What is the Solution for a Lossless Transmission Line with Given Parameters?

In summary, we have a lossless transmission line with characteristic impedance Z0 = 50Ω, propagation constant β = 5∏x10-3, and length l = 80 meters, terminated into a load ZL = 80Ω. The transmission line is powered by a source with 120 V and source impedance ZG = 12 Ω. We need to calculate the load voltage, but are facing difficulties with evaluating the imaginary arguments in the equations. One suggestion is to use a simple equation for E2, involving only β, l, Z0, and ZL. Another option is to use ABCD matrices.
  • #1
Mali
2
0

Homework Statement



A lossless transmission line with

characteristic impedance Z0 = 50Ω , β = 5∏x10-3,
and length l =80 meters, is terminated into a load ZL = 80Ω. The transmission line is powered by a source with 120 V and ZG = 12 Ω. Calculate the load voltage.



Homework Equations



V(z) = V+(e-jβz + [itex]\Gamma[/itex]ejβz

[itex]\Gamma[/itex] = (ZL - Z0)/(ZL + Z0)

Zinput = Z0(ZL+Z0tanh(jβL))/(Z0+ZLtanh(jβL))

V+ = VGZinput/((eL+[itex]\Gamma[/itex]e-jβL)(Zinput+ZG))

The Attempt at a Solution



[itex]\Gamma[/itex] = (80-50)/(80+50) = 3/13

Zinput = 50(80+50tanh(j*.4*pi)/(50+80tanh(j*.4*pi)

I can't find the input independence because my calculator can't evaluate the imaginary argument in the tanh function. I also have to same problem with solving for V+ and the imaginary exponents in the denominator. Does anyone know a way around this, or am I completely doing the wrong thing here?
 
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  • #2
I'd like to add that I'm very, very confused by the concepts, and I'm basically just juggling the equations derived in class.
 
  • #3
There is a simple equation for E2 for a lossless transmission line, involving only β, l, Z0 and load impedance ZLand E1.

What is tanh(jx) ) in terms of a non-hyperbolic trig function?

Unfortunartely for you, E1 is the voltage at the driven end of the line so you have to take care of the source impedance ZG. Hint: ZG forms a voltage divider with Zinput.

I can't post the equation for E2. Would violate the terms of use of this forum. It can be found in many places.

BTW the nice way to handle this problem is with ABCD matrices but I take it you haven't had those yet.
 
Last edited:

What is a lossless transmission line?

A lossless transmission line is a type of transmission line that has no resistance, capacitance, or inductance. This means that there is no energy loss as the signal travels through the line.

What are some examples of lossless transmission lines?

Examples of lossless transmission lines include coaxial cables, twisted pair cables, and waveguides.

What are the benefits of using a lossless transmission line?

The main benefit of using a lossless transmission line is that it allows for efficient and accurate transfer of signals without any loss of energy. This is important in applications where maintaining the integrity of the signal is crucial, such as in telecommunications and high-speed data transfer.

How do you calculate the characteristic impedance of a lossless transmission line?

The characteristic impedance of a lossless transmission line can be calculated using the formula Z0 = √(L/C), where Z0 is the characteristic impedance, L is the inductance per unit length, and C is the capacitance per unit length.

What are some factors that can affect the performance of a lossless transmission line?

Some factors that can affect the performance of a lossless transmission line include the length of the line, the frequency of the signal, and any impedance mismatches between the line and the connected components. Environmental factors such as temperature and humidity can also have an impact on the line's performance.

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