# A ##50\Omega## lossless transmission line impedance question

• David J
In summary: The equation provided in the conversation already includes the unit for Z0, so it is not necessary to include it again when substituting in values. However, when writing the final answer, Zin should have units of ohms.
David J
Gold Member

## Homework Statement

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a ##50\Omega## lossless transmission line of length ##0.4\lambda## is terminated in a load of ##(40+j30)\Omega##. Determine, using ##Zin=Z0\frac{ZL cos \beta l+j Z0 sin \beta l}{Z0 cos \beta l +j ZL sin \beta l}## the input impedance to the line.

##Z0=50\Omega##

##ZL=(40+j30)\Omega##

##\beta l =2.513## radians

##cos\beta l =-0.809## radians or ##0.999^0##

##sin\beta l=0.588## radians or ##0.0438^0##

## Homework Equations

[/B]
##Zin=Z0\frac{ZL cos \beta l+j Z0 sin \beta l}{Z0 cos \beta l +j Zl sin \beta l}##

## The Attempt at a Solution

[/B]
##Zin=50\Omega\frac{(40+j30)(-0.809) +j(50)(0.588)}{50(-0.809) +j(40+j30)(0.588)}##

So ##50\Omega\frac{(-32.36-j24.27)+(j29.4)}{(-40.45+(-17.64+j23.52)}##

So ##50\Omega\frac{(-32.36+j5.13)}{(-58.09+j23.52)}##

##=50\Omega(0.5093+j0.118)##

So ##Zin=25.465+j5.90## or ##26.14\angle13^0## or ##26.14\angle0.23## radians

I have been over this 5 times now to check and I feel I am correct but I would like a second opinion please, if possible, to identify any errors etc.

Thanks

The only problem I can see is that the sines and cosines should be unitless. The sine or cosine (or result of any trig function) is a pure number that represents a ratio. Otherwise, well done!

Appreciated. thanks again

Sorry to resurrect this post, just a quick simple one, Would Zin not have ohms as a unit?

Yes, Zin should be in ohms.

## 1. What is a lossless transmission line?

A lossless transmission line is a type of electrical cable that has no resistance, allowing for the transfer of electricity with minimal energy loss. This is achieved by using highly conductive materials and minimizing any interruptions or imperfections in the cable.

## 2. What is the impedance of a lossless transmission line?

The impedance of a lossless transmission line is a purely reactive value, meaning it has no resistance component. It is typically represented by the symbol ##Z_0## and has a value of 50 ohms.

## 3. How does a lossless transmission line affect signal transmission?

A lossless transmission line has a constant impedance along its length, which allows for efficient transfer of signals without any reflections or losses. This results in a high-quality transmission of signals with minimal distortion.

## 4. How is the impedance of a lossless transmission line calculated?

The impedance of a lossless transmission line can be calculated using the formula: ##Z_0 = \sqrt{\frac{L}{C}}##, where L is the inductance per unit length and C is the capacitance per unit length of the cable.

## 5. What factors can affect the impedance of a lossless transmission line?

The impedance of a lossless transmission line can be affected by the materials used, the dimensions of the cable, and the frequency of the signal being transmitted. Changes in any of these factors can alter the inductance and capacitance of the cable, thus affecting its impedance.

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