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  1. I followed the below procedure to design a 700MHz 1/4 wave monopole PCB antenna.

Is this correct?

  a. The trace length decided to match with Wavelength of the frequency 

   Wavelength (Lambda) = Wave Velocity (v) / Frequency (f)

                =299792458 /700000000
                =428.2/4
                =107mm
    So, the trace length =107mm.

 b.Trace width decided by microstrip 50 ohm impedance matching calculation

enter image description here

So, the trace width =51.9mils

  1. How to decide the gap between two trace line? Is there any calculation?

Please guide me.

enter image description here

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I don't design these kinds of antennas and my books on the topic are at work, so I'm not sure, but you will find you already got into trouble with your length calculation. You used the speed of light in a vacuum to calculate the length. (notice the 'in a vacuum' part). On a PCB the wave will travel more slowly. Your microstrip tool helpfully points out that in this case, it should be about 0.56 times the speed of light, so ballpark 150e6 meters per second.

And on top of that - you don't need a 50 ohm transmission line to make such an antenna in the first place. You don't want that ground underneath, because you want radiation of the monopole (with a ground, it wouldn't be a monopole)

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