Microwave Filter

RF Engineering – Passive Circuit Microstrip Filter Design

1.0 Introduction to FilterA filter is a network that provides perfect transmission for signal with frequencies in certain passband region and infinite attenuation in the stopband regions. Such ideal characteristics cannot be attained, and the goal of filter design is to approximate the ideal requirements to within an acceptable tolerance. Filters are used in all frequency ranges and are categorized into three main groups: Low-pass filter (LPF) that transmit all signals between DC and some upper limit c

and attenuate all signals with frequencies above c. High-pass filter (HPF) that pass all signal with frequencies above the cutoff value c

and reject signal with frequencies below c. Band-pass filter (BPF) that passes signal with frequencies in the range of 1 to 2 and

reject frequencies outside this range. The complement to band-pass filter is the band-reject or band-stop filter.

In each of these categories the filter can be further divided into active and passive type. The output power of passive filter will always be less than the input power while active filter allows power gain. In this lab we will only discuss passive filter. The characteristic of a passive filter can be described using the transfer function approach or the attenuation function approach. In low frequency circuit the transfer function (H()) description is used while at microwave frequency the attenuation function description is preferred. Figure 1.1a to Figure 1.1c show the characteristics of the three filter categories. Note that the characteristics shown are for passive filter.

Figure 1.1A – A low-pass filter frequency response.

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A Filter H()

V1() V2()

1

2

V

VH

1

21020

V

VLognAttenuatio

c

|H()|

1Transfer function

Attenuation/dB

0

c

3

10

20

30

40


Figure 1.1B – A high-pass filter frequency response.

Figure 1.1C – A band-pass filter frequency response.

2.0 Realization of FiltersAt frequency below 1.0GHz, filters are usually implemented using lumped elements such as resistors, inductors and capacitors. For active filters, operational amplifier is sometimes used. There are essentially two low-frequency filter syntheses techniques in common use. These are referred to as the image-parameter method (IPM) and the insertion-loss method (ILM). The image-parameter method provides a relatively simple filter design approach but has the disadvantage that an arbitrary frequency response cannot be incorporated into the design. The IPM approach divides a filter into a cascade of two-port networks, and attempt to come up with the schematic of each two-port, such that when combined, give the required frequency response. The insertion-loss method begins with a complete specification of a physically realizable frequency characteristic, and from this a suitable filter schematic is synthesized. Again we will ignore the image parameter method and only concentrate on the insertion loss method, whose design procedure is based on the attenuation response or insertion loss of a filter. The insertion loss of a two-port network is given by:

(2.1)

Where is the reflection coefficient looking into the filter (we assume no loss in the filter).

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Attenuation/dB

0

c

3

10

20

30

40c

|H()|

1 Transfer function

1

|H()|

1 Transfer function

2

Attenuation/dB

0

1

3

10

20

30

40 2


Design of a filter using the insertion-loss approach usually begins by designing a normalized low-pass prototype (LPP). The LPP is a low-pass filter with source and load resistance of 1 and cutoff frequency of 1 Radian/s. Figure 2.1 shows the characteristics. Impedance transformation and frequency scaling are then applied to denormalize the LPP and synthesize different type of filters with different cutoff frequencies.

Figure 2.1 – A normalized LPP filter network with unity cutoff frequency (1Radian/s).

Low-pass prototype (LPP) filters have the form shown in Figure 2.2 (An alternative network where the position of inductor and capacitor is interchanged is also applicable). The network consists of reactive elements forming a ladder, usually known as a ladder network. The order of the network corresponds to the number of reactive elements. Impedance transformation and frequency scaling are then applied to transform the network to non-unity cutoff frequency, non-unity source/load resistance and to other types of filters such as high-pass, band-pass or band-stop. Examples of high-pass and band-pass filter networks are shown in Figure 2.3 and Figure 2.4 respectively.

R

Figure 2.2 – Low-pass prototype using LC elements.

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A Filter H()

V1() V2()RS =1

RL =1

Attenuation/dB

0

c = 1

3

10

20

30

40

L1=g2 L2=g4

C1=g1 C2=g3RL= gN+1

1

L1=g1 L2=g3

C1=g2 C2=g4

RL= gN+1g0= 1


Figure 2.3 – Example of high-pass filter, note the position of inductor and capacitor is interchanged as compared with low pass filter.

Figure 2.4 – Example of band pass-filter, the capacitor is replaced with parallel LC network while the inductor is replaced with series LC network.

3.0 Brief Overview of Low-Pass Prototype Filter Design Using Lumped Elements There are a number of standard approaches to design a normalized LPP of Figure 2.3 that approximate an ideal low-pass filter response with cutoff frequency of unity. Among the well known methods are: Maximally flat or Butterworth function. Equal ripple or Chebyshev approach. Elliptic function.

We will not go into the details of each approach as many books have covered them. Interested reader can refer to reference [3], which is a classic text on network analysis or [4], a more advance version. The basic idea is to approximate the ideal amplitude response |H()|2 of an amplifier using polynomials such as Butterworth, Chebyshev, Bessel and other orthogonal polynomial functions. This is usually given as:

(3.1)

Here Ko and Co are constants and PN() is a polynomial in of order N. Ko and Co are usually dependent on the type of polynomial used. A comparison of approximating the LPP amplitude response with Butterworth, Bessel and Chebyshev polynomials is illustrated in Figure 3.1.

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L2L1

C1 CN

C2L2

L1 C1 L3 C3

CNLN


Figure 3.1 – Amplitude response of fourth order (N=4) Butterworth, Chebyshev and Bessel filters using (3.1).

Each approximation has its advantages and disadvantages, for instance the Chebyshev approximation provide rapid cutoff beyond 1.0 radian/second. However the user must compromise this with ripple in the pass band. The Bessel approximation has the slowest cutoff rate, but this is offset with a favourable linear phase response, which reduces phase distortion. A Butterworth approximation has a characteristic between the two. A ladder LC network with the number of reactive elements corresponding to the order of the polynomial PN in (3.1) is then compared with equation (3.1). The respective inductance and capacitance of the reactive elements can then be obtained. An alternative approach would be to synthesize the transfer function of (3.1) using standard techniques as listed in references [3] and [4]. It is suffice to say that for each approach, values of g 1, g2, g3 … gN for an Nth order LPP have been tabulated by many authors (For instance see [2]). Here we will demonstrate the design of a low-pass filter and a band-pass filter using the insertion-loss method and illustrate the implementation of the RLC lumped circuit using distributed elements such as microstrip and stripline in microwave region.

The Table 8.3 of reference [2] is repeated here. We will use this table to design a LPP Butterworth filter. The values of gi correspond to inductance and capacitance in the LPP Butterworth filter.

N g1 g2 g3 g4 g5 g6 g7 g8 g91 2.0000 1.00002 1.4142 1.4142 1.00003 1.0000 2.0000 1.0000 1.00004 0.7654 1.8478 1.8478 0.7654 1.00005 0.6180 1.6180 2.0000 1.6180 0.6180 1.00006 0.5176 1.4142 1.9318 1.9318 1.4142 0.5176 1.0000

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Amplitude in dB Bessel

Butterworth

Chebyshev


7 0.4450 1.2470 1.8019 2.0000 1.8019 1.2470 0.4450 1.0000

8 0.3902 1.1111 1.6629 1.9615 1.9615 1.6629 1.1111 0.3902 1.0000

Table 3.1 – Element values for Maximally flat (Butterworth) LPP (g0 = 1, c =1). Source - G.L. Matthaei, L. Young and E.M.T. Jones, “Microwave filters, impedance-matching networks, and coupling structures”. Artech House 1980.

4.0 Designing a Low Pass Prototype (LPP)We will now design a 4th order Butterworth LPP and use this design for the rest of the lab. The specification of the filter is as follows: RS = RL = 50. Cutoff frequency fc = 1.5GHz or c = 9.4248109 rad/s.

Step 1 – Design the LPP filter with c = 1 rad/s.Using Table 3.1, the schematic of the LPP filter is as shown in Figure 4.1.

Figure 4.1 – The 4th order Butterworth LPP filter.

Step 2 – Perform impedance and frequency scalingThe filter designed in Figure 4.1 supports load impedance of 1 and cutoff frequency of 1 radian/second. This filter can be converted into a low-pass filter, which meets arbitrary cutoff frequency and impedance level specification using frequency scaling and impedance transform. For a new load impedance of Ro and cutoff frequency of o, the original resistance Rn , inductance Ln and capacitance Cn are changed by the followings [3]:

(4.1a)

(4.1b)

(4.1c)

The transformation as shown in (4.1a) to (4.1c) implies that the schematic does not need to be changed, only the element values are scaled down or up to reflect the new specifications. Space does not permit us a detailed discussion of how equations (4.1a)-(4.1c) achieve this. But a qualitative justification is as follows.

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L1=0.7654H L2=1.8478H

C1=1.8478F C2=0.7654FRL= 1g0= 1

L1 = g1 = 0.7654HL2 = g3 = 1.8478HC1 = g2 = 1.8478FC2 = g4 = 0.7654F


The transfer function of a linear two-port network is a function of the impedance or admittance of the individual R, L and C in the network. This is because the transfer function is derived using circuit theory rules (Kirchoff’s voltage and current laws) involving the impedance or admittance. Furthermore the numerator and denominator of the transfer function involve combination of operations such as parallel of impedance/admittance and addition of the impedance/admittance. These operations have the characteristic that if each impedance/admittance is multiplied by a constant, the net effect is equivalent to multiplying the total impedance/admittance by the constant. For instance:

(4.2a)

(4.2b)

(4.2c)

There is no non-linear operation such as square or cube of the impedance/admittance. With this in mind the transfer function is written as:

(4.3)

If each impedance/admittance is multiplied by Ro:

(4.4)

However multiplying each impedance with Ro means we are scaling the impedance due to each R, L and C by Ro as seen in the following:

(4.5a)

(4.5b)

(4.5c)

Frequency scaling is achieved by using the transformation

(4.6)

Suppose the impedance of an inductor is jL. At = 1 the impedance is jL. Another inductor with inductance L/o will give similar impedance at = o. Thus we

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observe that the frequency response of the inductor is scaled by o. Similarly if a capacitor C is replace with capacitance C/o, its frequency response is also scaled by o. The resistor being independent of frequency is not affected by frequency scaling. Combining the frequency scaling and impedance scaling operation, one would arrive at the equations (4.1a) to (4.1c).

Using the transformation (4.1a) to (4.1c) with Ro = 50 and o = 2(1.5109) on the schematic of Figure 4.1, the new schematic of the low-pass filter is shown in Figure 4.2 below.

Figure 4.2 – The denormalized low-pass filter with cutoff frequency at 1.5GHz and impedance of 50.

5.0 Implementing the Low-pass Filter using Microstrip Line – Hi Z-Low Z Transmission Line FilterA relatively easy way to implement low-pass filters in microstrip or stripline is to use alternating sections of high and low characteristic impedance (Zo) transmission lines. Such filters are usually referred to as stepped-impedance filter and are popular because they are easy to design and take up less space than similar low-pass filters using stubs. However due to the approximation involved, the performance is not as good and is limited to application where a sharp cutoff is not required (for instance in rejecting out-of-band mixer products).

A short length of transmission line of characteristic impedance Zo can be represented by the equivalent symmetrical T network shown below (see reference [2], chapter 8):

Figure 5.1 – Equivalent T network for a transmission line with length l. Here Z11 and Z12 are the Z parameters of the two port network.

(5.1a)

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L1=4.061nH L2=9.803nH

C1=3.921pF C2=1.624pFRL= 50g0=1/50

Z11 - Z12 Z11 - Z12

Z12

l

Zo


(5.1b)and is the propagation constant of the transmission line. For EM wave propagation that is of TEM mode or quasi-TEM mode, the propagation constant can be approximated as:

(5.2)

where e is the effective dielectric constant of the transmission line structure. When l < /2, the series element of Figure 5.1 can be thought of as inductor and the shunt element can be considered a capacitor. This is illustrated in Figure 5.2 (a) with:

(5.3a)

(5.3b)

Assuming a short length of transmission line (l < /4) and Zo=ZH >> 1:(5.4a)(5.4b)

Assuming a short length of transmission line (l < /4) and Zo=ZL 1:(5.5a)

(5.5b)

Figure 5.2 – Approximate equivalent circuits for short section of transmission lines.

The ratio ZH/ZL should be as high as possible, limited by the practical values that can be fabricated on a printed circuit board. Typical values are ZH=100 to 150 and ZL=10 to 15. Since a typical ow-pass filter consists of alternating series inductors and shunt capacitors in a ladder configuration, we could implement the filter on a printed circuit board by using alternating high and low characteristic impedance section transmission lines. Using (5.4a) and (5.5b), the relationship between inductance and capacitance to the transmission line length at the cutoff frequency c are:

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jX/2

jB

jX/2

X Zol

B YolWhen Zo 0l < /4

When Zo >> 1l < /4

(a)

(b)

(c)


(5.6a)

(5.6b)

6.0 Designing with Microstrip lineCross section of microstrip and strip transmission line on printed circuit board (PCB) is shown in Figure 6.1. For stripline the propagation mode is TEM since the conducting trace is surrounded by similar dielectric material. Hence e = r, the dielectric constant of the medium. For microstrip line the propagation mode is a combination of TM and TE modes. This is due to the fact that the upper dielectric of a micostrip line is usually air while the bottom dielectric is the printed circuit board dielectric. A TEM mode cannot be supported as the phase velocities for electromagnetic waves in air and the PCB are different, resulting in mismatch at the air-dielectric boundary. However at frequency of 6GHz or lower, the axial E and H fields are small enough that we can approximate the propagation mode as TEM, hence the name quasi-TEM applies. For microstrip line the effective dielectric constant e falls within the range 1 and r. At low frequency most of the electromagnetic field is distributed in the air, while at high frequency the electromagnetic field crowds towards the PCB dielectric. This result in the curve shown in Figure 6.2, thus the microstrip line is dispersive.

Figure 6.1 – Cross section view of microstrip and strip transmission line as implemented on a printed circuit board.

Figure 6.2 – Effective dielectric constant of microstrip and strip transmission line.

6.1 Formulas for Effective Dielectric Constants and Characteristics Impedance

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Microstrip Line

Conducting trace (thickness = t)

Dielectric

Air

Ground Plane

H

W

Strip Line

r rH

f

1

r

e

Microstrip Line Strip Line

f

1

r

e

Region where (6.1) applies.


We will use the microstrip line to implement the low pass filter designed earlier. Microstrip line is popular, as it is easily fabricated and low cost as compared to stripline. There is no closed form solution for the propagation of electromagnetic wave along a microstrip line. The solution for wave propagation is usually obtained through numerical method. Parameters such as the effective dielectric constant, characteristic impedance and line attenuation are then obtained from the numerical solution as a function of frequency. Empirical formulas are obtained from the numerical solution by the methods of curve fitting. Assuming the conductors and dielectric are lossless, and ignoring the effect the conductor thickness t, an example of the empirical formulas for e and Zo are given by [2]:

(6.1)

(6.2)

Zo and e as a function of W/d is plotted in Figure 6.3 using equations (6.1) and (6.2). The dielectric constant of the PCB dielectric is assumed to be 4.2 (for FR4).

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1 2 3 4 5 6 7 8 9 10 11 1210

20

30

40

50

60

70

8072.573

12.902

Z0 s( )

15

121 s

1 2 3 4 5 6 7 8 9 10 11 123

3.25

3.5

3.75

43.731

3.044

e s( )

121 s

e

W/H

W/H

Zo

15

W/H0.1 0.2 0.3 0.4 0.5 0.6 0.7

80

100

120

140

160

Z0 s( )

150

s

0.1 0.2 0.3 0.4 0.5 0.62.7

2.8

2.9

2.976

2.745

e s( )

0.70.1 s

W/H

Zo

e

W/H

110


Figure 6.3 – Zo and e versus W/H for r = 4.2.

6.2 Implementing the 4th Order Butterworth Low Pass Filter using Step Impedance Microstrip LineConsider the schematic of Figure 4.2 again. The filter parameters are as follows: Cutoff frequency fc = 1.5GHz. Required ZL = 15. Required ZH = 110. L1=4.061nH, L2=9.083nH, C1=3.921pF, C2=1.624pF.

Implementation:A typical FR4 fiberglass PCB with r = 4.2 and H = 1.5mm is used. From Figure 6.3 the following trace parameters are obtained:

W/H H/mm W/mm e

Zo = 15 10.0 1.5 15.0 3.68Zo = 50 2.0 1.5 3.0 3.21Zo = 110 0.36 1.5 0.6 2.83Table 6.2 – Dimension of various microstrip line characteristic impedance.

Therefore

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Using equations (5.6a) and (5.6b):

Figure 6.4 – The top view of the layout for the Low Pass Filter on the printed circuit board.

7.1 Analysis of the step-impedance low pass filter using Agilent Advance Design System (ADS) software1. Log into the workstation.

2. Run the ADS version 2003A software (newer version may be used).

3. From the main window of ADS, create a new project folder named “step_imp_LPF” under the directory “D:\ads_user\default\” (Figure 7.1 and Figure 7.2).

Figure 7.1 – Opening a new project in ADS main window.

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l2l1

50 line 50 line

l4l3

0.6mm15.0mm

3.0mm

To 50 Load


Figure 7.2 – The New Project dialog box.

4. The new schematic window will automatically appear once the project is properly created. Otherwise you can manually create a new schematic window by double clicking the Create New schematic button on the menu bar.

5. From the component palette drop-down list, set the component palette to “TLines-Microstrip”. Draw the schematic as shown in Figure 7.5. The MSUB component is the general substrate characteristics of the printed circuit board. The MLIN components represent a short length of microstrip transmission lines used in our low pass filter. Here MLIN1 corresponds to transmission line section 1, MLIN 2 to

transmission line section 2 and so forth (Figure 7.3 to Figure 7.5).

Figure 7.3 – The Schematic Editor window of ADS (New version of ADS may be slightly different).

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Component Palette

Work Area

Palette List

Ground Node


Figure 7.4 – Select the “Tlines-Microstrip” component palette from the Palette List.

Figure 7.5 – Insert the microstrip line component MLIN and substrate component MSUB into the Work Area.

6. Set the characteristics of the substrate “MSUB1” as to H = 1.5mm, T = 1.38mils (typical), Er = 4.2 and Cond = 5.8E+07 (conductivity of copper). The rest of the parameters leave as default. The parameters dialog box for MSUB can be invoked by doubling clicking on the MSUB component.

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Microstrip Line

Substrate Component


7. Set the characteristic W and L of each MLIN components according to the table of Section 6.2.

8. Now change the component palette to “Simulation-S_Param”. Insert the components S parameter simulation control “S P” and the termination network “Term” into the schematics. The termination network components TERM1 and TERM2 are actually a sinusoidal voltage source in series with an ideal series of resistance as shown in the model during S parameter simulation. The S parameter simulation control SP1 determines the start, stop and frequency stepping. Use the wire to connect the components together and ground the outer terminals of the TERM1 and TERM2 components (Figure 7.7).

Figure 7.6 – Select the S parameter component palette.

9. Set the parameters in SP1 to Start = 100MHz, Stop = 4GHz and Step = 10MHz. The final schematic should be as shown in Figure 7.7. In Figure 7.7, since there is a step discontinuity between the transmission line sections, this has to be modeled by inserting a step element “MSTEP” at the junction between two transmission line sections, this will make the simulated result more accurate.

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Figure 7.7 – The final schematic for the low pass filter model.

10. Finally run the simulation.

11. Invoke the data display window. Insert a Rectangular Plot component in the data display.

12. Select the item to display as S21, with the dB option. The S21 represents the attenuation from terminal 1 (input) to terminal 2 (output) of the filter as sinusoidal signals from 100MHz to 4GHz are imposed.

13. Study the 3dB cut-off frequency of the low-pass filter. You can use the Marker feature of the ADS display window to show the value of the attenuation at specific frequency.

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To model step discontinuity in microstrip line


m1freq=1.410GHzdB(S(2,1))=-3.051

0.5 1.0 1.5 2.0 2.5 3.0 3.50.0 4.0

-20

-15

-10

-5

-25

0

freq, GHz

dB(S

(2,1

))m1

Figure 7.8 – A sample result from the Data Display window of ADS, illustrating the S21

of the step-impedance low pass filter.

14. Adjust the parameter of TL1, TL2, TL3 and TL4 until the 3dB cutoff frequency is within 100MHz of 1.5GHz. This can be done using the optimization feature of the software. But as a start you can manually tune the width and length of each transmission line section to achieve the desirable cut-off frequency at 1.5GHz.

Lab ProcedureFollowing the steps in Section 4 to Section 6, design a 4 th order Butterworth Low-Pass Filter using ladder LC network with cut-off frequency at 1.8GHz. Show the steps of how the inductance and capacitance in the network are determined from the Low-Pass Prototype. Also show the conversion of the LC circuit into microstrip circuit, tabulating the dimensions of each section of the transmission line. Upon completing the design, simulate the frequency response of the low pass filter using HP ADS software, again following the steps shown in Section 7. Use a frequency sweep from 100MHz to 5GHz, with a step of 10MHz.

ReportThe report MUST be hand-written, except for graphics, which can be computer generated. Please submit your report within 7 days from the experiment to the lab.

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References1. R.E. Collin, “Foundations for microwave engineerng”, 2nd edition, 1992 McGraw-

Hill. TK7876-C645.2. D.M. Pozar, “Microwave engineering”, 2nd edition, 1998 John-Wiley & Sons.

TK7876.P69.3. F.F. Kuo, “Network analysis and synthesis”, 2nd edition, 1966 John-Wiley & Sons.4. Temes G.C., LaPatra J.W., “Introduction to circuit systhesis and design”, 1977

McGraw-Hill, TK454.5.5. W.K. Chen (editor),”The circuits and filters handbook”, 1995 CRC Press.

TK7876.C4977.

AppendixHaving carried out a computer analysis of the design, an actual step-impedance low pass filter is built by the author, below are the photograph of the printed circuit board and the frequency response measured using a vector network analyzer (VNA).

Figure A1 – Picture of the step impedance 4th order Butterworth Low Pass Filter constructed by the author in Dec 2000.

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Figure A2 – Actual measurement of the attenuation using HP8720D Vector Network Analyser. Note that the cut-off frequency is less than required, at 1.42GHz or 5.3% error.

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1.42GHz Scale:Y axis: 3dB per divisionX axis:300MHz per division

0dB

-3dB

Microwave Filter

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