9-1

Chapter 9

MICROWAVE FILTERS

9.1 Insertion-loss Method of Filter Design 9.1.1 Butterworth Filter 9.1.2 Chebyshev Filter 9.1.3 Elliptic function filter response 9.2 Low-pass Prototype Filter Design 9.2.1 Butterworth Filter Design 9.2.2 Chebyshev Filter Design 9.3 Filter Transformations 9.3.1 Frequency Transformations 9.3.2 Load Transformation 9.4.1 Microwave Filter Realizations

9.4.1 High - low Low-pass Filters in Microstrip/Stripline Technology 9.4.2 Planar Low-pass Elliptic Function Filters

9.4.3 Bandpass Filters 9.4.4 Impedance/admittance Inverters

9-2

Chapter 9

MICROWAVE FILTERS

Microwave filter is one of the very useful components for microwave systems. It can be used for excluding undesired frequencies as in UWB systems, for combining or sorting power at different frequencies as in multiplexers, and for impedance matching over broad-bandwidths. Filters can be classified into four categories on the basis of their insertion loss as a function of frequency; low-pass, band-pass, band-stop and high-pass. The design of microwave filters starts with a prototype filter whose low-pass lumped-element ladder network is normalized to 1-ohm terminations and a cut-off frequency of 1 rad/sec. Equations and/or tables are used for determining the prototype element values. Frequency and load transformations are then used to derive high-pass, band-pass, band-stop filters and low-pass filters with higher cut-off frequency. The filter with new element values may be implemented in lumped element form. If the lumped element realization is not possible, relationship between the transmission line section and the equivalent lumped elements are used to realize the filter in the distributed element form. Suitable techniques may be employed for changing the microwave filter configuration thus arrived at into a form suitable for practical implementation. Schematic of the above procedure is shown in Fig. 9.1. There are two low-frequency synthesis techniques in common use. These are: image parameter method (and the variations thereof, such as the constant-k and m-derived filters) and the insertion loss method. We shall use the insertion loss method because it is based on complete specification of a physically realizable frequency characteristic. These characteristics are: (a) width of pass-band frequencies, (b) maximum insertion loss or VSWR in the pass-band, and (c) minimum insertion loss or isolation in the stop band at one or more than one frequency. We shall use the frequency variable for the prototype filter and ω is reserved for later use with microwave filters. 9.1 INSERTION-LOSS METHOD OF FILTER DESIGN This method of filter design has been very well described in the text book by Collin*. A filter is specified by the insertion-loss as a function of frequency. The insertion loss can also be expressed in terms of power loss ratio , defined as the ratio of the power delivered by the generator to the load when connected directly, to the power delivered when the filter is inserted in between. If Γ is the input reflection coefficient of the filter terminated in a matched load, then

ΓΓ | | (9.1) -----------

*R.E. Collin, Foundations for Microwave Engineering, Second edition, McGraw-Hill, 1992, Chapter 8.

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where |Γ| is the magnitude of reflection coefficient. The insertion loss is given by 10 . The or the corresponding ρ may be converted into equivalent VSWR using the expression VSWR = (1+ρ)/(1- ρ . From (9.1) we infer that either the power loss ratio or the magnitude of the input reflection coefficient as a function of frequency may be specified for filter design. However, a completely arbitrary value of ρ cannot be chosen because it may not correspond to a physically realizable network. It can be shown* that the necessary and sufficient condition for a network to be physically realizable is given by

1 (9.2) where and are real, positive and even polynomials. Although an unlimited number of power loss ratios can be specified satisfying (9.2), it has been found in practice

Frequency and load transformations

Design eqns and/or Tables

Low-pass lumped element prototype filter for desired response

Specs

Lumped element realization

Implementation

Distributed network conversion

Practical realization e.g. stripline

Fig. 9.1: Design schematic for a microwave filter

--------------------------------------------------------------------------------------------------- *R.E. Collin, Foundations for Microwave Engineering, Second edition, McGraw-Hill, 1992, Chapter 8.

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that for microwave filter design only three types are sufficient. These are: maximally flat or Butterworth response in the passband, an equi-ripple or Chebyshev response in passband, and an elliptic function filter. Design of first two types of filter is described next. The design of elliptic function filter is beyond the scope of the text.

9.1.1 Butterworth Filter The power loss ratio for a maximally flat or Butterworth filter is described by the following equation 1 / (9.3) The passband is the frequency range from ′ 0 to the cut-off value . The maximum value of power loss ratio in the passband is therefore 1 , and for this reason is called the passband tolerance and k<1. Outside the passband, the power loss ratio increases at a rate dependent on 2N, where N is the order of the filter which is equal to the number of reactive elements in the filter. Typical filter characteristics are shown in Fig. 9.2 for N = 2,3. In addition to the maximally flat characteristic the Chebyshev characteristic is also shown there which is described next. 9.1.2 Chebyshev Filter The power loss ratio for an equi-ripple or Chebyshev filter is given by 1 ′/ ′ ) (9.4) where . is the Chebyshev polynomial of degree N, and described as

for

for (9.5)

The behavior of . is shown in Fig. 9.3. . oscillates between 1 for and increases monotonically for . Therefore, power loss ratio will oscillate between 1 and 1 in the passband ( ), equals 1 at , and will increase monotonically for . The number of ripples in the pass-band equals N/2 (see Fig. 9.2). For large values of / , ⁄ ⁄ and the power loss ratio for the Chebyshev filter can be approximated as

(9.6) When compared with maximally flat filter, (9.3), this power loss ratio is larger by a factor 2 . It means that the Chebyshev filter has a sharper selectivity or a much smaller frequency range separating the passband and the stopband, and is a desirable feature. For example for N=15, 70 dB attenuation is reached at 1.7 for the maximally flat filter case and at 1.18 for the Chebyshev filter case.

9-5

Fig. 9.2: Low-pass filter response for maximally flat and Cheyshev filters; N =2,3. Passband tolerance 2k and VSWR Setting the power loss ratio (9.1) to 1 at , we obtain the following relationship between passband tolerance k and the input reflection coefficient ρ ρ

√ (9.7)

The corresponding input VSWR is given by

1 2

1 2 (9.8a)

For k = 0.509 the insertion loss equals 1 dB and the corresponding VSWR is 2.66; and VSWR = 5.83 for an insertion loss of 3dB. The passband tolerance may also be expressed in terms of passband ripple in dB using the following equation 10 1 2 . (dB)

1 10 dB /10 (9.8b) The insertion loss value in the stopband is called isolation or stopband attenuation, and is given by the following expression for the Butterworth filter case Isolation (dB) = 10 1 2 / for the Chebyshev filter case Isolation (dB) = 10 1 2

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Fig. 9.3: Plots of Chebyshev polynomials for N = 1,2,3 and 4.

9.1.3 Elliptic function filter response Maximally flat and equi-ripple filters are described by monotonically increasing attenuation in the stop-band. For some applications it is sufficient to guarantee minimum stop-band attenuation rather then increasing attenuation with frequency. The minimum attenuation in the stop-band is related to the cut-off rate. This type of filters are called elliptic function filters and are characterized by equi-ripple behavior in the pass-band and stop-band as shown in Fig. 9.4. The specification for elliptic function filters is the maximum attenuation in the pass-band and minimum attenuation in the stop-band. Elliptic function filters are difficult to synthesize and are beyond the scope of the text.

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Fig. 9.4: An elliptic function low-pass filter response. 9.2 LOW-PASS PROTOTYPE FILTER DESIGN The low-pass prototype filter is a ladder network of series inductors and shunt capacitors as shown in Fig. 9.5. The element values are the same in both circuits, a and b. Both the circuits can be designed to provide the same insertion loss for the maximally flat and equi-ripple filters. Element Values for the Prototype Filter: Elements of the ladder network are denoted by ℓ. For a given value of N, element values can be obtained by determining the power loss ratio of the filter and comparing with (9.3) or (9.4) as the case may be. This approach can be conveniently followed for N upto 3 or 4. As an example let us select N = 2 for the circuit of Fig. 9.6 and determine the values of L and C for a source impedance of 1 ohm and 1 rad/sec. The input impedance for this simple circuit is given by (9.9)

Fig. 9.5: Low-pass ladder prototype filter networks

⁄

9-8

L C

Fig. 9.6: Prototype filter network for N = 2. The corresponding reflection coefficient with source impedance of 1 ohm is Γ(ω)= and the power loss ratio

ΓΓ| |

=1+ (9.10)

9.2.1 Butterworth Filter Design For this type of frequency response is described by (9.3). Therefore, (9.10) should be equated with 1 4

since 1, i.e.

1 41

1 2 2 2 2 2 2 4 2 2 2

4 (9.11)

Comparison at 0 gives R = 1; comparison of coefficient of imply L = C and comparison of coefficient of yield 4 or L = C = √2 (9.12) 9.2.2 Chebyshev Filter Design For this type of frequency characteristics is described by (9.4). Therefore,

1 22 1

1 2 2 2 2 2 2 4 2 2 2

4 (9.13)

Since 2 1, (9.13) becomes

1 2 2 1 211 2 2 2 2 2 2 4 2 2 2

4 (9.14)

Comparison at 0 gives

or R = 1+2 2 √ 1 (9.15a) Equating the coefficient of 2ω′ and 4ω′ give coefficient of , 4 2

4 (9.15b)

coefficient of , 4 24 (9.15c)

9-9

The above equations can be solved to determine L and C. It may be noted that 1R ≠ unlike the case of Butterworth filter. The reason is that the value of Chebyshev function

( ) is not zero at 0, in general, and depends on the order N according to (0) = 0 for N odd

1 for N even (9.16) Therefore, Chebyshev filters require

!

1 for N odd

1 2 2 2 1 2 for N even (9.17)

If the Chebyshev filter is to be designed for load R = 1, the mismatch between R 1 and R=1 for N even can be bridged by employing an impedance transformer in between. An alternative is to increase the order of filter to make N odd. The analysis procedure becomes cumbersome for large values of N. Instead, one can use the Tables available or the general expressions given next. (a) Maximally flat lowpass filter element values [Collins, Matthaei]

! 1 (9.18a) and ℓ 2 ℓ ℓ 1,2,3, … (9.18b) where ℓ is the value either of inductance in henry or capacitance in farad. Table 9.1 gives the element values for = 1, 1 and k = 1 for N = 1 to 10. (b) Equi-ripple filter element values [Collins, Matthaei] The general expressions are given below for a low-pass filter with 1:

!

1 for N odd

1 2 2 2 1 2 or 4⁄ for N even (9.19a)

⁄ (9.19b)

and ℓ 4 ℓ

ℓ

ℓ

ℓ ℓ 2,3, … (9.19c)

where ℓ

ℓ ℓ 1,2,3, … (9.19d)

ℓℓ ℓ 1,2,3, … (9.19e)

with

ℓ1 2

1 2ℓ

. (9.19f)

Table 9.2 gives the element values for = 1, 1, and 0.5 and 3dB for N = 1 to 10. The element values are numbered from as the generator resistance or conductance to ! as the load resistance or conductance for a filter with N reactive elements. The elements are defined as:

9-10

=generator resistance if g is a shunt capaitor

generator conductance if g is a series inductor (9.20a)

= inductance in H for series element

inductance in F for shunt element 1,2,3, … (9.20b)

! =load resistance if g is a shunt capaitor

load conductance if g is a series inductor (9.20c)

With this definition, the circuits in Figs. 9.5(a) and 9.5(b) are dual of each other and will give the same response. Table 9.1: Element values for maximally flat low-pass filter prototype with k = 1, = 1,

1

Table 9.2: Element values for equi-ripple low-pass filter prototype with = 1, 1

0.5 , k = 0.3493

N 1 2.000 1.0 2 1.414 1.414 1.0 3 1.000 2.000 1.000 1.0 4 0.7654 1.848 1.848 0.765 1.0 5 0.6180 1.618 2.000 1.618 0.6180 1.0 6 0.5176 1.414 1.932 1.932 1.414 0.5176 1.0 7 0.4450 1.247 1.802 2.000 1.802 1.247 0.445 1.0 8 0.3902 1.111 1.663 1.962 1.962 1.663 1.111 0.3902 1.0 9 0.3473 1.000 1.532 1.879 2.000 1.879 1.532 1.000 0.3473 1.0 10 0.3129 0.908 1.414 1.782 1.975 1.975 1.782 1.414 0.908 0.3129 1.0

N 1 0.6986 1.0 2 1.4029 0.7071 1.9841 3 1.5963 1.0967 1.5963 1.0 4 1.6703 1.1926 2.3661 0.8419 1.9841 5 1.7058 1.2296 2.5408 1.2296 1.7058 1.0 6 1.7254 1.2479 2.6064 1.3137 2.4758 0.8696 1.9841 7 1.7372 1.2583 2.6381 1.3444 2.6381 1.2583 1.7372 1.0 8 1.7451 1.2647 2.6564 1.3590 2.6964 1.3389 2.5093 0.8796 1.9841 9 1.7504 1.2690 2.6678 1.3673 2.7239 1.3673 2.6678 1.2690 1.7504 1.0 10 1.7543 1.2721 2.6754 1.3725 2.7392 1.3806 2.7231 1.3485 2.5239 0.8842 1.9841

9-11

3 , k = 1

The filter design procedure may be described as follows:

(i) Determine the number of elements or order of the filter N from the isolation or stopband attenuation at the desired frequency / . Use the following expressions for the maximally flat and equi-ripple filters.

Maximally flat filter case

ℓ /

ℓ / for k = 1, (9.21)

Equi-ripple filter case

/ /

/ for (9.22)

(ii) For the value of N determined above, read the element values of the prototype filter from the appropriate Table 9.1 or 9.2. Otherwise, (9.18)-(9.19) can be programmed for the purpose.

(iii) Scale the element values according to the frequency transformation and load transformation discussed next, since 1 and = 1 for the prototype filter.

9.3 FILTER TRANSFORMATIONS The design of low-pass prototype filter was discussed in the last section. The element values correspond to = 1, 1 for Butterworth and Chebyshev characteristics. We need to transform this filter into another filter with either higher cut-off low-pass filter, or band-pass, band-stop or high-pass filter. The element values therefore need to be properly scaled. We employ frequency and impedance transformations for the purpose. 9.3.1 Frequency Transformations (a) Low-pass Cut-off Translation: If the cut-off frequency of the new low-pass filter is to be raised from 1 to , we apply the frequency translation ω’ ω/ (9.23) The frequency translation of (9.23) maps the points ω’ 1 to the points ω . This is shown in Fig. 9.7. The series reactances and shunt susceptances for the new filter are

N 1 0.6986 1.0 2 1.4029 0.7071 1.9841 3 1.5963 1.0967 1.5963 1.0 4 1.6703 1.1926 2.3661 0.8419 1.9841 5 1.7058 1.2296 2.5408 1.2296 1.7058 1.0 6 1.7254 1.2479 2.6064 1.3137 2.4758 0.8696 1.9841 7 1.7372 1.2583 2.6381 1.3444 2.6381 1.2583 1.7372 1.0 8 1.7451 1.2647 2.6564 1.3590 2.6964 1.3389 2.5093 0.8796 1.9841 9 1.7504 1.2690 2.6678 1.3673 2.7239 1.3673 2.6678 1.2690 1.7504 1.0 10 1.7543 1.2721 2.6754 1.3725 2.7392 1.3806 2.7231 1.3485 2.5239 0.8842 1.9841

9-12

obtained by replacing ω’ by ω/ . Thus, for the k-th filter element the new impedance value should be (9.24) Writing so that this impedance value does not disturb the impedance match with source and load gives the new inductance ⁄ (9.25a) Similarly, the new capacitance value is related to the prototype element value as ⁄ (9.25b) It may be noted that and for 1. The power loss ratio for the new filter is shown in Fig. 9.7(b). (b) Low-pass to High-pass Transformation: A high-pass filter comprises of series capacitors and shunt inductors because of their favourable impedance behaviour at high frequencies. This is proved next by using the following frequency transformation of the prototype low-pass filter ω’ /ω (9.26) This transformation maps the points ω’ 0 to the points ω ∞, the points ω’ 1 to ω , and the points ω’ ∞ to ω 0. The effect of this transformation is to interchange the passband and stopband regions as illustrated in Fig. 9.8. The new inductances and capacitances are obtained as follows: The series impedance of the prototype filter element j should be changed to

/ . This type of impedance behaviour can be realized by a series capacitance (without disturbing the existing impedance match) such that = -j (9.27)

Fig. 9.7: (a) Low-pass prototype filter response (b) Filter response after frequency translation ⁄

ω

ω/

9-13

Fig. 9.8: (a) Low-pass prototype filter response

(b) High-pass filter response after the transformation ⁄ Similarly, the shunt capacitance should be replaced by the shunt inductance such that (9.28)

The high-pass transformation of the prototype filter for N = 3 is shown in Fig. 9.9. (c) Low-pass to Band-pass Transformation: A band-pass filter allows a band of frequencies to pass through while rejecting the frequencies on either side. This response is similar to that of a resonator. Therefore, it is natural that a band-pass filter is realized using resonant elements. The appropriate frequency transformation of the low-pass filter is (9.29)

Fig. 9.9: Low-pass prototype to high-pass prototype transformation of N=3 filter.

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Fig. 9.10: Low-pass to band-pass frequency mapping using where and are the lower and upper frequencies of the desired passband, and is the geometric mean of and . Equation (9.29) can be solved for ω to give

ω ω (9.30) Since we have chosen √ , we obtain ω ω 4 (9.31) We can see from the above expression that 0 maps into two points ω ,

1 maps into four points ω , i.e. ω and ω . Thus the prototype filter passband (-1,0,+1) maps into passbands , , and

, , . This is illustrated in Fig. 9.10. The element values of the filter can be deduced as follows: Due to the frequency transformation (9.29) the series elements will have the new reactance as (9.32a) Similarly, the susceptance of the shunt elements of the filter will be transformed to (9.32b) These reactances and susceptances can be realized by the series and parallel resonant circuits of L and C, as shown in Fig. 9.11. For the series circuit we have,

√√

(9.33a)

and for the parallel circuit

√√

(9.33b)

9-15

If we choose √ 1⁄ , expressions (9.33) become

and (9.34)

The required series reactance of (9.33a) can now be seen to be realized if we replace the reactance (9.32a) of the prototype filter by a series circuit such that

or

for the series element (9.35a)

Similarly,

for the parallel element (9.35b)

For the resonant circuits, 1⁄ , and (9.35) give rise to the following expressions for the filter elements and for the series resonant circuit (9.36a) and and for the parallel resonant circuit (9.36b) Thus, in a band-pass filter the series inductor are replaced by series tuned L-C circuits and the shunt capacitors are replaced by parallel tuned L-C circuits as shown in Fig. 9.12. The relationship between the low-pass prototype element values and the band-pass proto-type filter element values is given by (9.36). 9.3.2 LOAD TRANSFORMATION The design of low-pass, high-pass and band-pass prototype filters worked out so far assumes load and source impedances of 1 ohm, except for Chebyshev filters with N even for which load resistance is not unity. For realistic filters with a source or load impedance of , the new values of inductances and capacitances should be such that impedances of series and shunt elements are multiplied by , i.e. (9.37a) / (9.37b)

Fig. 9.11: Series and parallel tuned circuits for the realization of band-pass filters.

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Fig. 9.12: A prototype bandpass filter configuration 9.4 MICROWAVE FILTER REALIZATIONS 9.4.1 High - low Low-pass Filters in Microstrip/Stripline Technology The low-pass filters in microstrip and stripline configurations can be obtained by realizing series L in the form of a section of high impedance line, and shunt C in the form of a section of low impedance line. In the printed form, the low-pass filter will appear as cascaded sections of alternating high impedance and low impedance sections as in Fig. 9.13. However, steps in width will give rise to junction discontinuity reactance. These should be accounted for in determining the line lengths for the filter elements. Approximate equivalent circuit of a short section of line A two-port transmission line section can be modeled as a T-network or π-network; the T-network model is shown in Fig. 9.14. Comparing the Z-matrices of the transmission line section and the T-network we can obtain the equivalent L and C values of T-network. The Z-matrix for a transmission line section is given as,

ℓ ℓℓ ℓ

Fig. 9.13: High - low low-pass filter configuration in microstrip/stripline technology.

9-17

0 ,Z βl

L/2 L/2, (Z11-Z12)

C, (Z12)

Fig. 9.14: T-network equivalent of a short length of transmission line

The equivalent T-network in terms of Z-parameters is given as: Series elements: ℓ ℓ ℓ/2 Shunt element: ℓ Correspondence 2⁄ ℓ/2 and 1⁄ ℓ gives ℓ and ℓ / (9.38) For ℓ , the above expressions may be approximated as ℓ and ℓ (9.39) The equivalent circuit of Fig. 9.14 may be approximated as lumped inductor or capacitor alone if the effect of the other element is made negligible. For this, if we choose high for the inductor, according to (9.39) we need small ℓ for a given value of L. Also, high

and small ℓ reduce the associated C. Therefore, we choose highest value of consistent with fabrication limitations to realize series inductor. Similarly, lowest value of is employed to realize shunt capacitor. The approximate expressions for the line lengths are ( ℓ /4 : ℓ / for the inductor ℓ for the capacitor (9.40) The line lengths given by (9.40) are not final, and should be corrected for the step-in-width discontinuity reactance on either side of the section as shown in Fig. 9.15. The step-in-width junction discontinuity has been modeled as a series inductance as shown in Fig. 9.15. The equivalent reactance may be approximated as [Matthaei, p.206]

for ⁄ 2 (9.41)

Example 9.1: Let us design an N = 3 maximally flat low-pass filter with cut-off at 2 GHz in stripline configuration. The maximum realizable is 118 ohm and the minimum realizable is 30.5 ohm. Solution: Let us choose π-configuration for the prototype filter with 1 ohm source and load impedances as shown in Fig. 9.16. From the Table 9.1 we obtain 1 and = 2H Frequency translation to 2 gives the following filter element values 0.159 and 79.5

9-18

1w 2w

T

T

jX 01Z

T

02Z

Fig. 9.15: A simple equivalent circuit of step-in-width discontinuity

Transforming these element values for the load and source impedances of 50 ohm yields the new values as 50 7.95 and 1.59 Stripline realization of s and s: Let us assume a substrate with dielectric constant

2.7 and substrate thickness b = 0.635cm. The guide wavelength λ= ⁄ at 2 GHz is found to be 9.129cm. For realizing the inductor, let us choose a high impedance section with 118 ohm. The limit on the realizable value of high is set by the printed circuit fabrication tolerance of line width. The corresponding strip width is 0.635 . For the capacitors we choose a low impedance section with 30.5 . The limit on the realizable value of low is set by large strip width W. When W becomes comparable to the minimum wavelength of operation, the definition of TEM mode does not hold because fields do not vary in the transverse direction for TEM mode and our design does not apply. In the extreme case of very large W, transverse resonance across the width may occur. The strip width corresponding to 30.5 ohm is 9.19 . For the assumed substrate parameters, the value of β=2π/λ at 2 GHz is 0.688 rad/cm. Expressions (9.40) give ℓ (inductor) = 1.23cm and ℓ (capacitor) = 8.86mm at 2GHz. The calculation of line lengths does not account for the junction reactances. Including the junction inductances in the filter design modifies the line lengths required for the inductors and capacitors. The new line lengths are: 9.59mm for each of the capacitors and 1.19cm for the inductor. The line length for the inductor has decreased from 1.23 cm to 1.19cm due to additional inductive reactance associated with step discontinuity. The line length for the capacitor has however increased from 8.86 mm to 9.59mm. The designed filter geometry is shown in Fig. 9.17.

Fig. 9.16: Prototype low-pass filter for N = 3, π-configuration.

9-19

Fig. 9.17: Low-pass filter design for N = 3 in stripline technology 2.7, b6.35mm, 2GHz

Fig. 9.18: Transfer characteristics of the low-pass filter shown in Fig. 9.12 2.7, b 6.35mm, 2GHz

9-20

The insertion loss versus frequency behaviour of the realized stripline filter is plotted in Fig. 9.18. The rate of increase in attenuation with frequency is slow in the stop-band. The stop-band attenuation is about 10dB at 4 GHz (2 whereas the expected value for lumped element design is 18dB. The distributed element design gives poor stop-band attenuation because the equivalent lumped values depart significantly from the design outside the pass-band (Pozar, p.473). 9.4.2 Planar Low-pass Elliptic Function Filter This type of filter can be realized using rectangular microstrips/stripline in cascade. A single section planar low-pass filter is shown in Fig. 9.19. The two dimensional analysis (see Chapter 10) of this circuit shows that its lumped element equivalent circuit has the same structure as that of a third order Cauer-Chebyshev (cc) or elliptic function lowpass filter*. Details of this filter are given below. Analysis of the filter geometry shown in Fig. 9.19 can be carried out using the two-dimensional approach wherein the geometry is modeled as a two-port resonator with magnetic wall along the periphery and electric walls at the top and bottom. The impedance matrix of a rectangular patch of dimensions bxl is given by (i,j = 1,2)

∑ ∑ 2⁄ 2⁄ (9.42)

where sin ,⁄ mπ , nπ ⁄ ⁄ and

1 for 0√2 for 0

Here, b, l, and are the effective dimensions of the filter geometry and are greater than the corresponding physical dimensions as they include the energy stored in the fringing fields, and can be determined using the planar waveguide model of microstrip/stripline. Since the dimensions of the filter element to be considered are very small compared to the minimum wavelength of operation we can reduce the number of modes contributing to at a cost of small error. By retaining only the first two terms, corresponding to the

and modes, the impedance matrix of the filter reduces to √ / /

⁄ 2⁄ 2⁄ (9.43) Writing this expression as

√ (9.44) gives 1 / 2⁄ i = 1,2 and is the resonance frequency for the mode and is given by

9-21

Fig. 9.19: Geometry of a planar elliptic function low-pass filter; b and l are effective dimensions.

Synthesis of the filter The lumped element equivalent π-network corresponding to the two-port impedance parameters given by (9.44) can be drawn as in Fig. 9.20. The relationship between the lumped elements and the impedance matrix elements can be derived by comparing the input impedances and the transfer impedances of the two networks. The comparison gives

1 ⁄ , 1 ⁄ , 21 2

where (9.45) It is seen from (9.45) that the area lb of the strip affects the values of capacitances. The inductance depends on the aspect ratio l/b, and is intuitively expected that narrower the strip the larger its inductance value. The values of filter elements , , and can be controlled independently by varying the dimensions of the strip and the location of the ports and . The physical realizability of the lumped circuit of Fig. 9.20 imposes -----------------------------------------------

* Dinzeo, et. al., “Novel microwave integrated low-pass filters,” Electron. Lett., vol. 15, 1979, pp. 258-260

9-22

Fig. 9.20: A simple lumped equivalent π-network of the filter geometry of Fig. 9.19. some constraints on the parameters of the strip and ports. For the case , , so that

2⁄ is nearly unity, the following conditions should be satisfied , (9.46) Also, the capacitors and can be positive only if the ports are at the opposite sides with respect to l/2. The mode resonance frequency also corresponds to the open circuit resonance of the equivalent circuit of Fig. 9.20, and is given by (9.47)

The parallel resonance produces a zero in the parameter of the filter and is responsible for the sharp increase in insertion loss about . Example 9.2: A low-pass Cauer-Chebyshev 03-02-41 type of filter using the two-dimensional circuit approach is described*. Here, the first two digits define the order of the filter (n = 03), the next two digits determine the reflection coefficient (in this case 20%), and the last two digits specify the modular angle θ, that describes the sharpness of the filter response. In this case Ω

°1.5243

where Ω ⁄ is the normalized frequency at which the specified minimum stopband attenuation occurs. From the Tables, the values of the prototype low-pass filter are found to be 0.9333 , 0.4318 , 0.7973 For a load and source impedance of 50 ohm and a cut-off frequency the lumped element values are given by , and We have implemented this filter on low dielectric constant substrate with 2.5 and h = 0.159cm using a microstrip line with 70.66Ω, physical dimensions

9-23

= 17.82mm, = 2.557mm, 12.03 4.51 for a cut-off frequency of 5 GHz. The computed insertion loss and return loss of the designed filter is plotted in Fig. 9.21. The stopband attenuation is about 15 dB at 7.5GHz (1.5 .. 9.4.2 Bandpass Filters A bandpass filter consists of series resonators alternating with parallel resonators as shown in Fig. 9.12 for the prototype filter. This arrangement is, however, difficult to realize using distributed parameters at microwave frequencies. It is much more practical to realize a circuit that employs only one type of resonant circuit either series or parallel. This limitation arises from the transmission line technology. The new configuration can be obtained from the configuration of Fig. 9.12 by using impedance or admittance inverters to convert one type of resonant circuit to another type.

. .

= 17.82mm, = 2.557mm

. , h = 1.6mm (a)

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Fig. 9.21: Frequency response of designed low-pass elliptic function filter of Fig. 9.19. (a) Insertion loss, (b) return loss

(b)

. .

. , .

= 17.82mm, = 2.557mm

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. 9.4.4 Impedance/admittance inverters: The word immittance is a common word used for both impedance and admittance. An immittance inverter in its simplest form is a quarter-wave transformer. It can be used to transform a series element into a shunt element with same frequency dependence, e.g. a shunt admittance such as jωC can be transformed to a series element like jωL by an impedance inverter K. Further, the impedance value jωL can be controlled by the value of K and is given by K = / . Similarly, a series resonant circuit may be transformed into a parallel resonant circuit and vice versa by an immittance inverter. For example, a shunt admittance Y preceded and followed by admittance inverters as in Fig. 9.22 will perform like a series impedance, i.e., the two networks will have identical transmission coefficient provided (9.48) where J is the characteristic admittance of the admittance inverters. For the dual circuit shown in Fig. 9.22 we have (9.49) where K is the characteristic impedance of the impedance inverters. The admittance (impedance) level of the transformed network can be controlled through the value of J (K). The quarter-wave section based inverters will have narrow bandwidth (1-2%) because the variation of electrical length with frequency. An alternative is to employ lumped element based immittance inverters. Impedance (admittance) inverter realized using a shunt inductor (series capacitor) and its equivalence is given in Fig. 9.23.

Fig. 9.22: Use of immittance inverters to convert one type of resonant circuit to another type of resonant circuit.

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| /2|

| /2|

Fig. 9.23: Impedance (admittance) inverters and their realization using shunt inductor (series capacitor) surrounded by line length /2. The relationships between the equivalent parameters X(B), ( ,), and K (J) can be determined by comparing the ABCD-parameters of the equivalent networks at a quarter-wave frequency. Some more inverters based on lumped elements are given in the book, Foundations for Microwave Engineering, Sec. 8.18.

ℓ λ/4

jX

θ/2θ/2

ℓ λ/4

jB

θ/2θ/2