ECE 5180/6180 Microwave Filter DesignLectures: Lumped element filters (also applies to low frequency filters)Stub FiltersStepped Impedance FiltersCoupled Line FiltersLumped Element FiltersText Section 8.3Portfolio Question: How do you design a lumped element filter using the Insertion Loss MethodPower Loss RatioInsertion Loss IL = 10 log PLR2-port network:Filter Design by insertion loss method controls () to control passband and stopband of filter.Filter parameters:Passband -- frequencies that are passed by filterStopband -- frequencies that are rejectedInsertion loss -- how much power is transferred to load in passbandAttenuation -- how much power is rejected (not transferred to the load) in the stopbandPPower Available from SourcePower Delivered to LoadPPLRincload    VVS SS SVVPsourceVZPloadVZFor Matched System Z Z ZThenPsourcePloadVVS SFor ciprocal System S SPin Lin L oLRin in1211 1221 2212122212221212221 122 21 11111    ( Re )( ) Cutoff rate or attenuation rate -- how quickly the filter transitions from pass-to-stop or stop-to-passbandsPhase response -- Linear phase response in the passband means that signal will not be distorted.Classes of Filters:Determined by .We have not proven this yet, but a useful mathematical proof (section 4.1) shows that |()|2 is an even function of . So |()|2 can be written as a polynomial in 2 .The class of filter is controlled by the type of polynomial used.Polynomials M and N can be  Binomial (Butterworth) -- Maximally Flat Chebyshev -- Equal Ripple Elliptic -- Specified Minimum Stopband Attenuation (faster cutoff ) Linear Phase Binomial / Butterworth / Maximally FlatLow Pass Filter Design:N = Filter Order = frequency of interestc = cutoff frequency At c , PLR = 1+k2 If the -3dB point is defined to be the cutoff point (common), k=1For >>c then PLR k2 (/c)2N which means Insertion Loss increases at a rate of 20N dB / decade.(This allows us to increase the steepness of the cutoff by adding more sections.)Chebyshev / Equal Ripple FiltersWhere TN are Chebyshev polynomialsRipples are equal-in-size = 1+k2 Cutoff Rate is 20N dB/decade, same as binomial.Insertion loss in the stopband is (22N)/4 greater than binomial.Elliptic and Linear Phase FiltersOther options, see textbook.Filter Design Method1. Design a LP filter for normalized Z,2. Scale Z. 3. Convert from LP to HP or BP as desired.4. Convert from lumped to distributed elements as desired.( )( )( ) ( )( )( ) 222 2221 MM NPMNLRP kLRcN 122P k TLR Nc 12 21. Binomial Design of LP Filter for Normalized Z,a. Determine how many elements are needed (N)Find /c and look at the figure for attenuation in the stopband (Fig.8.26, p.450)Example: How many elements are required to design a maximally-flat filter with a cutoff frequency of2 GHz if the filter must provide 20 dB of attenuation at 4 GHz? For this case, |/c|-1 = |4 / 2 | -1 = 1.0 (bottom axis). Find N line on filter that is ABOVE the desired attenuation. N=4.b. Find resistance or conductance values from Table 8.3Look at N=3.g1 = 1.0; g2 = 2.0; g3 = 1.0; g4 = 1.0c. Choose LP Filter PrototypeWhy choose one over the other? Available components. (Responses of both are identical.)Notes:1) Rg and RL must be REAL. What if they aren't? (Add a length of line, resonate, or absorb imaginary part.)2) The designs in our book always have Rg=RL. What if they are not equal? There are other tables… see handout. This is effectively matching and filtering simultaneously.3) Design so far has considered normalized impedances Rg=RL=1 and normalized frequency (c =1) … Use impedance and frequency scaling if they aren't 1.1. Chebyshev (Equal Ripple) Design of LP Filter for Normalized Z,Same steps as for binomial. There are only a few differences….(a) Determine number of elements (N). This will always be less than or equal to binomial.Use Figure 8.27, with choice of size of ripple.(b) Use table 8.4, with same choice of ripple as used in part a.(c) Same as binomial.2. Impedance and Frequency Scaling (normalization)To build the same filter for Zo = Rg = RL and a given cutoff frequency cUse the same filter prototypes but scale the values:(a) Top filter prototypeRg=(Zo)(go)=C1=g1 / (Zo c)L2=(Zo)(g2) / cC3=g3 / (Zo c)RL = (Zo)(g4)(b) Bottom filter prototypeRg=1/(Zo go)L1=(Zo)(g1) / cC2=g2 / (Zo c)L3=(Zo)(g3) / cRL=1/(Zo g4)Binomial and Chebyshev are the same here, except for one difference:RL for binomial is always matched. RL for odd-order Chebyshev filters is NOT matched. Use a quarter-wave transformer to match.Example:For 0.5 dB ripple and N=3, g3=1.9841. For top prototype, RL=1.9841 Zo, which is not matched. Quarterwave transformer would have Zq=(1.9841) Zo3. Convert from LP to HP High Pass ConfigurationsFor both prototypes:Ck = 1/ (Zo c gk)Lk = Zo/ (c gk)For top prototype:Rg = Zo goRL = Zo / gn+1For bottom prototype:Rg = 1 / (Zo go)RL = 1/ (Zo gn+1)3. Convert from LP to Bandpass or BandstopNormalized bandwidth1 = lower limit2 = upper limito = 12See Table 8.6 p. 461 for conversions.To use for unnormalized filters:L  L ZoC  C / Zo  2