S parameter calculations for RF/Microwave design. Some comments on ARCTAN.

S parameters are usually used in calculations of RF/wireless/microwave design and development. S parameters are complex numbers that can be written in a rectangular form with real and imaginary components, as well as in the polar form, with a radius ( magnitude) and angle.

Conversions between the polar form and the rectangular form are common . Some engineers like the polar form others like the rectangular form. When converting from the rectangular form to the polar form, the magnitude is calculated by taking the square root of the sum of the squares of the real and imaginary component. Associated with this calculation is the angle.

The angle is calculated by taking the ARCTAN ( imaginary/real). This is easy to do but there is a wrinkle in this calculation that needs to be understood, and taken care of, when calculating the angle. We have found that some calculators on the web do not take care of this issue. 

The value of the angle is also dependent on the quadrant the angle lies in. For example, lets say that the real component is negative and the imaginary component is negative. When the imaginary is divided by the real in this case, the quotient is positive. When the ARCTAN is taken of this positive quantity the result can be ( if care is not taken) an angle in the first quadrant. This is misleading.

The correct result of such a calculation, must take account of which quadrant the angle lies in. In the above example the angle really lies in the third quadrant. The ARCTAN calculator can calculate the angle in the first quadrant. To get the correct answer one should subtract 180 degrees from the result.

Please see the Wikipedia for a detailed explanation of this. The search term is Inverse trigonometric functions.

At Signal Processing Group Inc., most of our RF/wireless/microwave designs use the s-parameter approach as routine. Please visit our website and check out other articles and information.

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