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In telecommunication, an artificial transmission line is a four-terminal electrical network that has the characteristic impedance, transmission time delay, phase shift, and/or other parameter(s) of a real transmission line and therefore can be used to simulate a real transmission line in one or more of these respects.
This lab session is focused on the design of an artificial transmission line. It aims to study and confirm the behaviour of an artificial line, and then compare the theoretical performance to what is obtainable in practice. The design models the transmission line from the scratch. To achieve this, values for the various components are picked to make the design feasible.
To design the transmission line, a T-section of a series inductance and a pair of shunted capacitors which is a low-pass filter is used as the building block. Since there will be no power loss in the circuit because it is purely reactive, it can be used to simulate the behaviour of a lossless transmission line over a specified range of frequencies which will be ascertained in the laboratory exercise. A cascade of this structure could be used to build an artificial line of any length which could replicate the behaviour of a real transmission line.
By varying the frequency of the signal, and the load at the end of the line, the voltage distribution across different points on the line can be ascertained. This gives an insight into the behaviour of real transmission line, and lead to the building of more efficient transmission lines.
For the purpose of this report, every reference to a transmission line refers to a lossless transmission line. It is also assumed that the signal is propagating in an air-filled medium.
The classic low-pass filter topology of series inductors interconnected with shunt capacitors can be used to simulate travelling wave propagation on transmission lines. When the line is operating below the filter cut-off frequency, its nodal voltages represent discrete samples of the standing wave voltage on an equivalent uniform transmission line. The various transmission line theory concepts covered below will be used during the lab exercise.
The characteristic impedance of a transmission line is the impedance measured at the input of this line when terminated by a matched load. In this laboratory session, it is assumed that a lossless line is being modelled. All parameters are calculated for on the assumption that the line is lossless, hence the basic unit is made up from just capacitors and inductors, neglecting any resistance or conductance. For a lossless line, characteristic impedance = ------------------- (1)
Where R=resistance,G=conductance,L=inductance,and C= capacitance.
This equation assumes that R<<Ï‰L, and G<<Ï‰C, hence R and G are set to zero.
2.2. Voltage reflection coefficient. (Î“)
For a wave travelling on a lossless transmission line from the generator end to load end, if the characteristic impedance of the transmission line matches the load impedance, there will be no reflections along the line. If there is a mismatch, the wave is reflected back towards the generator. The voltage reflection co-efficient is a measure of the degree of mismatch between the transmission line, and the load impedance which terminates the line. If represents the voltage going towards the load, and represents the reflected wave from the load end, the voltage reflection co-efficient Î“ = ----------------------(2)
Î“ = ----------------(3) where is the load impedance, and is the characteristic impedance of the line.
2.3. Electrical length
The electrical length of a section of a transmission is the length expressed as the number of wavelength of the signal propagating along the transmission line. It is commonly expressed in radians.
For an artificial transmission line, the electrical wavelength per section = Ï‰ --------------------------(4)
Where Ï‰ = angular frequency, = total inductance per section, and = total capacitance per section.
2.4. Physical length(â„“)
The physical length of a transmission line is the length of the line measured in metres. It represents the physical distance of the line.
Physical length â„“ = c ------------------------(5)
Where c is the speed of light in air, and the value is 3Ã-m/s.
2.5. Standing waves
When power is applied to a transmission line by a generator, a voltage and current appear whose values depend on the characteristic impedance and applied power. The voltage and current waves travel to the load at a speed slightly less than . If = , the load absorbs all power and none is reflected. The only waves present are the voltage and current (in phase) travelling waves from the generator to load.
If is not equal to , some power is absorbed, and the rest reflected. We thus have one set of waves, V and I, travelling towards the load, and the reflected set travelling back to the generator. This creates a standing wave on the transmission line, and this wave pattern will be studied in this laboratory exercise.
2.6. Quarter and half wavelength behaviour
Sections of transmission lines that are quarter-wavelength and half - wavelength long have important impedance- transformation properties.
When a transmission line is a quarter of a wavelength from the load impedance , the impedance at that point of the line is given by
This relationship is sometimes called reflective impedance, i.e. the quarter-wavelength reflects the opposite of its load.
Bearing this in mind, a transmission line a half wavelength away from its load reflects the load impedance. This is true for all transmission lines whose loads are not resistive and equal to the characteristic impedance of the transmission line. This concept can be very useful in length determination of a transmission line, impedance matching, and also study of general voltage distribution along the line.
2.7. Error calculation.
The error associated with the performance of an artificial transmission when compared to that of a real transmission line is given by the expression : = = ----------------------(7)
Where N = number of sections , and Î² is the phase constant.
3. Parameters calculations
The capacitance of a section is first determined.
L = 300ÂµH
Zo = 250â„¦
From (1) C =
C = 4.8nF.
Since C represents the total capacitance per section,
c = = 2.4nF.
where c is the value of each capacitor in a section.
Next to be determined is the electrical length and â„“ the equivalent physical length of the circuit.
â„“ = c
where c represents speed of light which is equal to 3 x 10^(-8).
â„“ = 3 Ã-
â„“ = 360meters.
A line of 3600 meters is to be model, therefore ten sections will be needed.
N = 10.
Wavelength of signal in air at 40khz =
The electrical length per section
4.1.Part 1: AC simulation of discrete transmission line.
First of all, the value of the capicators to be used is calculated then noted. All values of components to be used are then put down.
The low-pass filter circuit used to simulate the propagation of waves on a transmission line is then built using ADS 2009 CAD software. This is done by a combination of components as shown in the circuit diagram which is a low pass filter. The generator AC source is set to a frequency sweep between 10khz and 400khz.Simulation is done and a plot of gain against frequency is obtained.
A model of the 3600m line is the built by cascading the basic low-pass filter unit. The frequency is set at 40khz, and the load varied through 250ohms(matched load),0ohms(short-circuit),and 10000ohms(open circuit). The response of the circuit is then simulated, and values of voltages V1 - V9 as indicated in the circuit diagram obtained. This is to study the distribution of voltages along the modelled transmission line.
4.2. Part 2 : Time domain simulation of discrete transmission line
For this exercise, the voltage source is changed to a voltage step, and the AC simulator replaced with a transient simulator. The circuit is then simulated for matched load, short-circuit and open circuit conditions. Plots are obtained for the voltage in and Voltage out against time are then obtained. From the plots, the length of the transmission line can be calculated, and compared to that obtained in part 1.
5. RESULTS AND ANALYSIS OF RESULTS
5.1. AC SIMULATION OF DISCRETE TRANSMISSION LINE(FREQUENCY DOMAIN)
On carrying out the above laboratory exercise, the following results were obtained:
Fig 1 (a) and (b): Single Î -section AC simulation, and frequency response of single Î -section respectively.
Figure 1 represents the schematic diagram, and a plot of the response of the system to frequencies over a range of 10khz to 400khz. It will be noticed that up until a frequency of about 100khz, which is the 3db point, the gain of circuit is relatively steady, but deteriorates after that. This goes to show that for signals within the range of 0-100khz, the above circuit could be used to simulate the behaviour of a section of a real transmission line.
Fig 2 (a)
Fig 2 (a) & (b): Multiple Î -section AC simulation of an Artificial Transmission line, and Voltage distribution plot for Zload = 250â„¦ respectively.
Table 1: Standing wave voltages at each Node for Zload = 250â„¦
From transmission line theory, when the characteristic impedance and load impedance of a transmission are perfectly matched, there will be no reflections along the line. The voltage reflection co-efficient here is 0. The amplitude of the sinusoidal sine wave is fairly constant along the line with only difference in phase at different points along the line. Table 1 shows the distribution of voltage along the simulated line when load impedance is matched to characteristic impedance of the line. It can be seen from the plot in Fig 2(b) and table 1 that this is true.
Fig 3 (a) and (b): Multiple Î -section AC simulation of an Artificial Transmission line, and Voltage distribution plot for Zload = 0â„¦ respectively.
Table 2: Standing wave voltages at each Node for Zload = 0â„¦
For a short-circuited transmission line, the reflection co-efficient is -1. Table 2 shows the distribution of voltage across the line. Point V5 shows a point on the where both the forward and reflected voltage wave are in phase. Here, they add up, and the amplitude of the voltage sinusoid is highest. At other points along the line, voltages take on different values which is dependent on the amplitude and phase of both incident and reflected waves. Also seen from the plot in Fig 3(b) is that the voltage along the line tends to zero at Vout which is a short circuit.
Fig 4 (b )
Fig 4 (a) & (b) : Multiple Î -section AC simulation of an Artificial Transmission line, and Voltage distribution plot for Zload = 10000â„¦ respectively.
Table 3: Standing wave voltages at each Node for Zload = 10000â„¦
Fig 4 and table 3 contain data which explains the behaviour of a the artificial transmission line with Zload = 10000â„¦ which mirrors an open-circuited condition. Because of the open circuit at the load end, the sent signal is reflected. Fig 3b shows a plot of the voltage distribution along the simulated transmission line in this case.At point V5, the voltage is least and almost zero. This shows that both incident and reflected waves are almost 180 degrees out of phase.
Fig 5 (a) and (b) : Multiple Î -section AC simulation of an Artificial Transmission line, and Frequency response of Multiple Î -section respectively .
Figure 5(b) shows the response of the circuit in Fig 5(a). From transmission line theory, at a point a quarter of a wavelength from the load, an open circuit load will appear as short circuit. From the plot in fig 4b, it can be seen that this occurs at a frequency of 21khz. The length of the transmission line can thus be estimated.
V = fÎ»
Î» = V/f
Î»/4 = 3571m.
This is approximate the length of the transmission line.
5.2. TIME DOMAIN SIMULATION OF A DISCRETE TRANSMISSION LINE
Fig 6 (a) & (b) : Multiple Î -section Transient simulation of an Artificial transmission Line, and Time step response of the Artificial line with Zload = 250â„¦ respectively.
Figure 6(b) shows the response of the schematic diagram when the transmission line is matched. It shows the signal at Vin and Vout. It can be seen that both shapes are alike. This is because the incident signal is completely absorbed by the load, and there are no reflections. The delay in the rise of Vout is indicative of the time it takes the input signal to get to the load. Knowing this time, the length of the line can be calculated.
V = D/T
Where D stands for distance travelled, and T stands for time taken.
D = VT.
= (3*10^8)*[(12.7*10^(-6)) - (7.0*10^(-9))]
This is the length of the artificial line.
Fig 7 (a) &(b) : Multiple Î -section Transient simulation of an Artificial transmission Line, and Time step response of the Artificial line with Zload = 0â„¦ respectively.
Figures 8(a) &(b) : Multiple Î -section Transient simulation of an Artificial transmission Line, and Time step response of the Artificial line with Zload = 10000â„¦ respectively.
Figures 7 & 8 show responses of the circuit to short circuit and open circuit loads respectively. It can be seen that the value of Vin double for an open circuit, and was pulled down to zero for a short. This can be explained by the fact that the reflection co-efficient is -1 for a short circuit, and 1 for an open circuit. In the case of the open circuit, the reflected wave combines with the incident wave, and the amplitude of the incident wave is double. For the short circuit, the value pulled down to zero because the reflected wave is 180 degrees out of phase with the incident wave.
5.3 Error calculations
The error associated with working with a signal of 40khz,and working with 10 sections could be obtained from the formula :
This shows that the error associated with this setup is about 1.5%.
The first setup was done to determine the useable frequency for the signal which will be transmitted on the transmission line. It was then determined that frequencies below 100khz could be transmitted with negligible signal distortion. This showed frequency limitation of signals to be transmitted on a transmission line. A signal with a frequency of 40khz was picked for the above laboratory exercise. On cascading the low-pass filter structure, a structure with the characteristic of a transmission line was obtained. This structure was then terminated with varying loads(matched(250 0hms),open-circuited(10000 ohms),and short-circuited(0ohm))to study the distribution of voltages along the line. The effect of voltage reflection co-efficient on the wave pattern was noted.Results obtained showed the standing wave behaviour along the line at varying loads.
From the experiment performed in the time-domain, the wavelength of the propagated signal was obtained and calculations for the length of the line obtained could be seen to be same as length obtained in the first experiment performed in the frequency domain.
The quarter and half wavelength behaviour of transmission lines was also examined, and it was discovered that this is key in impedance matching when designing transmission lines.
From the results and calculations obtained, It can be concluded that a cascade of a series inductor, and pair of shunt capacitor with the right capacitances, inductances and signal frequency fairly simulates the behaviour of a physical transmission. With this in mind, transmission lines of any could be designed using this concept and tested for performance before the actual physical design is executed . Of course the error involved with using the cascaded model will be taken into consideration in the design of the physical structure. The error for the laboratory was 1.5% and this can be improved upon be increasing the number of sections (N).