Part A (2 Marks) UNIT II TRANSMISSION LINE PARAMETERS 1. When does a finite line appear as an infinite line? (Nov / Dec 2011) It is an imaginary line of infinite length having input impedance equal to the characteristics impedance of the transmission line. A line of finite length, terminated in a load equivalent to its characteristics impedance, appears to the sending end as an infinite line. 2. If a line is to have neither frequency nor delay distortion, how do you relate attenuation constant and velocity of propagation to frequency? (Nov / Dec 2011) Attenuation constant and velocity of propagation will be independent of frequency. Attenuation constant, α = RG Velocity of propagation, v = 1/ LC. 3. What is meant by infinite line? (May/ Jun 2012) It is an imaginary line of infinite length having input impedance equal to the characteristics impedance of the transmission line. A line of finite length, terminated in a load equivalent to its characteristics impedance, appears to the sending end as an infinite line. 4. Briefly discuss the difference between wavelength and period of a sine wave. (Nov / Dec 2008) Distance covered by an wave of one cycle is wavelength. Time taken to have a phase change of 360 o is period. 5. Define phase distortion. (Nov / Dec 2009) For an applied voice voltage was the received waveform may not be identical with the input waveform at the sending end, since some frequency components will be delayed more than those of other frequencies. This phenomenon is known as delay or phase distortion. 6. What is meant by inductance loading of telephone cables? (Nov / Dec 2009) Distortionless line with distributed parameters is used to avoid the frequency and delay distortion experienced on telephone cables. It is necessary to increase the L/C to achieve distortionless condition L/C = R/G than side suggested that the inductance be increased and pup in suggested that this increase in the inductance by lumped inductors spaced at intervals along the line. This use of inductance is called the loading the line. 7. What is the relationship between characteristics impedance and propagation constant. (Apr / May 2010) Relationship between propagation constant and characteristics impedance, γ = Z 0 Y Z 0 characteristics impedance Y admittance γ propagation constant.
8. Define delay distortion. (Nov / Dec 2010) For an applied voice voltage, the received waveform will not be identical with the input since some frequency components will be delayed more than other frequencies. This is called delay distortion. 9. Write the expressions for the phase constant and velocity of propagation for telephone cable. (Nov / Dec 2010) For a telephone cable: Phase constant (β) = (ωrc /2) Velocity of propagation (v) = (2ω/CR). 10. How can distortion be reduced in a transmission line? (Apr / May 2011) Distortion can be reduced in a transmission line by maintaining the condition, RC = LG. 11. A transmission line has Z 0 = 745-12 0 Ω and is terminated is Z R = 100 Ω. Calculate the reflection loss in db. (Apr / May 2011) Z 0 = 745-12 0 Ω Z R = 100 Ω Reflection factor k = 2 Z R Z 0 / (Z R + Z 0 ) = 0.7645 ohm. Reflection loss = 20 log 1 / k = 3.7751 db. Part B (16 marks) 1. Obtain the general solution of a transmission line. (Nov / Dec 2011) (May/Jun 2012) (Apr/May 2010) General solution of the transmission line: It is used to find the voltage and current at any points on the transmission line. Transmission lines behave very oddly at high frequencies. In traditional (low-frequency) circuit theory, wires connect devices, but have zero resistance. There is no phase delay across wires; and a short-circuited line always yields zero resistance. For high-frequency transmission lines, things behave quite differently. For instance, short-circuits can actually have an infinite impedance; open-circuits can behave like shortcircuited wires. The impedance of some load (ZL=XL+jYL) can be transformed at the terminals of the transmission line to an impedance much different than ZL. The goal of this tutorial is to understand transmission lines and the reasons for their odd effects. Let's start by examining a diagram. A sinusoidal voltage source with associated impedance ZS is attached to a load ZL (which could be an antenna or some other device - in the circuit diagram we simply view it as an impedance called a load). The load and the source are connected via a transmission line of length L: In traditional low-frequency circuit analysis, the transmission line would not matter. As a result, the current that flows in the circuit would simply be:
However, in the high frequency case, the length L of the transmission line can significantly affect the results. To determine the current that flows in the circuit, we would need to know what the input impedance is, Zin, viewed from the terminals of the transmission line: The resultant current that flows will simply be: Since antennas are often high-frequency devices, transmission line effects are often VERY important. That is, if the length L of the transmission line significantly alters Zin, then the current into the antenna from the source will be very small. Consequently, we will not be delivering power properly to the antenna. The same problems hold true in the receiving mode: a transmission line can skew impedance of the receiver sufficiently that almost no power is transferred from the antenna. Hence, a thorough understanding of antenna theory requires an understanding of transmission lines. A great antenna can be hooked up to a great receiver, but if it is done with a length of transmission line at high frequencies, the system will not work properly. Examples of common transmission lines include the coaxial cable, the microstrip line which commonly feeds patch/microstrip antennas, and the two wire line:
To understand transmission lines, we'll set up an equivalent circuit to model and analyze them. To start, we'll take the basic symbol for a transmission line of length L and divide it into small segments: Then we'll model each small segment with a small series resistance, series The parameters in the above figure are defined as follows: R' - resistance per unit length for the transmission line (Ohms/meter) L' - inductance per unit length for the tx line (Henries/meter) G' - conductance per unit length for the tx line (Siemans/meter) C' - capacitance per unit length for the tx line (Farads/meter) We will use this model to understand the transmission line. All transmission lines will be represented via the above circuit diagram. For instance, the model for coaxial cables will differ from microstrip transmission lines only by their parameters R', L', G' and C'. To get an idea of the parameters, R' would represent the d.c. resistance of one meter of the transmission line. The parameter G' represents the isolation between the two conductors of the transmission line. C' represents the capacitance between the two conductors that make up the tx line; L' represents the inductance for one meter of the tx line. These parameters can be derived for
each transmission line. An example of deriving the paramters for a coaxial cable is given here. Assuming the +z-axis is towards the right of the screen, we can establish a relationship between the voltage and current at the left and right sides of the terminals for our small section of transmission line: Using oridinary circuit theory, the relationship between the voltage and current on the left and right side of the transmission line segment can be derived: Taking the limit as dz goes to zero, we end up with a set of differential equations that relates the voltage and current on an infinitesimalsection of transmission line: These equations are known as the telegraphers equations. Manipulation of these equations in phasor form allow for second order wave equations to be made for both V and I:
The solution of the above wave-equations will reveal the complex nature of transmission lines. Using ordinary differential equations theory, the solutions for the above differential equations are given by: 2. The solution is the sum of a forward traveling wave (in the +z direction) and a backward traveling wave (in the -z direction). In the above, traveling voltage wave, is the amplitude of the backward traveling is the amplitude of the forward voltage wave, is the amplitude of the forward traveling current wave, and is the amplitude of the backward traveling current wave. 2. Deduce the expressions for characteristics impedance and propagation constant of a line of cascaded identical and symmetrical T sections of impedance. (Nov/ Dec 2011) The characteristics impedance is One T section representing an incremental length Δx of the line has a series impedance Z 1 = Z Δx and shunt impedance Z 2 = 1 / Y Δx. The characteristics impedance of any small T section is that of the line as a whole.
The four line parameters resistance (R), inductance (L), capacitance (C) and conductance (G) are known as the primary constants of the transmission line. 3. If Z = R + jωl and Y = G + jωc, show that the line parameter values fix the velocity of propagation for an ideal line. (Nov/Dec 2011) (Nov/Dec 2009) (Apr/May 2011) The propagation constant (γ) and characteristics impedance (Z 0 ) are called secondary constants of a transmission line. Propagation constant is usually a complex quantity. The characteristics impedance of the transmission line is also a complex quantity.
Velocity : The velocity of propagation is given by, This is the velocity of propagation for an ideal line. Wavelength : The distance travelled by the wave along the line while the phase angle is changing through 2Л radians is called wavelength.
4. Deduce the expressions for characteristics impedance and propagation constant of a line of cascaded identical and symmetrical T sections of impedance. (Nov/ Dec 2011) The characteristics impedance is One T section representing an incremental length Δx of the line has a series impedance Z 1 = Z Δx and shunt impedance Z 2 = 1 / Y Δx. The characteristics impedance of any small T section is that of the line as a whole.
The four line parameters resistance (R), inductance (L), capacitance (C) and conductance (G) are known as the primary constants of the transmission line. 5. Derive the two useful forms of equations for voltage and current at any point on a transmission line. (Nov/ Dec 2011) (May/Jun 2012) (Nov /Dec 2008) (May/Jun 2009) (Nov/Dec 2009) (Apr/May 2010) The expressions of voltage and current at the sending end of a transmission line length l are given by,
The input impedance of a transmission line is given by, If short circuited, the receiving end impedance is zero.
6. Write brief note on frequency and phase distortions. (Nov /Dec 2008) (Nov/Dec 2009) (Apr/May 2011) Frequency distortion : A complex voltage transmitted on a transmission line will not be attenuated equally and the received waveform will not be identical with the input waveform at the transmitting end. This variation is known as frequency distortion. The attenuation constant is given as, α = { RG ω 2 LC + (RG ω 2 LC) 2 + ω 2 (LG + CR) 2 } / 2 α is a function of frequency and therefore the line will introduce frequency distortion. Delay distortion: For an applied voice voltage wave the received waveform may not be identical with the input waveform at the sending end, since some frequency components will be delayed more than those of other frequencies. This phenomenon is known as delay or phase distortion. The phase constant is, β = { RG ω 2 LC + (RG ω 2 LC) 2 + ω 2 (LG + CR) 2 } / 2 β is not a constant multiplied by ω and therefore the line will introduce delay distortion. Frequency distortion is reduced in the transmission of high quality over wire lines by the use of equalizers at the line terminals. Delay distortion is of relatively less importance to voice and music transmission. But it can be very serious for video transmission. This can be avoided by the use of co-axial cables. 7. Derive campbell s equation. (May/Jun 2009) An analysis for the performance of a line loaded at uniform time intervals can be obtained by considering a symmetrical section of line from the center of line from the center of one loading coil to the center of the next coil. The section of line may be replaced with an equivalent T section having symmetrical series arms. The series arm of T section including loading coil is given by,
Where Z 1 /2 is the series arm of T section. Where l is the distance between two loading coils. The shunt arm Z 2 of the equivalent T section is, For loaded T section, This equation is called as the campbell s equation and it is used to determine the value of γ of a line section consisting of partially lumped and partially distributed elements. 8. Derive the conditions required for a distortionless line. (Nov / Dec 2010) If a line is to have neither frequency nor delay distortion, then attenuation factor α and the velocity of propagation v cannot be functions of frequency. If v = ω/β β must be a direct function of frequency.
This is the condition for distortionless line. This is the same velocity for all frequencies, thus eliminating delay distortion. Attenuation factor To make α independent of frequency, the term (RG ω 2 LC) 2 + ω 2 (LC + RG) 2 is forced to be equal to (RG + ω 2 LC) 2.
This will make α and the velocity independent of frequency simultaneously. To achieve this condition, it requires a very large value of L, since G is small. It is independent of frequency, thus eliminating frequency distortion on the line. The characteristics impedance Z 0 is given by, It is purely real and is independent of frequency.
9. Derive an expression for propagation constant and the velocity of propagation for an ordinary telephone cable. In the telephone cable the wires are insulated with paper and twisted in pairs. This construction results in negligible values of inductance and capacitance. Therefore, Lω << R and G << Cω. It is found that the propagation constant γ and the velocity of propagation v are functions of frequency. Thus, the higher frequencies are attenuated more and travel faster than the lower frequencies resulting in considerable frequency and delay distortion.