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The arbitrariness is removed when a particular solution is constructed. The solution method just presented is the classical one.
It is possible also to employ another method. Let us now proceed in the opposite direction and derive the time domain signal from its spectrum 1. This representation underlines the importance of sinusoidal functions in the analysis of linear systems. A very useful property of Fourier transforms is the following: This implies that the following relation holds: Spectrum of a sinusoidal signal. This is obvious if we consuider eq.
Observe further that. It is important to remember. Because of the very close connection between phasors and Fourier transforms. Their counterpart in two or three dimensions are very important for the study of waveguides and resonators.
Proceeding in a similar way on the wave equation 1. We must remember. These equations can be called Helmholtz equations in one dimension. It is to be remarked that when the dielectric is homogeneous. It is a function of z and of t. It is clear that the value of the cosine function is constant if the argument is constant. Tree dimensional representation of a a forward wave. In the spacetime plot of Fig.
Consider now the second term of the expression of the voltage 1. Each wave is made of voltage and current that. Also in this case. In any case. When on a transmission line both the forward and the backward wave are present with the same amplitude.
Forward and backward waves on the line are the two normal modes of the system. Whereas Figs. The minus sign in the impedance relation for the backward wave arises because the positive current convention of the forward wave is used also for the backward one. The state is a function of z and the corresponding point moves on a trajectory in the state space.
In order to understand better the meaning of these equations. Further considerations will be made in Section 3. I z in an arbitrary point z. In the light of these considerations. It is useful to describe the propagation phenomenon on the transmission line in geometric terms. In algebraic terms this state vector is Obviously the two basis states are the forward and backward waves discussed before.
Geometric representation of the electric state of a transmission line. It is convenient to rewrite also eq. This matrix is known as transition matrix in the context of dynamical systems in which the state variables are real and the independent variable is time but coincides with the chain matrix ABCD of the transmission line length. Assuming for simplicity of drawing that in a point of the line voltage and current are real.
In the general case. Suppose we know voltage and current in the point z0 of the line and we want to compute the corresponding values in an arbitrary point z. For comparison. Hence these equations describe the change of basis. In other words. Notice that they coincide with the basis states of 1. Applying this property to the exponential of the matrix in 1. I to the modal basis of forward and backward waves.
In particular. If r denotes the relative permittivity of the insulator. The expressions that yield these parameters as a function of the geometry of the structure require the solution of Maxwell equations for the various cases. In this chapter we limit ourselves to a list of equations for a number of common structures: C capacitance p. The parameters related to the losses will be shown in chapter 4. G conductance p. L inductance per unit length. The two conductors.
L e C versus the ratio of the conductor diameters. R resistance p. We can observe that Chapter 2 Parameters of common transmission lines 2. The maximum frequency for which the coaxial cable is single mode is approximately 2vf.
If the operation frequency increases. Figure 2. Parameters of the coaxial cable vs. Two-wire transmission line. Hence the voltage of the inner conductor is referred to ground. This structure has a true TEM mode only if the dielectric that surrounds the conductors is homogeneous and the formulas reported hereinafter refer to this case.
The parameters of the two-wire transmission line. It is to be remarked that the coaxial cable is an unbalanced line. In practice. For this reason. On the contrary. The parameters of the two-wire line are: This is clearly an unbalanced structure. We report below an approximate expression for the characteristic impedance.
Since the two planes have the same potential. Note that this is a three conductor line two plus a grounded one.
The parameters for the symmetric mode can be computed from the following equations: Using the previous formulas we get: The relevant parameters cannot be expressed in terms of elementary functions. Stripline geometry. For the design activity. Since the transverse cross section is not homogeneous. Microstrip geometry. Even in this case. In an analysis problem. Characteristic impedance of a stripline vs.
From the second. Since this result is greater than 2. First of all. If we desire a more accurate model. The characteristic impedance at the operating frequency is then computed by 2.
Chapter 3 Lossless transmission line circuits 3. With this result in our hands. The relationship between these two quantities is displayed in graphic form by means of a famous plot.
Its transformation law is easily deduced from the previous equation: Example 1 Shorted piece of lossless transmission line of length l. It is clearly a closed curve. The intersections with the real axis. This curve is shown in Fig. Example 3 Length of lossless transmission line terminated with a reactive load. If we choose the line length conveniently. Example 2 Length of lossless transmission line terminated with an open circuit.
Example 4 Length of lossless transmission line. Circuit consisting of a generator and a load. Recall that the input impedance Zsc of this piece. Compute the impedance seen by the generator. Example 6 Analysis of a complete circuit. This is also the input impedance of a piece of transmission loaded by ZL. We can now perform the complete analysis of a simple circuit. Lumped equivalent circuit. As discussed in section 1. Its value can be determined only if we know an estimate of the wavelength on the line.
ZL z Scattering description of a load. The natural state variables are instead the amplitudes of forward and backward waves. If instead the load impedance is arbitrary.
We prove now that this is the case. By the way. We have seen that the transformation law of the local impedance on a transmission line is fairly complicated. Obviously also the forward and backward currents I0 e I0 could be used as state variables: This power.
This is obviously related to the fact that an ideal line is lossless. If in the point z voltage and current are V z e I z.
Recalling 1. We extend them to the realm of distributed circuits containing transmission lines. It is useful to express this power in terms of the amplitudes of the forward and backward waves.
Consider an ideal transmission line. The net power coincides with the incident one. This condition takes place when the load is a pure reactance. Hence the two waves are power-orthogonal i. Since there is no ambiguity. Both of them are very often used in practice see Table 3. The analytic expression of V z is then Ideal transmission line terminated with a generic load impedance.
Voltage and current on the line can be expressed in the following way in terms of forward and backward waves: Our goal now is to obtain plots of the magnitude and phase of voltage.
A quantity frequently used in practice to characterize a load is the return loss RL. Let us start with the magnitude plot.
As for the second. This shape is easily explained. Plot of the magnitude of voltage. Correspondence between values of return loss. Return loss. Table 3. Plot of the phase of voltage. The normalized impedance. The opposite It can be shown  that the bilinear fractional transformation 3.
Both of them are complex variables and in order to provide a graphical picture of the previous equations. An example of Smith chart. We have seen Eq. Using the standard curves. This property is clearly very useful when we have to analyze transmission line circuits containing series and parallel loads.
A more complex problem. Notice that the phase values in this equation must be expressed in radians. Regions of the Smith chart: Computation of impedances and admittances. The second scale. Appropriate scales are provided on the chart to simplify these operations.
In this way eq. The presence of the 0. Two scales drawn on the periphery of the chart simplify the evaluation of eq. In this way. Shunt connection of a lumped load Consider now the case of of a line with the lumped load Yp connected in shunt at A. Let us see how the analysis is carried out in such cases. As for the forward voltage. The very circuit scheme adopted implies that both the voltage and the current are continuous at point A: Notice that the picture uses the symbols of the transmission lines: Shunt connection of a lumped load on a transmission line.
It is interesting to note that the loads in the circuits above are lumped in the z direction but not necessarily in others. Zs AFigure 3. We will see examples of such circuits in Chapter 6 on impedance matching. Yp could be the input admittance of a distributed circuit positioned at right angle with respect to the main line.
Zs could be the input impedance of a distributed circuit. Transmission line length as a two-port device Two analyze more complex cases. See also Chapter 7 for a review of these matrices. Yp A Figure 3. Shunt connection of a distributed load on a transmission line. Note that, also in this case, the current I2 is assumed to be positive when it enters into the port.
This attenuation has two origins: In accordance with the circuit point of view, adopted in these notes, we limit ourselves to a qualitative discussion of the subject. A much more detailed treatment can be found in . The phenomenon of energy dissipation in insulators is the simplest to describe. Indeed, consider a metal wire of length L, and cross section S, for each point of which 4. We have seen in Chapter 1 that dielectric losses are accounted for in circuit form by means of the conductance per unit length G.
The formulas that allow the computation of G for some examples of lines are reported in Section 4. The complex dielectric permittivity can describe also a good conductor. This phenomenon has two consequences: Perfect conductor and surface current on it.
It can be shown that the current density per unit surface in the left conductor. Table 4. Planar transmission line. This corresponds to showing the frequency dependance. A case that lends itself to a simple analysis is that of a planar transmission line. Here we focus on the x dependance.
This behavior is analyzed in greater detail below. Plot of the current density Jz vs. The expression 4. The imaginary part of Z in 4. The expression of the conduction current density 4. Note the range on the vertical axis. If the conductor has width w. Figure 4. The normalization impedance is the surface resistance Rs in a and the dc resistance Rdc in b. Since wh is the conductor cross-section area. Normalized series impedance of the planar line.
We note that the normalized resistance becomes very 3 2. Solid line: Since in general d h. The frequency on the horizontal axis is normalized to the demarcation frequency fd.
As far as the series reactance is concerned. Note that the internal inductance is always small with respect to the external one. Real part of the series impedance per unit length. The same interpretation was already given in connection with Eq.
In such conditions. Note that the Chapter 5 Lossy transmission line circuits 5. Let us analyze now the properties of 5. As for the propagation constant. This choice is natural when transients are studied and the line equations are solved by the Laplace transform technique instead of the Fourier transform.
ABCD of a line length. In these notes we will always use the phase constant k. As for the characteristic admittance. We interpret 5. From the analysis of 5. Note that it is identical to the plot of Fig. If we express the voltage ratio in dB. It is in this direction. The same considerations can be carried out for the second term of 5.
The same conclusion can be reached by introducing the reference in which the backward wave is at rest. Lossy transmission line loaded with a generic impedance Hence. The presence in these expressions of an exponential that increases with z seems to contradict the dissipative character of the lossy line.
Space-time plots of the forward and backward voltage waves a Figure 5. Actually the generator power is only partially delivered to the load: This result has also an intuitive explanation. Length of lossy transmission line terminated with an arbitrary load impedance We have seen in Chapter 3 that when an ideal line is connected to a reactive load.
PB is also the power delivered to the load ZL. The answer is no. There is also a physical explanation: We can ask ourselves if also on a lossy transmission line.
G do not depend on frequency.
The amount of power dissipated in the line length AB is readily found by taking into account the energy conservation: In this section we analyze it. Considering the equations 5. As for the low frequency approximation of the characteristic admittance. In the intermediate frequency range no approximation is possible and the general expressions 5. The other plots are instead of semi-log type.
The imaginary part instead tends to zero in both regimes. If the spacing is much smaller than the wavelength. Figure 5. Actually there are two types of matching, one is matching to the line, the other is matching to the generator.
These two objectives can be reached by means of lossless impedance transformers, which can be realized either in lumped or distributed form.
As for the latter, several solutions will be described. Consider the circuit of Fig. We have already analyzed this circuit in Section 3. The power absorbed by the load can also be expressed in terms of the maximum voltage on the line. This remark is important in high power applications, since for every transmission line there is maximum voltage that must not be exceeded in order to avoid sparks that would destroy the line.
From 6. B Generator matching Suppose that in the circuit of Fig. Rewrite 6. In the rest of this chapter we will show how to design impedance transformers that allow the matching condition to be reached. The optimum operating condition for the circuit of Fig. It can be readily checked that it corresponds to a maximum. If the losses were not negligible. Scheme of impedance transformer. Zin is the complex conjugate of the generator internal impedance.
If the network contains more than two independent elements. In the case of conjugate matching. Zin is the charac- ZL Z ing Figure 6.
We have seen in the previous section that for several reasons it is useful to be able to design impedance transformers that perform as indicated in Fig. First we address the simplest case of single frequency matching. There are various solutions to this problem. In the case of matching to the line. In this case the condition to After some algebra.
Obviously the square root must be real: The susceptance B and the reactance X can be realized by lumped elements inductors and capacitors if the frequency is low enough. From eq. It is interesting to ascertain for which combinations of load and input impedance each form of the L circuit can be used. B e X can be realized with transmission line lengths. In this way we have solved the matching problem in the most general case. Assume It is interesting to note that the problem can also be solved graphically by means of the Smith chart.
For other values of desired input impedance. This means that with present day technology this matching technique can be used up to some GHz vedi Pozar p. This reactance is realized by another length of transmission line.
We see that matching is possible only with the circuit of type a for Zin inside the circle and only with the circuit of type b for Zin to the right of the vertical line. Suppose that a shunt stub matching network is to be designed. Using again the Smith chart. Matching network with shunt stub: The matching network is an impedance transformer: The data are: Realizability of L matching networks.
For Zin in the circle. Example 1 Design a line matching network. In fact. In this case the length of AB becomes 0. In this case the arrival point on the Smith chart is not the origin but a generic point. If the stub were to be connected in series to the main line. The procedure described above to design a line matching network can be generalized to solve the problem of designing a conjugate matching network.
A similar remark holds for the stub. The matching network structure is the same as before: Smith chart relative to the design of the matching network of Fig. The problem. In this case we would have employed an impedance Smith chart: The relevant Smith charts are shown in Fig. Let us make reference to a shunt stub. There is indeed a general rule: The length of the stub is found from the Smith chart of Fig. Example 2 Design a conjugate matching network with an open circuit shunt stub. Smith chart relative to the design of the stub for the matching network of Example 1 Rd equals the entire Smith chart.
Moreover the intersection of Rr e Rd is not empty. The length of AB is 0. In this case the solution are at most two. Because of their form. The points of the region Rd represent the input admittances of a matching network of the type of Fig.
The relevant Smith chart is shown in Fig. The reason for which only examples of shunt stubs have been discussed is that this type of connection is more common. In this case. This device can be useful in the laboratory: If we want to design such a matching network. If we start from the load.
The points belonging to the region Rr represent input admittances of a stub matching network loaded by yL. The detailed procedure is the following: The procedure for the design of the same matching network. This limitation is not present in the case of a triple stub matching network. From these nontrivial examples we can appreciate the power of the Smith cart as a design tool.
Smith chart relative to the design of the conjugate matching network reversed L discussed in Example 2. Design of a double stub matching network. The exercises develop this theme as a pivot point between the lecture in class, and the understanding that comes with applying the ideas of Calculus.
In addition, the table of contents has been refined to match the standard syllabus. Many of the examples have been trimmed of distractions and rewritten with a clear focus on the main ideas.
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