Introduction to RF Filter Design. RF Electronics Spring, 2016 Robert R. Krchnavek Rowan University

Transcription

1 Introduction to RF Filter Design RF Electronics Spring, 2016 Robert R. Krchnavek Rowan University

2 Objectives Understand the fundamental concepts and definitions for filters. Know how to design filters using tabulated parameters for common filter types. Know how to convert lumped-element filter designs into distributed-element filters.

3 Filter Configurations Ω= ω ω c where ω c is defined as the cutoff frequency for low-pass and high-pass filters and the center frequency for bandpass and bandstop filters.

4 Low-Pass Filters Profiles for Three Common Types Binomial or Butterworth - easy to implement; monotonic profile; requires numerous elements to get step profile. Chebyshev - equal amplitude variations; steeper profile than Butterworth. Elliptic or Cauer - amplitude variations in both stopband and passband; steepest profile; complicated to design.

5 Bandpass Filter - Profile

6 Filter Definitions Insertion loss - how much power is lost in going through the filter. Ripple - the flatness of the signal in the passband. Bandwidth - the width of the passband. Shape factor - the sharpness of the filter response. Rejection - the attenuation of the undesired signals. Quality factor - see next slide. IL = 10 log P in P L BW 3dB = f 3dB U SF = BW 60dB BW 3dB = f U 60dB fu 3dB f 3dB L fl 60dB fl 3dB

7 Q - Quality Factor The quality factor, or Q, is a parameter that is used to describe the selectivity of the filter. The unloaded Q is defined as Q =2πf C ( maximum energy stored in the filter at fc power lost in the filter ) The loaded Q is defined as Q LD =2πf C ( ) maximum energy stored in the filter at f C power lost in the filter and to the external circuit A higher Q indicates a more selective filter. Details to follow.

8 Series RLC Bandpass Filter Find V R1 for this circuit.

9 Series RLC Bandpass Filter Q =2πf C ( maximum energy stored in the filter at fc power lost in the filter 1 2 LI2 p = maximum energy stored in the filter at f C ) I 2 rmsr = power lost in the filter I 2 rms = 1 2 I2 p Q =2πf C L R = ω C L R The resonant frequency is the frequency where the imaginary component of the impedance is equal to 0: ȷω C L + 1 ȷω C C =0 ω C = 1 LC

10 Series RLC Bandpass Filter Solving for the frequencies at which V R1 is down 3 db yields ω U = R 2L + ( R 2L ) LC and ω L = R 2L + ( ) 2 R + 1 2L LC The bandwidth is given by BW = ω U ω L = R L and BW = f U f L = 1 2π R L And, using our previous result for unloaded Q, we see the relationship between Q and BW is given by Q =2πf C L R = f C BW or BW = f C Q

11 Series and Parallel Resonators

12 Quantitative Analysis of a Series RLC Bandpass Filter H(ω) = V L V G H(ω) = V L V G = Z L (Z G + Z L )+R + ȷ [ωl 1/(ωC)] This filter is different from the previous one because of the addition of Z G and Z L.

13 Series RLC Bandpass Filter where R E = Z G + Z L Define three different Q factors: Unloaded, internal, or filter Q External Q Loaded Q Q F = ω CL R Q E = ω CL R E Q LD = ω CL R + R E

14 Series RLC Bandpass Filter Note: Z L =Z G =50Ω, R=20Ω, L=5 nh, and C=2 pf.

15 Series RLC Bandpass Filter Series RLC bandpass filter is easy to analyze. Minimum attenuation at the resonance point. HOWEVER, the transition from passband to stop band is not very sharp (large shape factor.)

16 Butterworth Filter One of a series of special filter designs that consist of more elements than a simple RLC and give better control over the filter parameters. Also known as a maximally flat filter - no ripple. Strategy First, do the normalized, low-pass filter. Second, implement the desired form through a frequency scaling. Third, if necessary, create distributed elements.

17 Butterworth Filter g N+1 = g m = inductance for series inductor capacitance for shunt capacitor (m 1,...,N) { load resistance if the last element is a shunt capacitor load conductance if the last element is a series inductor } Two different networks that are used to implement the Butterworth filter.

18 Butterworth Filter IL = 10 log P in P L = 10 log ( 1+a 2 Ω 2N) The coefficient a is usually taken to be 1 so that the IL is 3 db at the cutoff frequency.

19 Butterworth Filter The attenuation vs frequency as a function of the number of stages. Note: this design does NOT result in a linear phase relationship.

20 Butterworth Filter Coefficients for a maximally flat response.

21 Butterworth Filter Coefficients for a linear phase response.

22 Comments Coefficients for a 3 db Chebyshev filter design are in Table 5.4 (a). Coefficients for a 0.5 db Chebyshev filter design are in Table 5.4 (b). The generic, multisection, normalized element circuits designs are the same for Butterworth and Chebyshev filters.

23 Butterworth Example Design a 4th-order, low-pass, standard (maximally flat), 3 db Butterworth filter.

24 Frequency and Impedance Transformations The normalized values need to be modified to produce The desired response (low-pass, high-pass, etc.) At the desired center frequency. With an impedance that is realistic.

25 Frequency Transformation All of the different filter types are derived from the low-pass filter. The key is to determine a transformation function that maps the normalized, low-pass design into the appropriate (low-pass, high-pass, etc.) at the desired frequency. New values for L and C are obtained by maintaining the same Z through the transformation.

26 Frequency Transformation Low Pass ω c is the new cutoff frequency! =! c The impedances should remain the same: Z L = L =!! c L =!L new L new = L! c Z C = 1 C =! c!c = 1!C new C new = C! c

27 Normalized Low-Pass Low-Pass ω =Ωω c Z L = ȷΩL = ȷ ω ω c L = ȷωL new L new = L ω c Z C = 1 ȷΩC = ω c ȷωC = 1 ȷωC new C new = C ω c

28 Frequency Transformation High Pass ω c is the new cutoff frequency! =! c Again, the impedances should remain the same giving....

29 Normalized Low-Pass High-Pass ω = ω c Ω C new = 1 ω c L L new = 1 ω c C

30 Frequency Transformations Bandpass and Bandstop These transformations are more complex. See the textbook for both.

31 Frequency Transformations Summary

32 Impedance Transformation For the Butterworth designs, the source and load resistances have a value of 1. For the Chebyshev designs, even-number ordered designs have a non-unity load. Impedance transformation is the process of adjusting all the elements to account for different source and load impedances.

33 Impedance Transformation Assume the source impedance, R G, is scaled from 1 in the original design to R G,new. Then, the new values are: R G, new =1R G, new L new = LR G, new C C new = R G, new R L, new = R L R G, new

34 Distributed-Element Filters Above approximately 1 GHz, lumped-element filter design is problematic because the elements are approaching a significant fraction of λ. Distributed-element filters are common. One approach is to design the lumped-element filter and then convert it to a distributedelement realization.

35 Distributed-Element Filters Assume you have a lumped-element filter design that you want to build as a distributed-element filter. Recall our expression for the impedance of a terminated (lossless) transmission line: Z(d) =Z 0 Z L + ȷZ 0 tan βd Z 0 + ȷZ L tan βd

36 If Z L = 0, then Z(d) =ȷZ 0 tan βd If Z L =, then Z(d) = ȷZ 0 cot βd The electrical length, βd, can be put in the following form βd = 2π λ d = 2π v p /f d = 2πf v p Assume we chose a line that is 1/8 of a wavelength d = λ/8 d

37 The expression for impedance explicitly in terms of frequency is then (Z L = 0) Z(d) =ȷZ 0 tan 2πf v p The impedance of the stub must equal the impedance of the lumped element v p ȷωL = ȷZ 0 tan π 4 f 0 8 = ȷZ 0 tan π 4 For a capacitive element, you could use the opencircuited transmission line. f f 0 f f 0 One significant difference is the frequency range is shortened because the tan function is periodic.

38 For a d = short-circuited line, we have the following for the Richard s transform: For a d= open-circuited line, we the following for the Richard s transform:

39 Distributed-Element Filters - Physical Realization Using transmission line sections to build the filter may require sections of line that separate elements from each other. These are called unit elements. The unit elements have an electrical length of f βd = π 4 f 0 We also need to be able to create distributed element sections for difficult-to-replace lumped elements such as series inductors. We use Kuroda s identities for this.

40 Distributed Filter Implementation Design a 4th-order, low-pass, standard (maximally flat), 3 db Butterworth filter. It should have a cutoff frequency of 1 GHz. 1. Select the normalized filter order and parameters to meet the design criteria. 2. Replace inductances and capacitances with equivalent λ/8 transmission lines. 3. Convert series stub lines to shunt stub lines through Kuroda s identities. 4. Denormalize and select equivalent microstrip lines.

41 Distributed Filter Implementation R=1Ω H H G=1 Ω F F 1. Select the normalized filter order and parameters to meet the design criteria.

42 Distributed Filter Implementation Z 0 = Z 0 = R=1Ω G=1 Ω Y 0 = Y 0 = Replace inductances and capacitances with equivalent λ/8 transmission lines.

43 Distributed Filter Implementation Z 0 = Z 0 = R=1Ω Z UE =1Ω Z UE =1Ω UE UE G=1 Ω Y 0 = Y 0 = Convert series stub lines to shunt stub lines through Kuroda s identities.

44 Distributed Filter Implementation Z 0 = Z 0 = R=1Ω Z UE =1Ω UE UE G=1 Ω Y 0 = Convert series stub lines to shunt stub lines through Kuroda s identities.

45 Distributed Filter Implementation Z 0 = R=1Ω UE UE G=1 Ω Y 0 = Convert series stub lines to shunt stub lines through Kuroda s identities.

46 Distributed Filter Implementation Z 0 = R=1Ω Z UE =1Ω UE UE UE G=1 Ω Y 0 = Convert series stub lines to shunt stub lines through Kuroda s identities.

47 Distributed Filter Implementation R=1Ω UE UE UE G=1 Ω Y 0 = Convert series stub lines to shunt stub lines through Kuroda s identities.