In our previous post in this section, we stated what we will study next to characterize the Distributed Model:
- The transmission lines equations
- Solve and exercise applying the equations
- Propagation constant and characteristic impedance of the line
- Solve and exercise calculating this parameters on a real line
In this post, we will see points 1 and 2.
Distributed Element Model
Recall the representation and meaning of the distributed elements in a transmission line:
These parameters vary according to the type of line. For example:
Note: click on the images to see a better resolution. Then, click to go back in your browser to come back to the post 😉
The transmission lines equations
When using the distributed element model, we can apply the Kirchhoff laws:
And the solutions are:
The distributed coefficients of a transmission line with w =104 rad/sec are:
R = 0.053 Ω /m L = 0.62 mH/m G = 950 pS/m C = 39.5 pF/m.
In the z-coordinate over the line, the instantaneous current is given by:
i(t) = 75 cos 10 4t mA
a) Obtain the expression for the voltage gradient along the line, in the point z.
b) What is the maximum value of the voltage gradient?
In the time domain, the voltage gradient is given by:
Substituting values, we have:
= – 0.053 ( 0.075 cos 10 4 t) + ( 0.62 x 10 -6 ) (0.075 x10 4 sin 10 4 t) =
= – 3.98 x10 -3 cos 10 4 t + 0.465 x 10 -3 sin 104 t =
= 4.006 x10 -3 cos( 10 4 t – 3.03) Volts/meter.
= 4.006 x10 -3 cos( 10 4 t – 173.4 ° ) Volts/meter.
The maximum voltage gradient is equal to the amplitude, 4 mV. This happens when:
cos(104 t – 3.03) = 1
This implies that,
104 t – 3.03 = 0 , 2π , 4π … Radians.
This occurs in the following instants of time:
t0 = 3.03 / 104 = 3.03 x 10-4 sec ,
t0 = 303 ms
t1 = (2π + 3.03) / 104 = 9.31 x 10-4 sec , t1 = 931 ms, …
tn = (nπ + 3.03) / 104 sec, with n = 0, 2, 4 …
We hope this post was useful 🙂 In the next post in this section, we will cover the points:
3. Propagation constant and characteristic impedance of the line
4. Solve and exercise calculating this parameters on a real line
After that, we will move to the the parameters when the line is in a transient state which analyses what happen when there is a sudden change in the conditions of the line.
Does this sound interesting? 😀