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1 EEE 244 -- What Is a Microstrip Transmission Line and How Do You

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1 EEE 244 -- What Is a Microstrip Transmission Line and How Do You EEE 244 -- What is a microstrip transmission line and how do you design one? A microstrip transmission line is a "high grade" printed circuit construction, consisting of a track of copper or other conductor on an insulating substrate. There is a "backplane" on the other side of the insulating substrate, formed from a similar conductor. Looked at on end, there is a "hot" conductor, which is the track on the top, and a "return" conductor, which is the backplane on the bottom. A microstrip is therefore a variant of a two-wire transmission line. If one solves the electromagnetic equations to find the field distributions in the vicinity of a microstrip, one finds very nearly a completely TEM (transverse electromagnetic) wave pattern. This means that there are only a few regions in which there is a component of electric or magnetic field in the direction of (as opposed to perpendicular to the direction of) wave propagation. This field pattern is commonly referred to as a Quasi TEM pattern. Since some of the electric energy that is stored in this conductor configuration is in the air, and some is in the dielectric, the effective dielectric constant for the waves on the transmission line will lie somewhere between that of the air and that of the dielectric. Typically, the effective dielectric constant will be 50-85% of the substrate dielectric constant, depending on the 1 geometry of the microstrip. This effective dielectric constant determines the phase velocity of electromagnetic waves on the microstrip transmission line. As an example, in (nominally) an air spaced microstrip, the velocity of waves would be c = 3 * 10^8 meters per second. We have to divide this figure by the square root of the effective dielectric constant to find the actual wave velocity for the real microstrip line. At 10 GHz the wavelength on that nominally air spaced microstrip is therefore 3 cm; however, on a substrate with relative dielectric constant of 10, the effective dielectric constant of the microstrip design may be 7, and the wavelength is only 3/(sqrt{7}) = 1.13 cm. Thus, for example, the maximum length for a microstrip "stub" to be used in stub impedance matching, which is no more than half a wavelength, will be only 5.6 mm when fabricated using this substrate. Alternately, the pulse delay time between point “a” and point “b” will be increased by nearly a factor of 3 using this substrate. It is therefore quite important to accurately design microstrip transmission lines in order to achieve appropriate impedance matching and/or pulse delay time. A set of detailed design formulae and algorithms for microstrip design is presented in T C Edwards, "Foundations for Microstrip Circuit Design", Wiley-Interscience, New York, 1981. This reference also has a rough nomogram for calculating the impedance of a microstrip line using the dielectric properties and the geometry in the picture above. The following analysis is developed from that source. Microstrip parameters The basic configuration of the microstrip is shown in the picture above. One of the most challenging problems associated with this configuration arises from the fact that the small strip is not immersed in a single dielectric. On one side there is the board dielectric, and on the top is usually air. The technique that has been developed to handle this challenge uses, as was mentioned above, the concept of effective relative dielectric constant, εeff. This value represents some intermediate value between the relative dielectric constant of the board material, εr, and that of air (assumed equal to 1) that can be used to compute microstrip parameters as though the strip were completely surrounded by material of that effective relative dielectric constant. One obvious advantage of the microstrip structure is the "open" line which makes it very easy to connect components. On the other hand, the configuration doesn't provide the "shielded" signal line advantage of the stripline. Another advantage is that microstrips can be packed together with fairly high density (multiple channels) with only minimal "crosstalk" interference, and therefore lends itself well to RF and microwave IC design. Aside from the difficulty of calculating the value of εeff, there is another important effect. It is clear that εeff will depend on both W and h. Hence, the phase velocity along the microstrip will depend on these parameters. Assuming the relative permeability of all materials in the line design is well approximated by µr = 1, the phase velocity will be given by: c up = εeff Since the characteristic impedance (Zo) of the line will also depend on these parameters, every time we need to design a microstrip with a new characteristic impedance, we will be faced with 2 the additional complication of having to deal with a change in phase velocity (or delay time) and consequently of the wavelength of waves on that microstrip. Note that this is not a problem with coaxial cable or stripline design. To get an idea of the range of εeff, consider the cases of a very wide W and then a very narrow W. For a wide microstrip, nearly all of the electric field lines will be concentrated between the metal planes, similar to the case of a parallel plate capacitor that you studied in physics. Thus: maximum εeff = εr On the other extreme, for narrow W the electric field lines will be about equally divided between the air and the board dielectric so that: minimum ε = 1 (ε + 1) eff 2 r This gives you a range: 1 (ε + 1) ≤ ε ≤ ε 2 r eff r Several different equations have been developed for use in calculating characteristic impedance for microstrip design. Probably the most useful are the following which are reported to be accurate to within about 1%: 60 h W Zo = ln 8 + εeff W 4h + 1 - 1 -1/2 2 where ε = εr + εr 1 + 12 h + 0.04 1 - W for W ≤ 1 eff 2 2 W h h or 120π Zo = ε W + 1.393 + 0.667ln W + 1.444 eff h h + 1 - 1 -1/2 where ε = εr + εr 1 + 12 h for W ≥ 1 eff 2 2 W h Notice that these are relatively straightforward equations for the calculation of characteristic impedance, given W, h, and εeff. However, the more useful calculation involves determination of the W/h ratio, given a required characteristic impedance. Here, then, is the design challenge since the equations are transcendental (don't have a closed form solution) for the W/h parameter. As you probably guessed, this is a job for MATLAB and its powerful root-finding algorithms. Now, just to make things a bit more challenging, we'll introduce a further "correction" to the above equations which is a consequence of considering the finite thickness (t) of the microstrip. This correction is in the form of an "effective" microstrip width (We), which is used to replace W in those equations: W = W + t 1 + ln 2h for W ≥ 1 e π t h 2π or W = W + t 1 + ln 4πW for W ≤ 1 e π t h 2π 3 These corrections are further subject to the restrictions that: t ≤ h and t < W 2 Both of these restrictions are usually easily satisfied in practice. We are now ready to proceed with a design example. Microstrip design example It is necessary to connect two microwave ICs using a microstrip transmission line. The line needs to be 10 cm long; to be constructed using copper microstrip and backplane conductors with a thickness (t = 0.15 mm) separated by an epoxy fiberglass (εr = 4.9) circuit board with a thickness (h = 0.8 mm). Impedance matching requires the line to have a characteristic impedance (Zo = 50 Ω). Determine the following: 1. The actual width (W) of the microstrip 2. The effective dielectric constant (εeff) for the microstrip design 3. The pulse delay time (td) for the microstrip transmission line A set of MATLAB plots of the characteristic impedance as a function of W/h (without correction for finite t) for the epoxy fiberglass circuit board is presented in the figures below. You will observe that characteristic impedance decreases as microstrip W/h ratio increases. You'll also observe, as you probably can guess, that the two sets of equations yield the same characteristic impedance when W/h = 1. 4 % Microstrip characteristic impedance plot, W/h ≤ 1 echo on er=4.9 r=[0.01:0.01:1]’; eff=(er+1)/2+((er-1)/2)*((1+12./r).^(-1/2)+0.04*(1-r).^2); Zo=(60./sqrt(eff)).*log(8./r+r/4); plot(r,Zo) xlabel('W/h ratio') ylabel('Zo (ohms)') grid on title('Microstrip Characteristic Impedance, W/h ≤ 1') echo off % Microstrip characteristic impedance plot, W/h ≥ 1 echo on er=4.9 r=[1:0.01:5]’; eff=(er+1)/2+((er-1)/2)*(1+12./r).^(-1/2); Zo=(120*pi)./(sqrt(eff).*(r+1.393+0.667*log(r+1.444))); plot(r,Zo) xlabel('W/h ratio') ylabel('Zo (ohms)') grid on title('Microstrip Characteristic Impedance, W/h ≥ 1') echo off 5 Let's begin by calculating the characteristic impedance for our case if we make W/h = 1. Using the first set of equations (W/h ≤ 1) for this calculation, we find that: ε = 4.9 + 1 + 4.9 - 1 1 + 12 -1/2 = 3.491 eff 2 2 and 60 1 Zo = ln 8 + = 67.765Ω 3.491 4 Since this value is greater than the desired characteristic impedance of 50 Ω, it is clear that the W/h ratio must be increased in order to realize the design.
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