1. Hanyang University 1/24 Microwave Engineering Chapter 8.8 Wonhong Jeong 2015.04.28

2. Hanyang University 2/24 8.8.1. Bandstop and Bandpass Filters Using Quarter-Wave Resonators From chapter 6, we studied about quarter-wave transmission line and stubs for matching techniques. Quarter-wavelength sections of line between the stubs act as admittance inverters to effectively convert alternate shunt resonators to series resonators. For narrow bandwidths the response of such a filter using N stubs is essentially the same as that of a coupled line filter using N + 1 sections. The internal impedance of the stub filter is Z 0, while in the case of the coupled line filter end sections are required to transform the impedance level. This makes the stub filter more compact and easier to design. A disadvantage, however, is that a filter using stub resonators often requires characteristic impedances that are difficult to realize in practice. λ/4

3. Hanyang University 3/24 Bandstop filter using N open circuited stubs Transmission line of characteristic impedance Z 0n is And this impedance can be approximated as for frequencies in the vicinity of the center frequency. The impedance of a series LC circuit is where The impedance of a series LC circuit is

4. Hanyang University 4/24 Equating (8.122) and (8.123) gives the characteristic impedance of the stub in terms of the resonator parameters: If we consider the quarter-wave sections of line between the stubs as ideal admittance inverters, the bandstop filter can be represented by the equivalent circuit. Next, the circuit elements of this equivalent circuit can be related to those of the lumped-element bandstop filter prototype.

5. Hanyang University 5/24 The admittance at the corresponding point in the circuit is The admittance Y seen looking toward the L 2 C 2 resonator is

6. Hanyang University 6/24 Equating with (8.125) and (8.126), We can obtain following conditions. Since, these results can be solved for L n ; Using (8.124) and the impedance-scaled bandstop filter elements from Table 8.6 gives the stub characteristic impedances as (General form of characteristic impedances of a bandstop filter using open-circuited stub)

7. Hanyang University 7/24 Bandstop filter using N shorted circuited stubs For a bandpass filter using short-circuited stub resonators the corresponding result is These results only apply to filters having input and output impedances of Z 0 and so cannot be used for equal-ripple designs with N even.

8. Hanyang University 8/24 Example 8.8 Bandstop Filter Design Design a bandstop filter using three quarter-wave open-circuit stubs. The center frequency is 2.0 GHz, the bandwidth is 15%, and the impedance is 50 Ω. Use an equal-ripple response, with a 0.5 dB ripple level. The fractional bandwidth is Δ = 0.15. When the values for N = 3, g n value is “right red box”. Using equation (8.130), we can obtain Z 0. Solution]

9. Hanyang University 9/24 Continued] The calculated attenuation for this filter is shown in Figure 8.49; the ripple in the passbands is somewhat greater than 0.5 dB as a result of the approximations involved in the development of the design equations. The calculated attenuation for this filter is shown in Figure 8.49

10. Hanyang University 10/24 8.8.2. Bandpass Filters Using Capacitively Coupled Series Resonators Another type of bandpass filter that can be conveniently fabricated in microstrip or stripline form is the capacitive-gap coupled resonator filter. [1] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, Artech House, Dedham, Mass., 1980. An Nth-order filter of this form will use N resonant series sections of transmission line with N + 1 capacitive gaps between them. These gaps can be approximated as series capacitors; design data relating the capacitance to the gap size and transmission line parameters is given in graphical form in reference [1]. The filter can then be modeled as shown in Figure as below. The resonators are approximately λ/2 long at the center frequency, ω 0.

11. Hanyang University 11/24 We can redraw negative-length transmission line sections on either side of the series capacitors. The lines of length ϕ will be λ/2 long at ω 0, so the electrical length θ i of the i th section is with ϕ i < 0. The reason for doing this is that the combination of series capacitor and negative-length transmission lines forms the equivalent circuit of an admittance inverter. In order for this equivalence to be valid, the following relationship must hold between the electrical length of the lines and the capacitive susceptance. In this condition, the equivalent circuit with negative-length transmission line sections on either side of the series capacitors can be drawn as below.

12. Hanyang University 12/24 The capacitive-gap coupled filter can then be modeled as shown in below. By using the design equations (8.121) for a bandpass filter with N + 1 coupled line sections, we can find admittance inverter constants, J i, from the low-pass prototype values, g i, and the fractional bandwidth, Δ. As in the case of the coupled line filter, there will be N + 1 inverter constants for an N th-order filter. Then (8.134) can be used to find the susceptance, B i, for the i th coupling gap. Finally, the electrical length of the resonator sections can be found from (8.132) and (8.133):

13. Hanyang University 13/24 Example 8.9 Capacitively coupled series resonator bandpass filter design 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. At least 20 dB of attenuation is required at 2.2 GHz. We first determine the order of the filter to satisfy the attenuation specification at 2.2 GHz. Using (8.71) to convert to normalized frequency gives Solution] Attenuation versus normalized frequency for equal-ripple filter prototypes at 0.5 dB ripple level. From the figure, we see that N = 3 should satisfy the attenuation specification at 2.2 GHz. Then,

14. Hanyang University 14/24 The low-pass prototype values are given in Table 8.4, from which the inverter constants can be calculated using (8.121). Then the coupling susceptances can be found from (8.134), and the coupling capacitor values as Continued]

15. Hanyang University 15/24 Finally, the resonator lengths can be calculated from (8.135). The following table summarizes these results. Continued]  Results

16. Hanyang University 16/24 8.8.3. Bandpass Filters Using Capacitively Coupled Shunt Resonators A related type of bandpass filter is shown in Figure 8.52, where short-circuited shunt resonators are capacitively coupled with series capacitors. An Nth-order filter will use N stubs, which are slightly shorter than λ/4 at the filter center frequency. The short-circuited stub resonators can be made from sections of coaxial line using ceramic materials having a very high dielectric constant and low loss, resulting in a very compact design even at UHF frequencies [9]. [9] M. Sagawa, M. Makimoto, and S. Yamashita, “A Design Method of Bandpass Filters Using Dielectric-Filled Coaxial Resonators,” IEEE Transactions on Microwave Theory and Techniques, vol. MTT-33, pp. 152–157, February 1985. Such filters are often referred to as ceramic resonator filters and are among the most common types of RF bandpass filters used in portable wireless systems. Most cellular telephones, GPS receivers, and other wireless devices employ two or more filters of this type.

17. Hanyang University 17/24 Below the figure indicates the general bandpass filter circuit where shunt LC resonators alternate with admittance inverters. The function of the admittance inverters is to convert alternate shunt resonators to series resonators. The extra inverters at the ends serve to scale the impedance level of the filter to a realistic level. Using an analysis similar to that used for the bandstop filter, we can derive the admittance inverter constants and coupling capacitor values as

18. Hanyang University 18/24 Replace the admittance inverters with the equivalent π-network for obtaining equivalent lumped element circuit. effective resonator capacitor values are given by where C n = −C n-1,n − C n,n+1 represents the change in the resonator capacitance caused by the parallel addition of the inverter elements.

19. Hanyang University 19/24 Note that the resonant frequency of the stub resonators is no longer ω 0, since the resonator capacitor values have been modified by the ΔC n. This implies that the length of the resonator is less than λ/4 at ω 0, the filter center frequency. A short-circuited length of line with a shunt capacitor at its input has an input admittance of If the capacitor is replaced with a short length, Δl, of transmission line, the input admittance would be

20. Hanyang University 20/24 Comparing (8.139b) with (8.139a) gives the change in stub length in terms of the capacitor value: if C < 0, then Δl < 0, indicating a shortening of the stub length. Thus the overall stub length is given by Dielectric material properties play a critical role in the performance of ceramic resonator filters. Materials with high dielectric constants are required in order to provide miniaturization at the frequencies typically used for wireless applications. Losses must be low to provide resonators with high Q, leading to low passband insertion loss and maximum attenuation in the stopbands.

21. Hanyang University 21/24 Example 8.10 Capacitively coupled shunt resonator bandpass filter design 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? We first calculate the attenuation at 3.0 GHz. Solution] Attenuation versus normalized frequency for equal-ripple filter prototypes at 0.5 dB ripple level. From the figure, we can find the attenuation as 35 dB. Then,

22. Hanyang University 22/24 Then we use (8.138), (8.140), and (8.141) to find the required resonator lengths: Continued] Next we calculate the admittance inverter constants and coupling capacitor values using (8.136) and (8.137): Note that the resonator lengths are slightly less than 90 ◦ (λ/4).

23. Hanyang University 23/24 Continued] Note that the resonator lengths are slightly less than 90 ◦ (λ/4). The stopband rolloff at high frequencies is less than at lower frequencies, and the attenuation at 3 GHz is seen to be about 30 dB, while our calculated value for a canonical lumped-element bandpass filter was 35 dB.

24. Hanyang University 24/24 Thank you for your attention