ω

18 V

18 A 2 A

18 V 9 Ω

10 V

aV = 1 θ =ln(2)

aI = 4

1 Ω

5 V

5 A 1.25 A

5 V 4 Ω

NOTE: at problems 31 – 42, the values from the gray boxes are given, the other boxes contain the solutions.


17.

18.

19.

20.

21.

22.

aV = 0.25 θ =ln(0.5)

aI = 1

3 Ω

12 Ω

aV = 1 θ =ln(0.5)

aI = 0.25

8 Ω

2 Ω

aV = 1 θ =ln(4 )

aI = 16

2 Ω

32 Ω

aV = 4 θ =ln(2)

aI = 1

12 Ω

3 Ω

aV = 1 θ =ln(2)

aI = 4

2 Ω

8 Ω

72 V

aV = 1 θ =ln(1/3)

aI = 1/9

9 Ω

36 V

4 A 36 A

36 V 1 Ω

aV = 1 θ =ln(0.5)

aI = 0.25

8 Ω

2 Ω


V. MATCHING CIRCUITS THEORY 1. Define the ideal transformer. It is a transformer that simultaneously fulfills the following conditions: (1) the winding resistance is zero, (2) the coupling is perfect and (3) the inductances tend to zero, with their ratio equaled to the square of the winding ratio. 2. Write the winding ratio of an ideal transformer that matches the resistances Rg and Rs.

g sn R / R= 3. Define the matching condition of a two-port chain. In any section of the chain, the downstream impedance must be equal to the upstream impedance. 4. Write (without the relations) the principle of designing the Γ matching circuits. An LC two-port in Γ is inserted, with the coil connected to the load. After the transfiguration of the series-parallel L–Rs connection, the measurement conditions are imposed: (1) the compensation of the reactances and (2) the equality of the resistances. 5. Write the reactances of the Γ matching circuits in case Rg > Rs.

( ) g sL g s s g CL

R RX R R 1 R / R ; X

X= − = −

6. Write the reactances of the Γ matching circuits in case Rg < Rs.

g s g sL C

g s L

R R R RX ; X

1 R / R X= = −

−

7. Define the coupling factor for the T matching circuits.

sg

cCT RR

XK =

8. Define the coupling factor for the Π matching circuits.

sg

cC RR

XK =Π

9. Define the coupling condition for the T matching circuits.

sgcCT RRXsau1K ≥≥

10. Define the coupling condition for the Π matching circuits.

sgcC RRXsau1K ≤≤Π

11. Write the relation between the phase difference and the coupling factor for the T matching circuits.

ICT sin

1Kϕ

=

12. Write the relation between the phase difference and the coupling factor for the Π matching circuits. IC sinK ϕΠ −= 1. What can be replaced in a matching circuit in order to reject a frequency that is less than the working frequency? We can replace a transversal coil by a resonant LC series circuit or a longitudinal capacitor by a resonant LC parallel circuit. 2. What can be replaced in a matching circuit in order to reject a frequency that is grater than the working frequency?


We can replace a longitudinal coil by a resonant LC parallel circuit or a transversal capacitor by a resonant LC series circuit. 3. Write the conditions that have to be imposed when finding the rejection circuits.. (1) at the working frequency, the reactance of the element that will be replace must be equal to the equivalent reactance of the LC circuit and (2) the LC circuit must be resonant at the frequency that will be reject. PROBLEMS 1. The ideal transformer from the figure matches the resistances Rg = 4 Ω and Rs = 1 Ω. Fill in the empty box with the value of the winding ratio. 2. The ideal transformer from the figure matches the resistances Rg = 9 Ω and Rs = 1 Ω. Fill in the empty box with the value of the winding ratio. 3. The ideal transformer from the figure matches the resistances Rg = 4 Ω and Rs = 2 Ω. Fill in the empty box with the value of the winding ratio. 4. The ideal transformer from the figure matches the resistances Rg = 9 Ω and Rs = 3 Ω. Fill in the empty box with the value of the winding ratio. 5. The ideal transformer from the figure matches the resistances Rg = 1 Ω and Rs = 4 Ω. Fill in the empty box with the value of the winding ratio. 6. The ideal transformer from the figure matches the resistances Rg = 1 Ω and Rs = 9 Ω. Fill in the empty box with the value of the winding ratio. 7. The ideal transformer from the figure matches the resistances Rg = 2 Ω and Rs = 4 Ω. Fill in the empty box with the value of the winding ratio. 8. The ideal transformer from the figure matches the resistances Rg = 3 Ω and Rs = 9 Ω. Fill in the empty box with the value of the winding ratio. MCQ 1. In order to obtain matching in an non-matched section, one must insert:

A. an ideal transformer. B. a matching Γ-circuit. C. a symmetric two-port. D. an asymmetric two-port.

Solution: A, B, D 2. In order to maintain the matching in a matched section, one must insert:

Rg Rs

2 : 1

Rg Rs

3 : 1

Rg Rs

2 : 1

Rg Rs

3 : 1

Rg Rs

0.5 : 1

Rg Rs

0.33: 1

Rg Rs

1 : 2

Rg Rs

1: 3


A. an ideal transformer. B. a matching Γ-circuit. C. a symmetric two-port. D. an asymmetric two-port.

Solution: A, C 3. The winding ratio of the ideal transformer that matches the resistances gR 4= Ω and sR 1= Ω is:

A. 2 B. 0.5 C. 0.25 D. 4 Solution: A

4. The winding ratio of the ideal transformer that matches the resistances gR 9= Ω and sR 1= Ω is:

A. 9 B. 3 C. 31 D.

91

Solution: B 5. The winding ratio of the ideal transformer that matches the resistances gR 4= Ω and sR 2= Ω is:

A. 0.5 B. 2

1 C. 2 D. 2

Solution: C 6. The winding ratio of the ideal transformer that matches the resistances gR 9= Ω and sR 3= Ω is:

A. 3

1 B. 9 C. 3 D. 3

Solution: D 7. The winding ratio of the ideal transformer that matches the resistances gR 4= Ω and sR 1= Ω is:

A. 2 B. 0.5 C. 0.25 D. 4 Solution: B

8. The winding ratio of the ideal transformer that matches the resistances gR 9= Ω and sR 1= Ω is:

A. 9 B. 3 C. 31 D.

91

Solution: C 9. The winding ratio of the ideal transformer that matches the resistances gR 4= Ω and sR 2= Ω is:

A. 0.5 B. 2

1 C. 2 D. 2

Solution: B 10. The winding ratio of the ideal transformer that matches the resistances gR 9= Ω and sR 3= Ω is:

A. 3

1 B. 9 C. 3 D. 3

Solution: A 11. If a Γ-circuit matches the resistances gR 4= Ω and sR 2= Ω and its inductive reactance is LX 2= Ω , then the capacitive reactance is:

A. – 4 Ω B. – 2 Ω C. 2 Ω D. 4 Ω Solution: A

12. If a Γ-circuit matches the resistances gR 3= Ω and sR 6= Ω and its inductive reactance is LX 6= Ω , then the capacitive reactance is:

A. 6 Ω B. – 3 Ω C. 3 Ω D. – 6 Ω Solution: B

13. If a Γ-circuit matches the resistances gR 8= Ω and sR 4= Ω and its inductive reactance is lX 4= Ω , then the capacitive reactance is:

A. – 4 Ω B. 8 Ω C. – 8 Ω D. 4 Ω Solution: C

14. If a Γ-circuit matches the resistances gR 2= Ω and sR 4= Ω and its inductive reactance is lX 4= Ω , then the capacitive reactance is:

A. 2 Ω B. 4 Ω C. – 4 Ω D. – 2 Ω Solution: D

15. A Γ-circuit matches the resistances gR 6= Ω and sR 3= Ω . What is its inductive reactance and how is it


connected to the load?

A. 3 Ω in series. B. 6 Ω in parallel. C. 4

18 Ω in series. D. 184 Ω in parallel.

Solution: A 16. A Γ-circuit matches the resistances gR 4= Ω and sR 8= Ω . What is its inductive reactance and how is it connected to the load?

A. 12 Ω in series. B. 8 Ω in parallel. C. 4 Ω in series. D. 2 Ω in parallel. Solution: B

17. A Γ-circuit matches the resistances: gR 13= Ω and sR 9= Ω . What is its inductive reactance and how is it connected to the load?

A. 9 Ω in series. B. 13 Ω in parallel. C. 6 Ω in series. D. 3 Ω in parallel. Solution: C

18. A Γ-circuit matches the resistances: gR 4= Ω and sR 13= Ω . What is its inductive reactance and how is it connected to the load?

A. 13 Ω in series. B. 4 Ω in parallel. C. 2

39 Ω in series. D. 3

26 Ω in parallel.

Solution: D 19. If a T-circuit induces the phase difference 30ϕ = ° , then the coupling factor and the sign of the radical from the expression of Xa and Xb are:

A. TK 2= ; + B. TK 2= ; – C. TK 2= − ; – D. TK 2= − ; + Solution: A

20. If a T-circuit induces the phase difference 30ϕ = − ° , then the coupling factor and the sign of the radical from the expression of Xa and Xb are:

A. TK 2= ; + B. TK 2= ; – C. TK 2= − ; – D. TK 2= − ; + Solution: C

21. If a T-circuit induces the phase difference 150ϕ = ° , then the coupling factor and the sign of the radical from the expression of Xa and Xb are:

A. TK 2= ; + B. TK 2= ; – C. TK 2= − ; – D. TK 2= − ; + Solution: B

22. If a T-circuit induces the phase difference 150ϕ = − ° , then the coupling factor and the sign of the radical from the expression of Xa and Xb are:

A. TK 2= ; + B. TK 2= ; – C. TK 2= − ; – D. TK 2= − ; + Solution: D


VI. Filters THEORY

1. Define the constant-k filters. The constant-k filters are the passive filters having the product of longitudinal and transversal inductances equaled to a constant. 2. Write the advantages of using the constant- k filters. (1) the configurations and the sizing relations are simple; (2) the attenuation in the stop band tends to infinity at the frequencies that are away from the cutoff frequency. 3. Write the disadvantages of using the constant- k filters. (1) the delimitation between the pass band and the stop band is not clear; (2) in the pass band, the characteristic impedance varies with the frequency, therefore the attenuation in the pass band is not zero. 4. Write the transversal and longitudinal impedances of an m derived filter, in terms of the longitudinal and

transversal impedances of the corresponding k filter, for the T configuration. 2

lm l tm t l1 1 mZ mZ ; Z Z Zm 4m

−= = +

5. Write the transversal and longitudinal admittances of an m derived filter, in terms of the longitudinal and

transversal admittances of the corresponding k filter, for the Π configuration. 2

lm l t tm t1 1 mY Y Y ; Y mYm 4m

−= + =

6. Write the values of the elements of a k LPF, in terms of the load resistance and the cutoff frequency.

s

t t s

2R 2L ; CRω ω

= =

7. Write the values of the elements of a k HPF, in terms of the load resistance and the cutoff frequency.

s

t t s

R 1L ; C2 2 Rω ω

= =

8. Write the relation between the normalized variable x and the longitudinal and transversal impedances of a non-

dissipative filter. 2 l

t

Z2x

2Z= −

9. Write the phase difference in the pass band of a non-dissipative filter in terms of the normalized variable x. ( ) ( )b x 2 arcsin x=

10. Write the attenuation in the pass band of a non-dissipative filter in terms of the normalized variable x. ( )a x 2 arg ch x=

11. Write the characteristic impedance of a constant-k filter in T, in terms of the normalized variable.

2CTZ R 1 x= − .

MCQ 1. For an m derived LPF, in T, the transversal inductance is given by:

A. 2

s

t

R1 m4m 2ω−

⋅ B. s2t

2R4m1 m ω

⋅−

C. 2

s

t

2R1 m4m ω−

⋅ D. s2t

R4m21 m ω⋅

−

Solution: C 2. For an m derived HPF, in T, the transversal capacitance is given by:


A. 2

s t

1 m 14m 2R ω−

⋅ B. 2s t

4m 2R1 m ω⋅

− C.

2

s t

1 m 24m R ω−

⋅ D. 2s t

4m 12R1 m ω⋅

−

Solution: D 3. For an m derived LPF, in T, the transversal capacitance is given by:

A. s t

1 2m R ω⋅ B.

t s

2mRω

⋅ C. s t

1 1m 2R ω⋅ D.

t s

1m2 Rω⋅

Solution: B 4. For an m derived HPF , in T, the transversal inductance is given by:

A. st

R1m 2ω⋅ B. s

t

2R1m ω⋅ C. s

t

Rm

2ω⋅ D. s

t

2Rm

ω⋅

Solution: A

i. flow graphs...f o r e x a m a s c 1 i. flow graphs theory 1. state the definition of the graph...

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