1. ELCT564 Spring 2012 9/17/20151ELCT564 Chapter 8: Microwave Filters

2. Filters 9/17/20152ELCT564 Two-port circuits that exhibit selectivity to frequency: allow some frequencies to go through while block the remaining In receivers, the system filters the incoming signal right after reception Filters which direct the received frequencies to different channels are called multiplexers In many communication systems, the various frequency channels are very close, thus requiring filters with very narrow bandwidth & high out-of band rejection In some systems, the receive/transmit functions employ different frequencies to achieve high isolation between the R/T channels. In detector, mixer and multiplier applications, the filters are used to block unwanted high frequency products Two techniques for filter design: the image parameter method and the insertion loss method. The first is the simplest but the second is the most accurate

3. Periodic Structures 9/17/20153ELCT564 Passband Stopband Bloch Impedance

4. Terminated Periodic Structures 9/17/20154ELCT564 Symmetrical network

5. Analysis of a Periodic Structure 9/17/20155ELCT564 Consider a periodic capacitively loaded line, as shown below. If Zo=50 Ω, d=1.0 cm, and Co=2.666 pF, compute the propagation constant, phase velocity, and Bloch impedance at f=3.0 GHz. Assume k=k 0.

6. Image Parameter Method 9/17/20156ELCT564

7. Constant-k Filter 9/17/20157ELCT564 m-derived section

8. Composite Filter 9/17/20158ELCT564

9. Summary of Composite Filter Design 9/17/20159ELCT564

10. Example of Composite Filter Design 9/17/201510ELCT564 Design a low-pass composite filter with a cutoff frequency of 2MHz and impedance of 75 Ω, place the infinite attenuation pole at 2.05 MHz, and plot the frequency response from 0 to 4 MHz.

11. Insertion Loss Method 9/17/201511ELCT564 Filter response is characterized by the power loss ratio defined as: Where Γ(ω) is the reflection coefficient at the input port of the filter, assuming the the output port is matched. Low-pass & Band-pass filter Insertion Loss:

12. Filter Responses 9/17/201512ELCT564 Maximally Flat, Equal Ripple, and Linear Phase Maximally Flat: Provides the flattest possible pass band response for a given complexity. Cutoff frequency is the freqeuncy point which determines the end of the pass band. Usually, where half available power makes it through. Cut-off frequency is called the 3dB point Equal Ripple or Chebyshev Filter: Power loss is expressed as Nth order Chebyshev polynomial T N (ω) T N (x)= cos (Ncos -1 x), |X| ≤1 T N (x)= cosh (Ncosh -1 x), |X|≥ 1 Much better out-of-band rejection than maximally flat response of the same order. Chebyshev filters are preferred a lot of times.

13. Filter Responses 9/17/201513ELCT564 Linear Phase Filters Need linear phase response to reduce signal distortion (very important in multiplexing) Sharp cut-off incompatible with linear phase– design specifically for phase linearity Inferior amplitude performance If φ(ω) is the phase response then filter group delay

14. Filter Design Method 9/17/201514ELCT564 Development of a prototype (low-pass filter with fc=1Hz and is made of generic lumped elements) Specify prototype by choice of the order of the filter N and the type of its response Same prototype used for any low-pass, band pass or band stop filter of a given order. Use appropriate filter transformations to enter specific characteristics Through these transformations prototype changes – low-pass, band-pass or band- stop Filter implementation in a desired from (microstrip or CPW) use implementation transformations.

15. Maximally Flat Low-Pass Filter 9/17/201515ELCT564 g 0 =1,ω c =1, N=1 to 10

16. Equal-Ripple Low-Pass Filter 9/17/201516ELCT564 g 0 =1,ω c =1, N=1 to 10

17. Maximally-Flat Time Delay Low-Pass Filter 9/17/201517ELCT564 g 0 =1,ω c =1, N=1 to 10

18. Filter Transformations 9/17/201518ELCT564 Impedance Scaling Frequency Scaling for Low-Pass Filters Low-Pass to High-Pass Transformation

19. Filter Implementation 9/17/201519ELCT564 Richards’ Transformation Kuroda’s Identities Physically separate transmission line stubs Transform series stubs into shunt stubs, or vice versa Change impractical characteristic impedances into more realizable ones

20. Design Steps 9/17/2015 20 ELCT564 Lumped element low pass prototype (from tables, typically) Convert series inductors to series stubs, shunt capacitors to shunt stubs Add λ/8 lines of Zo = 1 at input and output Apply Kuroda identity for series inductors to obtain equivalent with shunt open stubs with λ/8 lines between them Transform design to 50Ω and fc to obtain physical dimensions (all elements are λ/8).

21. Low-pass Filters Using Stubs 9/17/2015 21 ELCT564 Distributed elements—sharper cut-off Response repeats due to the periodic nature of stubs Design a low-pass filter for fabrication using microstrip lines. The specifications include a cutoff frequency of 4GHz, and impedance of 50 Ω, and a third-order 3dB equal-ripple passband response.

22. Bandpass and Bandstop Filters 9/17/2015 22 ELCT564 A useful form of bandpass and bandstop filter consists of λ/4 stubs connected by λ/4 transmission lines. Bandpass filter

23. Stepped Impedance Low-pass Filters 9/17/2015 23 ELCT564 Use alternating sections of very high and very low characteristics impedances Easy to design and takes-up less space than low-pass filters with stubs Due to approximations, electrical performance not as good – applications where sharp cut-off is not required

24. Stepped Impedance Low-pass Filter Example 9/17/2015 24 ELCT564 Design a stepped-impedance low-pass filter having a maximally flat response and a cutoff frequency of 2.5 GHz. It is necessary to have more than 20 dB insertion loss at 4 GHz. The filter impedance is 50 Ω; the highest practical line impedance is 120 Ω, and the lowest is 20 Ω. Consider the effect of losses when this filter is implemented with a microstrip substrate having d = 0.158 cm, ε r =4.2, tanδ=0.02, and copper conductors of 0.5 mil thickness.

25. Coupled Line Theory 9/17/2015 25 ELCT564

26. Coupled Line Bandpass Filters 9/17/2015 26 ELCT564 This filter is made of N resonators and includes N+1coupled line sections dn ≈ λg/4 = (λge + λgo)/8 Find Zoe, Zoo from prototype values and fractional bandwidth From Zoe, Zoo Calculate conductor and slot width N-order coupled resonator filter N+1 coupled line sections Use 2 modes to represent line operation

27. Coupled Line Bandpass Filters 9/17/2015 27 ELCT564 1. Compute Zoe, Zoo of 1st coupled line section from 2. Compute eve/odd impedances of nth coupled line section 3. Compute even/odd impedances of (N+1) coupled line section 4. Use ADS to find coupled line geometry in terms of w, s, & βe, βo or ε eff,e, ε eff,o 5. Compute

28. Coupled Line Bandpass Filters Example I 9/17/2015 28 ELCT564 Design a 0.5dB equal ripple coupledline BPF with fo=10GHz, 10%BW & 10-dB attenuation at 13 GHz. Assume Zo=50Ω. From atten. Graph N=4 ok But use N=5 to have Zo=50 Ω go=ge=1, g1=g5=1.7058, g2=g4=1.229, g3=2.5408

29. Coupled Line Bandpass Filters Example II 9/17/2015 29 ELCT564 Design a coupled line bandpass filter with N=3 and 0.5dB equal ripple response. The center frequency is 2GHz, 10%BW & Zo=50Ω. What is the attenuation at 1.8 GHz

30. Capacitively Coupled Resonator Filter 9/17/2015 30 ELCT564 Convenient for microstrip or stripline fabrication Nth order filter uses N resonant sections of transmission line with N+1 capacitive gaps between then. Gaps can be approximated as series capacitors Resonators are ~ λg/2 long at the center frequency

31. Capacitively Coupled Resonator Filter 9/17/2015 31 ELCT564 Design a bandpass filter using capacitive coupled series resonators, with a 0.5 dB equal-ripple passband characteristic. The center frequency is 2.0 GHz, the bandwidth is 10%, and the impedance is 50 Ω. At least 20 dB of attenuation is required at 2.2GHz

32. Bandpass Filters using Capacitively Shunt Resonators 9/17/2015 32 ELCT564

33. Bandpass Filters using Capacitively Shunt Resonators 9/17/2015 33 ELCT564 Design a third-order bandpass filter with a 0.5 dB equal-ripple response using capacitively coupled short-circuited shunt stub resonators. The center frequency Is 2.5 GHz, and the bandwidth is 10%. The impedance is 50 Ω. What is the resulting attenuation at 3.0 GHz?