There are times when you can't get the values you need in inductances and capacitors when developing in high frequency situations. Sometimes, this isn't a big deal but when you get up to 1 GHz and need small values of Inductors with high Q, problems occur. So one way around this is to turn your needed inductor value into a microstrip equivalent. You can accurately develop an inductor with whatever value you desire with this method.
Keep in mine, this is best understood by those that have done some form of PCB development.
Method of Conversion
In order to convert a value you must first have one. So once you have determined your L value you must then find the equivalent impedance Zo. This is a simple step. Use the following equations to accomplish this:
where L is Inductance in Henrys, f is frequency in Hz, and Theta is Electrical Length in Degrees. This will give you the appropriate Characteristic Impedance for a particular frequency. If your circuit has to have a high bandwidth, this may not be the best method.
Now use Equation 2 if your inductor is in between 2 nodes. If one of the inductor nodes are going to ground it's preferable to use Equation 3. These numbers won't usually make that big of a difference but it can matter at extremely high frequencies.
Once you have the impedance you then need to convert this impedance[Zo] to a width[W] and length for a standard microstrip. (Note: I prefer to use a wave-guide for most of my circuits for ease of use with ground, but I won't cover the equations here). You will need the substrates thickness[H] (or distance from the top plane to the ground plane), substrate relative dielectric constant[Er]. Since finding the width uses an iterative algorithm it's best to just use software to find it. Use this site, Microstrip Calculator.
Normally I just use the Linecalc software that is embedded in Agilent's ADS. This is probably some of the better simulator software on the market. I recommend it whenever possible to use. I have extensive knowledge of the program, so if you have questions feel free to get in touch with me. Hope this has been helpful. Below I'll place the iterative equation used to find the impedance of a line.
To find the length[L] based on electrical length (degrees) use the following:
[Mu_r] relative permeability (usually 1 for circuits)
[epsilon_r] relative permitivity (Rogers 4003=3.55)
[c] speed of light in free space (2.99792e8)
[Theta] Electrical Length in Degrees
[L] Physical Length in Meters
The most common unit for length is mils especially in High Frequency Applications. To convert from Meters to Mils:
I hope this has been somewhat useful. If you have questions, feel free to write a comment or email me. Later
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Justin Coulston
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