Transmission Lines: Part 1 An Introduction

 Suppose we have a long wire with 2 points on it A and B. at t=t1, If we apply a voltage pulse to the point A, assuming the wire being perfect conductor, What is the voltage at point B at the same instant of time?

From circuit theory, since the wire has no resistance associated with it, there’s no voltage drop across the wire and the voltage at point B is equal to that of point A at the same instant of time.

But this is not true. This is valid only if the signal travels across the wire with an infinite speed! In reality, signals cannot travel faster than the speed of light. So the voltage pulse will reach the point B at instant t2 after some delay. That delay is known as transit time.

Transit time results in a voltage difference between point A and B. There is always voltage difference between any 2 points in the line due to the finiteness of the speed of the wave.

We can minimize that difference in voltage by decreasing the wave frequency. In other words; The higher the wave period with respect to the transit time, The less would be the voltage difference.

Transit time is equal to the distance travelled by the wave over propagation speed, While the period is equal to 1 over frequency. Hence to minimize the voltage difference between point A and B on the wire, the length of the wire between point A and B must be very small compared to the wavelength. This concept (the transit time effect) is applied to any electrical component.

If the length of the component is large compared to the wavelength of the signal, then we can’t ignore the transit time effect. To make KVL and KCL valid we must meet the condition (circuit element must be much small compared to the wavelength of the signal).

If I divide this wire into smaller and smaller sections so that each section is small compared to the wavelength, the transit time effect would be negligible over each section so we can apply low frequency circuit analysis (KVL,KCL).

That wire is called transmission line. A path carrying electrical energy from source to load.

Consider a trans-Atlantic cable running from the US to the Great Britain. A good example of a uniform two-line transmission line.

If we short the other end and measure the input impedance, it looks like a big inductor with series resistor modeling the wire losses.

Similarly if we leave the other end open, The input impedance will look like a substantial shunt capacitor (two very big conductors separated by dielectric).

Due to the conductivity of the shields and the sea-water, there is also a shunt conductance associated with the cable, even when left open.

Since the distance is too large, Even at very law frequencies the signal wavelength would be too small compared to the wire and the delay would be significant.

So how do signals propagate through this cable?

Well each transmission line as we just see is composed of series inductance and resistance and parallel capacitance and conductance. Let’s model the cable by them then analyze the circuit.

In other words, lumping the total inductance, capacitance, resistance and conductance along the wire into idealized concentrated components with infinitesimal dimensions and connecting between them with perfect conductors. This is known as lumped element model.

Using lumped element model here, gives us completely wrong answer. Even at very low frequencies, The large series inductance would act as an open circuit, whereas the large shunt capacitance would make the wire a short circuit.

For the lumped element model to be valid, the signal wavelength must be very large compared to the circuit element. So we can divide the total series and shunt impedances into smaller and smaller sectionsof very small dimensions compared to the wavelength.

Let’s ignore the losses for the time being so that we can remove the series resistance and the parallel conductance from the model.

And here comes the problem. For 1cm long section we would need more than 500 Million inductors and capacitors! Even the most efficient simulator running on the most powerful computer can’t deal with that.

So how did people solve this problem when the first Transatlantic Cable was laid out in 1857?

Well, have you heard about infinite ladder networks before?

This model is an infinite ladder network! With Z1 and Z2 being the impedances of inductors and capacitors respectively.

Using the input impedance equation of the infinite ladder network, then the input impedance of such a cable is having that form.

Since each section is only 1cm long, inductors and capacitors are too small and can be neglected making the input impedance equal to the square root of l over c.

Observe that if we choose to terminate the ladder network at any point with an impedance of Zin, then the input impedance remain the same. The behavior of a finite ladder section terminated with Zin is indistinguishable from that of the infinite network.

So for lossless transmission line the voltage wave will see a real impedance while travelling through the wire. This impedance is known as characteristic

impedance. If the line is terminated with a load of that impedance, the characteristic impedance wouldn’t change.

We started by lossless elements (inductors, capacitors) and end up with real impedance does that means that the lossless transmission line dissipate energy?

Actually not. The circuit is made up of lossless components. We have 2 scenarios here. If the line is infinite, the signal will travel through it indefinitely. It is dissipated from the generator perspective.

If the line is terminated by resistance equal to the characteristic impedance, the signal travel through the wire and get dissipated in the load. In both cases the generator feels injecting the signal to resistance equal to the characteristic impedance.

But how the injected signal itself propagate through the line.

Now to answer that question let’s analyze one section from the line.

Even at very high frequencies, the signal wavelength will be very large compared to the section dimensions. hence lumped element model is valid and Kirchhoff laws can be applied at each section.

The analysis is very simple Just applying voltage signal to the input of the section and measuring the change in voltage and current across the element.

I’m not gonna bother you by the equations. The final result is these forms: voltage and current wave equations.

Where gamma is called propagation constant a characteristic quantity of transmission lines.

It depends only on the line parameters (L,C,R,G). Wave equations are simple differential equations telling us that at each instant of time each point in the line has different voltage and current based on its location.

If we assume that the voltage and current vary sinusoidally with respect to time, the solution of wave equations is consisting of forward travelling wave and backward travelling wave.

Where alpha and beta are the real and imaginary parts of gamma.

Let’s ignore the backward one, we’re gonna cover that in a separate video when we talk about reflection.

Now alpha represents the losses in the wire. Lets ignore it by ignoring R and G in each section to have a lossless line.

Taking the real part, we notice that the voltage and current vary sinusoidally with respect to time and distance along the line.

Notice that the current wave is the voltage wave divided by the characteristic impedance.

At each instant of time, each point in the line has different voltage and current.

By fixing the distance and varying the time, it feels like the signal is moving along the line.

What we are doing here is just plotting the voltage variation along the line at successive instants of time.

Here we assume that the line is lossless. Meaning there’s no attenuation in the signal while travelling through the line.

Voltage and current at each point in the line vary sinusoidally with respect to time you

can notice that by seeing the movement of the dot up and down. Where the height of each dot represents the voltage at that point in the line.

Don’t forget that the forward travelling current wave is just the voltage wave scaled by real number the characteristic impedance.

So it varies the same way as the voltage. 

Notice that when we change distance and fixing the time, it turns out that the signal is moving with respect to time meaning for a lossless line each point in the line sees a delayed version of the signal depending on it’s location on the line.

The larger the distance from the generator, The higher will be the delay. That delay is a function of distance along the line and is linearly proportional to (beta) the phase constant.

If beta is zero, there wouldn’t be delay. Beta is equal to 2 pi over the wave length.

The larger the wavelength of the signal, The less would be the delay