DISTRIBUTED ELEMENT MODEL IN TRANSMISSION LINES (PART II)

In our previous post in this section, we stated what we will study next to characterize the Distributed Model:

1. The transmission lines equations
2. Solve and exercise applying the equations
3. Propagation constant and characteristic impedance of the line
4. 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:

 

Exercise Explained:

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?

Solution:

a)

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.

 

b)

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 , 2Ï€ , 4Ï€ … Radians.

This occurs in the following instants of time:

 

t₀ = 3.03 / 10⁴ = 3.03 x 10^-4 sec ,

Then

t₀ = 303 ms

t₁ = (2Ï€ + 3.03) / 10⁴ = 9.31 x 10^-4 sec , t₁ = 931 ms, …

t_n = (nÏ€ + 3.03) / 10⁴ sec, with n = 0, 2, 4 …

What’s next?

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? 😀