ELEctromagnetic DIAgnostics Lab.DIT Università di TrentoDIT ‐ Università di Trento

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Microwave Filters The insertion loss method

Master Master DegreeDegree Electronic and TelecommunicationElectronic and TelecommunicationA.A. A.A. 20122012‐‐20132013

The insertion loss method

A perfect filter should have zero insertion loss in the pass‐band, andinfinite attenuation in the stop‐band, and a linear phase response inthe pass‐band to avoid signal distortion. Unfortunately this kind offilter doesn’t exist , so compromises must be made and this is theart of filter design A method that permits to obtain good filterart of filter design. A method that permits to obtain good filterapproximations is the INSERTION LOSS METHOD.Different trade off could be reached with the ILM method to meetDifferent trade off could be reached with the ILM method to meetthe applications requirements. In particular if a minimum insertionloss is required a Chebyshev response would satisfy.

However in all cases the insertion loss method allows filterperformance to be improved in a straightforward manner (you mustperformance to be improved in a straightforward manner (you mustonly increase the filter ORDER.

The insertion loss method

In the insertion loss method a filter response is defined by itsinsertion loss, or power loss ratio Plr:

This quantity is the reciprocal of |S12|2 if both load and source areThis quantity is the reciprocal of |S12| if both load and source arematched. The insertion loss (IL) I dB is

It is possible to express the denominator of the power loss ratio vsthe angular frequency as follows:the angular frequency as follows:

The insertion loss method

Where M and N are real polynomials function of the angularfrequency. The power loss ratio Plr can be expressed as follows:

The specification of this function will characterize the filterbehavior. In the following some practical filter characteristics will bebehavior. In the following some practical filter characteristics will bediscussed.MAXIMALLY‐FLAT this characteristic is also called the binomial orButterworth response, and it provides the flattest possible pass‐band response for a filter complexity. In particular for a low‐passfilter it is specified by the following relation:filter it is specified by the following relation:

The insertion loss method

Usually the power loss ratio at the cut‐off is chosen equal to ‐3dB.For angular frequencies far away from the cutoff the attenuationincrease monotonically with frequency at the rate of 20NdB/decade.

The insertion loss method

EQUAL‐RIPPLE if a Chebyshev polynomial is used to specify theinsertion loss of an N‐order low‐pass filter:

Then a sharper cutoff will result, although the passband responseThen a sharper cutoff will result, although the passband responsewill have ripples of amplitude 1+k2 . Since Tn(x) oscillates between+/‐1, k2 determines the passband ripple level, for large x theinsertion loss become:

Which also increases at the rate of 20N dB/decade. But the/insertion loss for the Chebyshev case is (22N)/4 greater than thebinomial response.

The insertion loss method

The insertion loss method

EQUAL‐RIPPLE the maximally flat and equal‐ripple response both havemonotonically increasing attenuation in the stop band. In many applications it isrequired to specify a minimum stop band attenuation Such filter are calledrequired to specify a minimum stop‐band attenuation. Such filter are calledelliptic filters. The maximum attenuation in the pass‐band Amax can be specifiedas well as the minimum attenuation in the stop band Amin.

The insertion loss method

LINEAR‐PHASE in some applications (multiplexing filters for telecommunicationsystems) it is important to have a linear phase response in the pass‐band to avoidsignal distortion A linear phase characteristics can be achieved with thesignal distortion. A linear phase characteristics can be achieved with thefollowing phase response:

The following relation specify the group delay defined as follow:

Which shows that the group delay for the linear phase filter is a maximally flatfunctionfunction

The insertion loss method

The following schema shows the process of filter design by the insertion lossmethod. The method start with the design of a low pass filter that then is scaledto the desired frequency and impedanceto the desired frequency and impedance .

At the end of the method a lumped elements low pass filter is obtained and youcan obtain the other filters (High‐pass/Pass‐Band/Stop‐Band) with suitabletransformations.transformations.

Maximally flat low‐pass filter prototype

Let us consider the two element low pass filter reported below:

It is a second order low‐pass filter. The source impedance is 1 Ohm, and the cut‐It is a second order low pass filter. The source impedance is 1 Ohm, and the cutoff frequency ωc The power loss ratio for N=2 is given by the following:

Maximally flat low‐pass filter prototype

The input impedance of this filter is

Since

The power loss ratio can be written considering the reflection coefficient.The power loss ratio can be written considering the reflection coefficient.

Maximally flat low‐pass filter prototype

Considering that

And

The power loss ratio can be written as follow:

Maximally flat low‐pass filter prototype

Notice that this expression is a polynomial in ω2 since R=1, and PLR=1 for ω=0The coefficient of ω2 must vanish so:

Or L=C

with

Maximally flat low‐pass filter prototype

This procedure can be extended to find the element values for filters with anarbitrary number of elements N, but clearly this is not a practical procedure forl N F li d l d i h h i d i 1 Ohlarge N. For a normalized low‐pass design where the source impedance is 1 Ohm,the cutoff frequency ωc=1. The elementsvalues for the following lowpass filternetworks:

Could be derived considering the following table.

Maximally flat low‐pass filter prototypeThe following table reports the normalized value for maximally flat low‐pass filterup to order N=10. If you need N>10 you can cascade two filters.

Maximally flat low‐pass filter prototype

Equal Ripple low‐pass filter prototypeFor an equal ripple low‐pass filter with a cutoff frequency ωc=1 the power lossratio is:

Where 1+k2 is the ripple level in the passband. Since the Chabyshev polynomialshave the following property:

Shows that the power loss ratio at ω=0 for N odd, but a power loss ratio of 1+k2 ifN is even.

Equal Ripple low‐pass filter prototypeFor the two elements filter reported below and for a Chebishev function

The following function should be considered:The following function should be considered:

The equation can be solved for C, L if the ripple level is K is known, to obtain theelement values

Equal Ripple low‐pass filter prototypeThe following equations gave the ripple value vs. the load:

Starting from the ripple value we can easily obtain the values of eacht id i th f ll i l ticomponents, considering the following relation:

However to simplify the design there are Table with the tabulated values forHowever to simplify the design there are Table with the tabulated values fordifferent ripple values.

Equal Ripple low‐pass filter prototype

Table with the tabulated values for a Equal ripple low‐pass filter prototypecharacterized by a ripple equal to 0.5 dB in the pass band.

Equal Ripple low‐pass filter prototype

Table with the tabulated values for a Equal ripple low‐pass filter prototypecharacterized by a ripple equal to 3.0 dB in the pass band.

Equal Ripple low‐pass filter prototype

Attenuation vs. normalized frequency for equal ripple filter prototypecharacterized by a ripple equal to 0.5 dB in the pass band.

Equal Ripple low‐pass filter prototype

Attenuation vs. normalized frequency for equal ripple filter prototypecharacterized by a ripple equal to 3.0 dB in the pass band.

Maximally flat low‐pass filter prototypeThe design of a maximally flat filter follows the same design rules. Also in thiscase there are table that permit you a fast filter design.

DenormalizationIn the prototype design the source and load resistance are unity this to permit theIn the prototype design, the source and load resistance are unity this to permit thechoice of different input/output impedance values. A source resistance of R0 can beobtained by multiplying the impedances of the prototype design by R0. Consideringthe input impedance the scaled quantities are given by the following:the input impedance the scaled quantities are given by the following:

Where L, C and Rl are the components values of the original prototype. This is thedenormalization for the input impedance.

Denormalization – Frequency ScalingTo change the cutoff frequency of a low pass prototype from unity to ω weTo change the cutoff frequency of a low-pass prototype from unity to ωc wemust scale the frequency dependence of the filter by the factor 1/ ωc.

Where ωc is the new cutoff frequency. The new element values aredeterminated by applying the substitution of the scaled angular frequency toth t f th filt bt i ithe susceptance of the filter obtaining:

Denormalization – Frequency + Impedance ScalingWh b th f l i d li i d th f ll iWhen both frequency plus impedance scaling are required the followingdenormalization formula can be used to obtain the final filter design:

This is the complete design of a low pass-filterThis is the complete design of a low pass filter.

Low-Pass High-Pass transformationTh i ti l d i th d l t t ith l filt d i tThe insertion loss design method always starts with a low pass filter design toobtain the other filter such as high pass and pass/stop band filters, suitabletransformation are considered. The first transformation is the low-pass High

t f tipass transformation.If we use the following transformation we’ll obtain a transformation of a lowpass filter into an high pass filter:

This change maps the angular frequency ω=0 to ±� and vice versa. The cutoffoccour when ω=±ωc The negative sign is needed to convert inductors andccapacitor into realizable capacitors and inductors.

Low-Pass High-Pass transformationTh t f ti h th t i i d t t b l d ithThe transformation shows that series inductors must be replaced withcapacitance and shunt capacitors must be replaced with inductors. The newcomponent values are given by the following:

Low-Pass Band-Pass transformationTh f ll i t f ti it t t l filt i tThe following transformation permits to convert a lowpass filter into apassband filter.

Where

Is the fractional bandwidth of the passband. The central frequency betweenthe two cutoff frequencies could be chosen as the aritmetic or geometricalthe two cutoff frequencies could be chosen as the aritmetic or geometricalmean.

Low-Pass Band-Pass transformationIf th t i l i id d bt i th f ll iIf the geometrical mean is considered we obtain the following:

Where when

Low-Pass Band-Pass transformationThi t f ti h th t i i d t i t f d t i LCThis transformation shows that a series inductor is transformed to a serie LCcircuit with element values:

And the shunt capacitor is transformed into a shunt LC circuit with element

Low-Pass Stop-Band transformationTh i t f ti b d t t f l filt i tThe inverse transformation can be used to transform a low pass filter into astop band filter.

Low-Pass Stop-Band transformationThi t f ti h th t i i d t i t f d t ll l LCThis transformation shows that a series inductor is transformed to a parallel LCcircuit with element values:

And the shunt capacitor is transformed into a serie LC circuit with element

Summary of the prototype filter transformations