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S-55.3120 Passive filters, exercise problems

1. Realize driving point impedance

Z(s) =s2 + 4s + 1

s2 + 2s + 1

applying resistance, conductance and reactance reductions (in that order).

2. Realize driving point impedance

Z(s) =5s4 + 7s3 +

17

2s2 +

7

2s + 2

(2s2 + 1)(2s2 + s + 1).

3. Realize driving point impedance

Z(s) =2s3 + 2s

2s2 + 1

applying theorems Foster I/II and Cauer I/II.

4. Determine the scattering matrix and the characteristic function of the circuit below.

rrr

1rrr

1

1

1

2

r

r r

r

5. The transducer power gain of a reciprocal and lossless two-port is

|S21(j)|2 = 4(4 22 + 1)

96 24 72 + 4 .

Determine the scattering matrix S of this two-port.

6. Determine the characteristic function, , of the two-port of the previous problem. Drawj(j), |(j)|2 and |S21(j)|2. How does the characteristic function describe the propertiesof the filter ? Hint:

(j)

= 0, when

= 1 0.36260 = 2 1.59222

j(j1) 0.12640j(j2) 3.4255

7. Design a filter obeying attenuation 1/|S21|2 = 1 + 6. The termination impedances are 1 .

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8. Design a lossless Butterworth lowpass filter of order three such, that the 3 dB cutoff fre-quency is p = 1 and the termination impedances are equal to 1 .

9. Design a 2nd order lossless Chebyshev lowpass filter such, that the 3 dB cutoff frequency isp = 1 and the load impedance is 1 .

11. Design a Butterworth filter such, that the attenuation satisfies the condition shown in thefigure below. The termination impedances are 50 .

T

E5 9 11 20 f [MHz]

80

0.1

Attn. [dB]

12. Design a Chebyshev filter that satisfies the conditions of the previous problem.

14. Design, using Brunes synthesis, a filter obeying transducer power gain

|S21(j)|2 = 4(4 22 + 1)

96 24 72 + 4such, that the termination impedances are 1 .

15. Determine the open-circuit impedance matrix, when

|S21(j)|2 = 4(2 1)2

96 24 72 + 4and the termination impedances are 1 .

16. Design a filter of the previous problem using zero-shifting technique.

17. Design a TL11 filter obeying transducer power gain

|S21(j)|2 = 64(2

1)2

(2

2)2

14410 4888 + 1936 + 7414 7642 + 256with termination impedances equal to 1 . Realize the resonators in order = 1 and =

2

starting from the generator side. Hint:

S =1

12s5 + 20s4 + 37s3 + 45s2 + 26s + 16

12s5 + 23s3 + s2 + 2s 8(s2 + 1)(s2 + 2)

8(s2 + 1)(s2 + 2) 12s5 + 23s3 s2 + 2s

Remove the negative inductance using tightly coupled transformer.

18. Show, that the filter of the previous problem can be realized with a TL11 circuit withoutnegative elements (and design the filter).

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19. What is the minimum order of an elliptic filter that satisfies the |S21(j)|2 in the figurebelow ? Compare the order with that of the Chebyshev filter.

T

E0 1 2

0

1

65

|S21(j)|2 [dB]

20 Determine |S21(j)|2 of the previous problem finding first the reflection and transmissionzeros. Determine the other extreme points also. If |S21(j)|2 is realized with a L00 circuitsuch that the passband is 0...10 kHz, what are the resonance frequencies of the parallel LCsections ?22. Design an elliptic filter that satisfies the requirements shown in the figure below. Thetermination impedances are 600 .

T

E0 10 15.6 f [kHz]

0

0.2

45

|S21(j)|2 [dB]

23. Design an elliptic type bandpass filter that obeys the requirements of the figure below.The generator impedance is 150 . Draw the attenuation (approximately), and calculate theresonance frequencies of the parallel LC sections.

T

E

0

1.5

35

|S21(j)|2 [dB]

0 30 50 20000 30000 f [Hz]

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24. Derive the Nortons transform shown in the figure below.C1

C2

r r

r r

C2K

C1/K

r

r r

r

K : 1

K = C1 + C2

C1

25. In the figure below is elliptic prototype filter C062021b such that the cutoff frequencyof the stop band is s = 2.882384, attenuation As = 100.04 dB and the transmission zeroesare 1 = 4.160091 and 2 = 2.985065. Design a coil-saving bandpass filter with a cen-ter frequency 468 kHz and bandwidth 35 kHz. The generator impedance is 5 k and the load

impedance is 500 .

rrr

1 1.283

1.324

2.099

1.371

1.987

0.8837

rrr 2

3

1 2

0.04365 0.08185

26. In the figure below is an elliptic bandpass filter (passband ripple is 0.25 dB in the fre-

quency range 800.0 kHz ... 800.4 kHz).The attenuation at the stopband is at least 50 dB,when f 799.75kHz and f 800.65 kHz. Design a crystal filter such, that the inductancevalue of the crystal is L = 74.40mH and r 150. The termination impedances are 1 k.f1 = 799.06 kHz, f2 = 799.71 kHz,f3 = 800.69 kHz, and f4 = 801.34 kHz. The units of theelements in the figure are L[mH], C[pF] and R[].

rrr

1000

390.7

74.4

0.5323

383.7

0.005241

7526

21.53

8.287

4.787

1674

0.0005838

67680

93.91

92.22rrr

1098

f1f2

f3f4

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27. Determine the ABCD-matrix and the driving point impedance Z() of the circuit below.What is the general form of the ABCD-matrix, when n transmission lines are connected inseries. Calculate Z(1) and compare the result to the case when the length of the lines is

infinite. What is S21() ?

rrr

1 Z0 = 4, Z0 = 2, rrr

1E

Z()

r

r

29. Show (using ABCD-matrices), that the two-ports below are equivalent. L represents an

inductance in the plane (short circuited transmission line) with delay and characteristicimpedance L.

L

Z0,

r

r

r

r

Z0, 1

Z0, = 1 + L

Z0

r

r

r

r

30. In the figure below is a Chebyshev lowpass filter with a 3 dB cutoff frequency equal to one.Using this prototype, design a transmission line bandpass filter with center frequency of 3 GHzand bandwidth 10 %. The termination impedances are 50 . Draw |S21(j)|2 (approximately)in the frequency range 0...8 GHz.

rrr

1 3.48

0.762

4.54

0.762

3.48rrr

1

r

r

r

r

r

r

31. Design using elliptic lowpass prototype filter an transmission line filter with center fre-quency 10 GHz and the reflection coefficient less than 10% in the passband frequency range9.5...10.5 GHz. The stopband attenuation must be at least 60 dB at frequency 11.4 GHz. Thetermination impedances are 50 . Draw |S21(j)|2 (approximately) in the frequency range0...25 GHz.

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28. Determine Zn-1() using Zn(). When the degree ofZn-1 is the same as the degree of Zn? On what condition is the degree of Zn-1 lower than the degree of Zn ?

Z0

Zn-1()

EZn()E

r

r

32. Realize Z() with a circuit having transmission lines connected in series.

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r

rr

r

Z()E

33. Determine constants a, b, and c such that Z() can be realized with a circuit havingtransmission lines connected in series and terminated with a 1 resistor. Realize Z() withsuch circuit.

Z() = a2

+ b + c42 + 5 + 6

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34. Determine scattering matrix S() and the driving point impedance Z() such that Z(0) =1, when

S21S

21 =1

1 + x4 ,

where a) x = cosh(s) and b) x = sinh(s). What is the ratio of the termination impedancesand the greatest stopband attenuation in each case ? Draw (approximately) |S21|2 a) on thecos() axis, b) on the sin() axis and on the j axis in both cases. What are the relative1 dB bandwidths, when in case a) the filter is a bandpass filter and b) the filter is a lowpassfilter and the bandwidth is compared to the lowest frequency that gives the greatest stopbandattenuation.35. Design a Butterworth lowpass filter using one 5cm long transmission line having 3 dBbandwidth equal to 1 GHz. The termination impedances are 50 . What is the greatest value

of the attenuation in the stopband ?

36. Design a Chebyshev type transmission line filter (transmission lines are connected in series)with a center frequency of 3 GHz and the 3 dB band in the frequency range 2 ...4 GHz. Theattenuation in the stopband must be at least 15 dB. The load impedance is 50 .