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How to derive characteristic impedance of the transmission line

By Applied Electronics - Sunday, March 5, 2017 No Comments
Any transmission line has characteristic impedance measured in unit of Ohm. In the previous blog post Derivation of Transmission Line Equations we derived the Telegraph equations or the transmission line equations. In this blog post we want to show how to derive characteristic impedance of a transmission line from telegraph equation. The characteristic impedance Zo is derived from the Transmission line equation.

To make you easier to follow, the model diagram of two parallel line transmission line which was used in the earlier tutorial is reshown below.


We started by modeling this transmission line and created the following mathematical model,


  By taking derivative w.r.t z we derived the transmission line or the telegraph equation.


where,

Now in order to derive the characteristic impedance of the transmission line we have to solve this telegraph equation. We can solve this 2nd order differential equation for the voltage V(z) which is function of distance along the transmission line and time e-jwt. Then after knowing the solution of this equation for voltage V(z) we can use it to solve for current I(z).

The solution is composed of two voltages traveling in opposite direction V(-z) and V(+z) as follows,



where A1 and A2 are constant of the differential equations. These constant are determined by the boundary conditions.

To find the solution for the current I(z) of the telegraph equation we use the solution V(z). We do that by substituting V(z) into the first order equation of the transmission line. We obtain the following solution,

Let,
Then the current I(z) is,



In the above derivation, the Z is called the characteristic impedance of the transmission lines. It id defined as the ratio of forward voltage to forward current.


 where we have used the fact that,

 

The unit of characteristic impedance Z is Ohm. It is a parameter of transmission line which depends on frequency (via w) and the transmission line R, L, G and C. The industry standard of transmission line characteristic impedance is 50 Ohm and 75Ohm.

See rf circuit design: theory and applications.

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