I have 2 questions related to Microstrip Line characteristic impedance:
1) What would be the relation of characteristic impedance of a given microstrip line with frequency?
As effective permittivity increases with the frequency and gets closer to that of the substrate, according to formulae in Pozar Microwave Engg the characteristic impedance should decrease in my opinion. Could someone please clarify this?
An updated issue on the same is that according to a LineCalc tool in ADS, the Zc increases for a given line with increasing frequency which is exactly opposite to what I expected. Am I missing out on something here?
2) Why are there are 2 formulae for calculating Zc based on W/h greater or lesser than 1? (Reference Pozar Microwave Engg)
Does this mean that we can have 2 different possibilities to get say a 50 Ohm line on the same substrate for values of line width greater and lesser than that of the substrate respectively?
Hi Arka,
within the frequency range where microstrips are used, the permittivity of most materials decreases but the loss increases with increasing frequency.
According to 3.195 in Pozar, the effective permittivity is not a function of frequency but of the ratio d/W, and of the permittivity of the substrate, of course. So, for given d and W, with increasing frequency the effective permittivity decreases, too.
There are formulae including the dependency of the effective permittivity of the frequency, too, which result in slightly rising effective permittivity with increasing frequency, provided the permittivity of the substrate is constant. However, below about 10 GHz the effective permittivity is virtually constant. See Fig. 7.1 in this article:
Article A Basic Rectangular Microstrip Patch Antenna Design and Anal...
At a few GHz, the decreasing of the permittivity of the substrate dominates the behavior of the effective permittivity, or at least compensates the increasing. See for example Fig. 2 here:
https://chemandy.com/technical-articles/measuring-track-characteristic-impedance/measuring-track-characteristic-impedance-article3.htm
Since with usual substrates and usual impedances of microstrips, the contribution of R and G (causing the loss) to the impedance is much smaller than that of C and L, with increasing frequency the impedance increases, even with increasing loss.
I don't have the LineCalc tool at hand but the microstrip calculator by Rogers shows the same tendency: For example, with substrate RO4350, thickness 30 mil, conductor width 45 mil it gives 61.87 ohm at 1 GHz, and 63.55 ohm at 10 GHz.
The formulae 3.196 in Pozar are approximations (as many engineering formulae are) with limited validity. Near W/d = 1, both should yield about the same result. The attached picture shows the dependency of Z of W/d = 0.3 .. 4, continous line = first formula, dashed line = second formula, for the case eps_r = 4, d = 1. (In this case, both results match at about W/d = 1.3) So, you don't have a choice between two configurations of microstrips for given impedance, permittivity, and thickness of the substrate.
Hi Arka,
within the frequency range where microstrips are used, the permittivity of most materials decreases but the loss increases with increasing frequency.
According to 3.195 in Pozar, the effective permittivity is not a function of frequency but of the ratio d/W, and of the permittivity of the substrate, of course. So, for given d and W, with increasing frequency the effective permittivity decreases, too.
There are formulae including the dependency of the effective permittivity of the frequency, too, which result in slightly rising effective permittivity with increasing frequency, provided the permittivity of the substrate is constant. However, below about 10 GHz the effective permittivity is virtually constant. See Fig. 7.1 in this article:
Article A Basic Rectangular Microstrip Patch Antenna Design and Anal...
At a few GHz, the decreasing of the permittivity of the substrate dominates the behavior of the effective permittivity, or at least compensates the increasing. See for example Fig. 2 here:
https://chemandy.com/technical-articles/measuring-track-characteristic-impedance/measuring-track-characteristic-impedance-article3.htm
Since with usual substrates and usual impedances of microstrips, the contribution of R and G (causing the loss) to the impedance is much smaller than that of C and L, with increasing frequency the impedance increases, even with increasing loss.
I don't have the LineCalc tool at hand but the microstrip calculator by Rogers shows the same tendency: For example, with substrate RO4350, thickness 30 mil, conductor width 45 mil it gives 61.87 ohm at 1 GHz, and 63.55 ohm at 10 GHz.
The formulae 3.196 in Pozar are approximations (as many engineering formulae are) with limited validity. Near W/d = 1, both should yield about the same result. The attached picture shows the dependency of Z of W/d = 0.3 .. 4, continous line = first formula, dashed line = second formula, for the case eps_r = 4, d = 1. (In this case, both results match at about W/d = 1.3) So, you don't have a choice between two configurations of microstrips for given impedance, permittivity, and thickness of the substrate.
Characteristic impedance of microstrip line is frequency dependent. The input impedance, looking into the front of a transmission line, is strongly frequency dependent. It is related to the capacitance and inductance, per unit length, of the line. Normally at low frequency characteristic impedance is at higher side and decrease to a constant value at higher frequency. At fix frequency, characteristic impedance is the intrinsic and instantaneous property of the geometry cross section and not a function of the traveling length. Trace width and dielectric height have the greatest role in the characteristic impedance.
The formulae I have seen give exactly the same result for W/h=1
It is easy to check this. As Dennis says, they are simple formulae, each fitting a more complicated curve over a limited range, and designed to give the same result, and perhaps even gradient, where they meet.
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