Microstrip Filter Design - Engineering Seminar

Introduction to Filter

A 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 w_c and attenuate all signals with frequencies above w_c.

·         High-pass filter (HPF) that pass all signal with frequencies above the cutoff value w_c and reject signal with frequencies below w_c.

·         Band-pass filter (BPF) that passes signal with frequencies in the range of w₁ to w₂ 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(w)) 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. 

Realization of Filters

At 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:

            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 1W 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.

            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.

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(w)|² of an amplifier using polynomials such as Butterworth, Chebyshev, Bessel and other orthogonal polynomial functions.  This is usually given as:

            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 P_N 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 g₁, g₂, g₃ … g_N 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 g_i correspond to inductance and capacitance in the LPP Butterworth filter.

Designing a Low Pass Prototype (LPP)

We will now design a 4^th order Butterworth LPP and use this design for the rest of the lab.  The specification of the filter is as follows: R_S = R_L = 50W.  Cutoff frequency f_c = 1.5GHz or w_c = 9.4248´10⁹ rad/s. 

Step 1 – Design the LPP filter with w_c = 1 rad/s.

Step 2 – Perform impedance and frequency scaling

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.

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. 

            Suppose the impedance of an inductor is jwL.  At w = 1 the impedance is jL.  Another inductor with inductance L/w_o will give similar impedance at w = w_o.  Thus we observe that the frequency response of the inductor is scaled by w_o.  Similarly if a capacitor C is replace with capacitance C/w_o, its frequency response is also scaled by w_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 R_o = 50W and w_o = 2p(1.5´10⁹) on the schematic of Figure 4.1, the new schematic of the low-pass filter is shown in Figure 4.2 below.

5.0 Implementing the Low-pass Filter using Microstrip Line – Hi Z-Low Z Transmission Line Filter

A relatively easy way to implement low-pass filters in microstrip or stripline is to use alternating sections of high and low characteristic impedance (Z_o) 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).

            The ratio Z_H/Z_L should be as high as possible, limited by the practical values that can be fabricated on a printed circuit board.  Typical values are Z_H=100 to 150W and Z_L=10W to 15W.  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 w_c are:

Designing with Microstrip line

Cross 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_e = 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_e falls within the range 1 and e_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.

Formulas for Effective Dielectric Constants and Characteristics Impedance

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_e and Z_o .

Implementing the 4^th Order Butterworth Low Pass Filter using Step Impedance Microstrip Line

Consider the schematic of Figure 4.2 again.  The filter parameters are as follows:

·         Cutoff frequency f_c = 1.5GHz.

·         Required Z_L = 15W.

·         Required Z_H = 110W.

·         L₁=4.061nH, L₂=9.083nH, C₁=3.921pF, C₂=1.624pF.

Implementation:

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

Analysis of the step-impedance low pass filter using Agilent Advance Design System (ADS) software

1.      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).

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).                                                  

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.

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).

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.

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.

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.