In transmission lines if the line is terminated with a load equal to the characteristic impedance, then the incident signal is fully dissipated in the load.
If the load impedance is different than the characteristic impedance, part of the signal is dissipated in the load and the other part is reflected back.
The reflected wave is zero at the matched condition and increases as the mismatch gets higher and higher.
The resultant signal travelling through the line is the superposition of the forward and the backward travelling waves.
The reflected signal interfere the incident one making the resultant looks like a standing wave.
The amplitude is changing in a limited range between maximum and minimum values determined by how big the reflection is.
As the mismatch between the load and the characteristic impedance increase the reflected signal increases as well and hence the interference gets bigger and bigger.
If the transmitted signal get reflected completely, the resultant is then a completely standing wave not moving forward or backward.
Today we will analyze transmission lines under sinusoidally time varying excitation.
And since any type of signal can be broken down into sinusoids from Fourier analysis, knowing how the transmission line or any system behaves under sinusoidal excitation, we can predict the behavior of the system when driven by any kind of signal.
In our discussion of transmission lines, we derived that the voltage wave travelling through the line is govern by the solution of the voltage wave equation.
A forward travelling wave and a backward travelling wave.
The current is just the voltage wave divided by the characteristic impedance of the line with minus sign here.
When the input voltage signal is varying sinusoidally with respect to time the signal is also varying sinusoidally with respect to distance along the line.
The forward and backward travelling waves are just 2 complex exponentials.
Complex exponential is nothing more than a vector in the complex plane That vector has some amplitude and some amount of phase angle measured from the positive x axis.
If the phase angle is variable, the complex exponential function is a rotating vector as we sweep that variable.
The phase angle of the forward and backward waves is function of 2 variables: time and the length of the line.
So each one of these variables is able to rotate the vector Omega and beta are two constants representing the angular frequency and the phase constant respectively.
You will feel the meaning of them by the end of this video. For the forward wave At each instant of time for instance at t=0, the phase angle changes as we sweep the L, Hence it rotates as we move along the line.
Larger values of beta (the phase constant) speed up the rotation as we move along the line. We’re interested in the real part, so the forward voltage wave is distributed sinusoidally along the line.
As we sweep the time the phase angle changes also.
It feels like voltage wave is moving through the line as time goes on. The angular frequency determines by how much the phase would change as we sweep the time.
If there is mismatch between the load and the characteristic impedance, then there is a reflected signal also distributed sinusoidally along the line.
As time goes on, the reflected signal moves in the opposite direction. Due to the reverse of polarities here These 2 waves are added up together and since they are moving in the opposite direction, the resultant is continuously swinging between a maxima and minima while moving in the direction that corresponds to the larger signal.
The forward direction. As the reflected wave gets bigger and bigger, the resultant is becoming more and more standing and becoming completely standing when the reflected wave is equal to the incident one.
We can look at this standing wave phenomenon from another perspective. Complex exponential function is just a rotating vector in the complex plane.
The forward travelling wave is a vector in the complex plane whose phase angle depends on time as well as the location on the line.
At t=0, at each point on the line the forward travelling wave is a vector with constant magnitude and a phase angle that depend on that location on the line.
As beta increase, the phase angle associated with each point on the line increase. As time goes on, the vectors along the line rotate with constant speed omega.
As you can see: as time goes on the forward travelling wave is moving along the line from the source toward the load.
Same thing applies for the backward travelling wave.
But unlike the forward one, It moves in the backward direction as time goes one.
Now addition of two vectors in the complex plane is just adding the tail of the second vector on top of the head of the first.
So at t=0, the whole voltage wave at each point on the line is the addition of the two vectors associated with the incident and the reflected waves.
As time goes on the vectors rotate and the resultant amplitude is swinging between maximum and minimum values.
This maxima and minima happens due to the constructive and destructive interference between forward and backward waves.
As the reflected wave gets bigger and bigger this maxima and minima becomes more and more significant.
The ratio between the maxima and minima is called voltage standing wave ratio(VSWR) If the reflected wave is equal to the incident wave(the incident wave get reflected completely), the resultant is completely standing wave.
So if we freeze the time, the full voltage wave is the superposition of two vectors representing the forward and backward waves.
And the voltage happens to be distributed sinusoidally along the line.
By varying the time, the voltage is propagating through the line with amplitude swinging up and down.
The current flowing through the line is also a wave and behaving the same way as the voltage.
It consists of forward travelling wave as well as backward travelling one if there is reflection.
In fact, the current wave is the voltage wave divided by the characteristic impedance of the line.
The reflected current wave is 180 degrees out of phase of the reflected voltage one as depicted by the minus sign here.
So the amplitude variation of the resultant current along the line is in the opposite direction to that of the voltage.
Whenever there is a voltage maxima there is a current minima and whenever there is a voltage minima the current is maximum.
Here the voltage and current waves travelling through the line. If we take the forward voltage and current waves as common, what’s left is one plus and one minus the same factor in the voltage and current.
The ratio of the magnitude of the reflected over incident waves is the reflection coefficient
at the load end and is equal to the load impedance minus the characteristic impedance over the load impedance plus the characteristic impedance. A complex quantity with magnitude and phase that represents how much the mismatch between the load impedance and the characteristic impedance.
So the whole voltage and current waves are just the forward wave with a variable magnitude. Now this is a vector in the complex plane with magnitude of one and phase of zero.
Added to it another vector with magnitude equal to the reflection coefficient and phase depends on the distance on the line.
As we move along the line the phase becomes more and more negative and hence the vector rotates in the clockwise direction.
The current amplitude is also a sum of constant and rotating vectors.
Notice that the minus sign here shifts the vector 180 degrees from the one associated to the voltage.
The resultant amplitude is the sum of the constant and the rotating vectors. As you can see the magnitudes of the whole voltage and current are changing between maximum and minimum values as we move along the line.
Wherever there is a maximum voltage there is minimum current and vice versa. as the reflections get higher and higher, the voltage and current amplitudes vary significantly while propagating through the line.
So what these equations tell us is at the input of the line if we apply a sinusoidally time varying sinusoid with certain amplitude and frequency. If the line is terminated with load equal to the characteristic impedance of the line then there is no reflected wave. There’s only a forward wave.
And the voltage and current waves are travelling through the line with constant amplitude. What changes is only the phase.
The signal keeps shifting in time as we move along the line. Meaning at each point on the line we see a delayed version of the signal based on that location on the line but the amplitude of the voltage and current is fixed.
Hence the voltage over the current at each point on the line is constant and equal to the characteristic impedance.
If we measure the impedance at any location on the line it will be a constant value. The characteristic impedance.
If there’s mismatch between the load and the characteristic impedance. then there is a reflected wave.
Added to the forward wave changing the amplitude based on the location on the line. As we move along the line, the amplitude of the voltage and current waves is changing between maximum and minimum value.
The maximum voltage amplitude occurs at the minimum current and the minimum voltage is at the maximum current. That amplitude variation becomes bigger and bigger as the mismatch between the load and the characteristic impedance goes higher and higher.
The extreme case happens when the full wave is reflected; the reflection coefficient is one.
In that case, the reflected wave is equal to the incident wave hence the amplitude varies from zero to two times the incident wave amplitude.
The ratio of the maximum and minimum values is called voltage standing wave ratio and current standing wave ratio.
The SWR is a measure of the depth of the standing wave pattern.
Hence a measure of the matching of load to the transmission line. A matched load would result in an SWR of 1 implying no reflected wave. An infinite SWR represents the full reflection of the incident wave.
So at each point on the line the amplitude of the voltage and current is different than that of the previous one. Hence the voltage over current or the impedance measured is different as we move along the line.
This concept is known as impedance transformation. If we terminate the line with impedance equal to the characteristic impedance, then the impedance measured at each point on the line is constant value (the characteristic impedance).
If there is a mismatch between the load and the characteristic impedance, depending on how large the mismatch, the amplitude of the wave is changing between peaks and valleys. Hence the impedance measured at each point on the line changes from point to point.
The voltage and current amplitude as well as impedance variations along the line go higher and higher as the mismatch increase and disappear when the load is matched to the line (the load is equal to the characteristic impedance)
You may noticed that the amplitude of the voltage and current waves is changing in a periodic manner.
The amplitude variation repeats itself every half of the wavelength of the signal. So the impedance measured on any point on the line is equal to the impedance measured after moving half wavelength from that point.
Voltage and current amplitudes as well as impedance repeat themselves as we move along the line. So if we terminate the line with some impedance, that impedance is transformed to different value In each point on the line.
If we connect a source to a load through a transmission line; the length of the line should be taken carefully since based on that length the source see different load impedances.
That problem as we’ve just seen can be solved by using matched load so that the source always sees the characteristic impedance irrespective of the length of the line.
Impedance transformation is very tedious to analyze mathematically.
This relationship gives us the transformed impedance at any location on the line based on the load impedance, the characteristic impedance and the length of the line from the load to the point.
Most textbooks use normalized impedances instead. Normalized impedance is just the impedance divided by the characteristic impedance.
Absolute load impedance doesn’t have any meaning. What’s more important is the load impedance relative to the characteristic impedance as this is what determines the mismatch.
Based on that mismatch the standing wave pattern will change. Hence the impedance transformation would change.
So we always deal with normalized impedances. Fortunately most of transmission line calculations aren’t done by math.
There’s a fancy tool called smith chart can do most of the work for us.
So voltage and current are transmitted from the source to the load through the transmission line in the form of waves.
Since the line is large compared to the wavelength of the signal, the voltage and current take some time to travel from source to load.
If the load is matched to the line, voltage and current on each location on the line is shifted on time meaning the voltage and current on any point is a delayed version and that delay is based on the location of the measurements.
The voltage and current take some time to reach a certain point on the line. If the load isn’t matched to the characteristic impedance. then not only there’s a delay but also the amplitude is changing from point to point along the line.
The amplitude variation is govern by the standing wave pattern. The effect of reflection and standing wave isn’t desirable but can be exploited to solve many many problems using transmission lines.
We will dive into that. But we must introduce the concept of smith chart at first
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