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Even if an electromagnetic eld can have an arbitrary time dependance, the time harmonic sinusoidal regime with frequency f is very important, both from a theoretical and from an application point of view.

It is known from Mathematics that a eld with arbitrary time dependence can be represented as a summation of sinusoidal elds with frequencies contained in a certain band Fourier theorem. In this case 0 denotes the minimum wavelength, i. The size L of the structures with which the electromagnetic eld interacts must always be compared with wavelength. The solution technique of electromagnetic problems and even their modeling is dierent depending on the regime of operation.

Lumped parameter circuit theory deals with the dynamics of systems made of elements of negligible electrical size. The state variables employed in the model are the potential dierence vrs t between two nodes Pr and Ps of a network and the electric current irs t that ows in the branch dened by the same two nodes.

Rigorously, these quantities are dened uniquely only in static conditions, i. This condition can be reformulated in terms of transit time. Hence, an electromagnetic system can be considered lumped provided the propagation delay is negligible with respect to the period of the oscillations.

For this reason one says that a lumped parameter circuit operates in quasi-static regime. Consider now one of the transmission lines shown in Fig.

Typically, their transverse size is small with respect to wavelength but their length can be very large.

Then, while a lumped parameter circuit is modeled as point like, a transmission line is a one dimensional system, in which voltage and currents depend on time and on a longitudinal coordinate that will always be indicated with z. The state variables of such a system are then v z,t and i z,t.

A circuit containing transmission lines is often called distributed parameter circuitto underline the fact that electromagnetic energy is not only stored in specic components, e.

Thomas' Calculus 11th Ed. Solution Manual

As a consequence, a transmission line is characterized by inductance and capacitance per unit length. The equations that determine the dynamics of a transmission line could be obtained directly from Maxwell equations, but for teaching convenience we will proceed in circuit terms, by generalizing the properties of lumped parameters networks.

In Fig. It is evident that all two conductor transmission lines have the same circuit symbol shown in Fig. As previously remarked, a transmission line can be long with respect to wavelength, hence its behavior cannot be predicted by Kirchho laws, that are applicable only to lumped parameter 7 1 Transmission line equations and their solution a b a Length of coaxial cable and b its symbolic representation circuits.

However we can subdivide the line in a large number of suciently short elements z , derive a lumped equivalent circuit for each of them and then analyze the resulting structure by the usual methods of circuit theory. This is actually the modeling technique used in some circuit simulators.

Table 3. Correspondence between values of return loss, magnitude of the reection coecient, VSWR and reection loss. We have seen in Fig. On the basis of these results, we nd that the circumference of Fig. It is clear that the circumference degenerates in the imaginary axis if 1, i.

With a little algebra it is possible to nd the expressions of the phase of voltage, current and normalized impedance: The phases of voltage and current are decreasing functions for increasing z, that tend to resemble a staircase when 0 1, i.

The normalized impedance, as already shown by 3. The Smith Chart is a graphical tool of great importance for the solution of transmission line problems. Nowadays, since computers are widespread, its usefulness is no longer that of providing the numerical solution of a problem, but that of helping to set up a geometrical picture of the phenomena taking place on a transmission line.

Mathematically, the Smith chart consists of a portion of the complex V plane, on which suitable coordinate curves are displayed. In particular, it is based on the two relations, shown in Section 3. Both of them are complex variables and in order to provide a graphical picture of the previous equations, we can draw two sets of curves in the complex V plane: In this way the transformation V and its inverse are geometrically straightforward.

It can be shown [2] that the bilinear fractional transformation 3. For this to be true without exceptions we must regard straight lines as degenerate circumferences of innite radius.

An example of Smith chart, equipped with all the necessary scales, is shown in Fig. Because of the form of the evolution law of the reection coecient on a line, the complex number V is always given in polar form, i.

The Smith chart is equipped with scales to measure magnitude and phase of V. We have seen Eq. Hence, the Smith chart can be considered equally well as: The complex V I. The complex plane, on which constant conductance and constant susceptance curves are drawn. Indeed, if we know the normalized impedance A , we can place it on the chart by viewing the set of lines as constant resistance and constant reactance circles: The opposite.

Constant reactance lines in the plane and their image in the to the region inside the unit circle. This property is clearly very useful when we have to analyze transmission line circuits containing series and parallel loads.

A more complex problem, which is solved with the same simplicity is the following. Suppose we must nd in the V A plane the set of impedances with conductance greater than one. Using the standard curves, labeled now with resistance and reactance values, to read the coordinates of the points, solves the problem.

Now let us see how the use of the Smith chart simplies the analysis of the circuit of Fig. From the load impedance ZL and the line characteristic impedance Z , compute the normalized impedance B at point B.

Place B on the Smith chart, so that V B is determined. The reection coecient at point A is given by V. Regions of the Smith chart: Notice that the phase values in this equation must be expressed in radians.

From a graphical point of view, it is straightforward to draw the plots of magnitude and phase of voltage and current, taking into account that it is just necessary to study the behavior of 1 V z , as explained. This quantity is called Transmission coecient for a reason explained in the next section. Appropriate scales are provided on the chart to simplify these operations.

In particular, the magnitude of 1 V z , in the range [0,2], is to be read on the scale with the label Transmission coecient E or I. Two scales drawn on the periphery of the chart simplify the evaluation of eq. The outer one has the label Wavelengths toward generator and displays the quantity l TG eqB def. A second one, concentric with the rst, is labeled Wavelengths toward load and displays the quantity l TL eqB def. The presence of the 0.

Moreover, the rst is a clockwise scale, the second a counterclockwise one. In this way eq. The rotation sense on the chart is clockwise, as specied by the sign of the exponent in 3. Note that the generator in the label has nothing to do with the one present in the circuit, but is the driving point impedance generator that one imagines to connect in the point of interest of a circuit to dene the relevant impedance.

The second scale, wavelengths toward load, has values that increase counterclockwise and is useful when the input impedance is known and the load impedance value is desired: Sometimes two transmission lines with dierent characteristics are cascaded or lumped loads are connected in series or in shunt with the transmission line. Let us see how the analysis is carried out in such cases. Cascade connection of transmission lines Consider rst the cascade connection of two lines with dierent characteristic impedance.

Notice that the picture uses the symbols of the transmission lines:. The very circuit scheme adopted implies that both the voltage and the current are continuous at point A: Shunt connection of a lumped load Consider now the case of of a line with the lumped load Yp connected in shunt at A.

Apply Kirchho laws at the node A:. Ip Yp A Figure 3. Shunt connection of a lumped load on a transmission line. Series connection of a lumped load Consider now the case of a lumped load Zs connected in series on a transmission line at A.

Kirchho law at node A yield. Series connection of a lumped load on a transmission line. To nd the link between forward and backward waves, it is convenient to work on the current, which is continuous: Note that in these cases the use of Kirchho laws is completely justied, since they have been applied to lumped elements. It is interesting to note that the loads in the circuits above are lumped in the z direction but not necessarily in others.

In other words, Zs could be the input impedance of a distributed circuit, positioned at right angle with respect to the main line, as shown in Fig. Likewise, Yp could be the input admittance of a distributed circuit positioned at right angle with respect to the main line, as in Fig. We will see examples of such circuits in Chapter 6 on impedance matching. Transmission line length as a two-port device Two analyze more complex cases, it may be convenient to represent a transmission line length as a two-port device, characterized via its matrices Z, Y , or ABCD, and then apply the usual lumped circuit theory.

We derive now the expression of these matrices for a length l of transmission line with characteristic impedance Z and propagation constant k.

Table of Contents

See also Chapter 7 for a review of these matrices. Note that, also in this case, the current I2 is assumed to be positive when it enters into the port. Note that, dierently from before, the current I2 is assumed to be positive when it goes out of the port. This is the reason of the minus signs in the dening equations. Also useful are the T and equivalent circuits, shown in Figure 3.

Energy dissipation in transmission lines Wave propagation in real world transmission lines is always aected by attenuation. This attenuation has two origins: The detailed study of these phenomena requires the solution of Maxwells equations in the structures of interest. 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 [3]. The phenomenon of energy dissipation in insulators is the simplest to describe.

In every real dielectric there are electrons that are not strictly bound to atoms and are set in motion by an applied electric eld: It is useful to note that this equation is the microscopic form of Ohms law. Indeed, consider a metal wire of length L, and cross section S, for each point of which 4. Usually the conduction current, which is in phase with the applied electric eld because d in 4. In very straightforward way we have introduced a complex equivalent permittivity , whose real part is the usual dielectric constant and whose imaginary part is related to the conductivity.

Thomas Calculus 11th [Textbook + Solutions]

Generally, the loss angle , is introduced: Observe that if the frequency behavior of is of interest, we must keep in mind that both r , and d are functions of frequency. Finally it is to be noted that the symbol has been introduced only for clarity. Indeed is always understood to be complex unless specic indications are given. 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 computation of this quantity, as well as that of all line parameters, starting from the geometry and the physical parameters of the materials, requires the solution of Maxwells equations for the structure under consideration. From the knowledge of the elds it is possible to derive the values of the line parameters. This procedure will be briey illustrated in the next section, where conductor losses are analyzed.

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. Actually, in conductors such as copper, for frequencies up to the millimeter wave range, the displacement current is negligible with respect to the conduction current, so that is assumed to be pure imaginary. In a transmission line in which the conductors can be assumed perfect, the electromagnetic eld is dierent from zero only in the insulators.

In these conditions, on the very surface of the conductors there is an electric current strictly related to the electromagnetic eld. It is a surface current, whose density per unit length J , measured along the boundary of the conductor cross-section, has a magnitude equal to that of the tangential magnetic eld in the points of the dielectric facing the conductor, see Fig.

Its direction is orthogonal to that of the magnetic eld. If now we imagine that the metal conductivity is very large but nite, it can be shown that the current is no longer conned to the conductor surface but is distributed also inside the metal, with a density per unit surface of the cross-section that decays exponentially toward the inside of it. Also the magnetic eld penetrates the metal, with a similar exponential decay. This phenomenon has two consequences: Figure 4.

Perfect conductor and surface current on it. Its density J is the current that ows through the line element ds. A case that lends itself to a simple analysis is that of a planar transmission line, shown in Fig. Here we focus on the x dependance, since we want to obtain the line parameters per unit length.

In the right conductor the current ows in the opposite direction. This corresponds to showing the frequency dependance, since the skin depth can be shown to be related to frequency by 2 4. In these conditions, the current ows essentially in a thin lm, adjacent to the interface between the metal and the insulator, which justies the name of the phenomenon. This behavior is analyzed in greater detail below. The skin depth is inversely proportional to the square root of frequency and of metal conductivity.

Table 4. Plot of the current density Jz vs. This surface resistance depends on frequency and is measured in. Finally, the factor 2 in Eq. It is to be noted that this impedance per unit length coincides with the series impedance of the equivalent circuit of an element z of transmission line see Fig.

The imaginary part of Z in 4. Note the range on the vertical axis, which is much smaller than in the left gure. The expression of the conduction current density 4. The expression 4. In these conditions the current ows with almost uniform density in the whole conductor cross-section see Fig. Since wh is the conductor cross-section area, this result coincides, as is to be expected, with the direct current resistance Rdc.

On the contrary, at high frequency, the resistance per unit length of each conductor is given by 4. We note that the normalized resistance becomes very. Normalized series impedance of the planar line. Solid line: The normalization impedance is the surface resistance Rs in a and the dc resistance Rdc in b.

Actually, the absolute resistance tends to the nite value Rdc as it is evident from. The frequency on the horizontal axis is normalized to the demarcation frequency fd. As far as the series reactance is concerned, Eq. Since is a function of frequency, this condition determines the demarcation frequency fd that separates the low and high frequency regimes. Since depends on , this equivalent inductance depends on frequency. Note that the internal inductance is always small with respect to the external one.

Indeed, the external inductance is given by d 4. Since in general d h, the internal inductance is negligible with respect to the external one. Real part of the series impedance per unit length, normalized to Rdc. The asymptotic behaviors are also plotted and dene the demarcation frequency. Metal losses Low frequency Resistance per unit length: This formula has a simple interpretation. When the skin eect is well developed, the series impedance is the same as the one we would have if the whole current ew with uniform density in a layer one skin depth thick.

The same interpretation was already given in connection with Eq. We said that the surface impedance is the same we would have for a uniform current ow in a layer of thickness.

Hence, every square, with arbitrary side, has the same resistance. After explaining in detail the analysis technique of circuits containing ideal transmission lines, i. We could now repeat step by step the analysis carried out for the ideal lines, but it is simpler to resort to the trick of introducing a complex inductance and capacitance per unit length Lc Cc in such a way that Eq.

It is just enough to take the solution of the ideal case and obtain its analytic continuation from the real values L e C to the complex ones Lc and Cc. Note that the. Obviously the time constants s , p go to innity for an ideal transmission line. Observe that even if the solution 5. Let us analyze now the properties of 5. As for the propagation constant, the quantity below the square root sign in 5.

Hence, by continuity, in the lossy case k belongs to the fourth quadrant. As for the characteristic admittance, the radicand in 5.

Moreover, we recall that Y is the input admittance of a semi-innite line: This choice is natural when transients are studied and the line equations are solved by the Laplace transform technique instead of the Fourier transform. In these notes we will always use the phase constant k. To understand better the meaning of the solution of lossy transmission line equations, we compute the time evolution of voltage and current relative to the rst term of 5.

We interpret 5. Note that it is identical to the plot of Fig. From the analysis of 5. The rst term of 5. The wave amplitude has an exponential decrease vs. The current is proportional to the voltage, but shows the phase shift arg Y with respect to it.

Note also that Y depends on frequency. The same considerations can be carried out for the second term of 5. The plot of the backward voltage wave vs. The presence in these expressions of an exponential that increases with z seems to contradict the dissipative character of the lossy line. Actually, in Fig. It is in this direction, in which the natural evolution of the phenomenon takes place, z that the amplitude of the backward wave reduces.

In Chapter 3 we have seen that in the analysis of circuits containing transmission lines, it is useful to introduce the notion of reection coecient: We can observe that if we move from the load toward the line input, both the phase and the amplitude of the reection coecient decrease, so that V traces a logarithmic spiral in the complex plane, with the origin as pole, as shown in Fig.

For this reason, we can say that the input impedance of a semi-innite real lossy line coincides with its characteristic impedance. This fact justies the use of the symbol Z. This result has also an intuitive explanation.

Indeed, the fact that the input impedance of a line is dierent from Z means that in A, apart from the forward wave, originally produced by the generator, there is also an appreciable contribution of the backward wave, created in B by the load mismatch.

If the product of the attenuation constant times the line length is very large 0 l , the backward wave in A is negligible and the line appears to be matched. Actually the generator power is only partially delivered to the load: We have seen in Chapter 3 that when an ideal line is connected to a reactive load, a purely stationary wave is established on it and the net power ux is zero.

We can ask ourselves if also on a lossy transmission line, connected to a reactive load, a purely stationary wave can be formed. The answer is no, because it is algebraically impossible to write the voltage on the line as the product of a function of t times a function of z.

There is also a physical explanation: This power, obviously, is dissipated in the line length comprised between the point under consideration and the load. The general formula that allows the computation of the power ow in each point of any transmission line has been derived in Chapter 3 and is reported her for sake of convenience: The power ow in this case can be computed by the formula that, rigorously, holds only in the case of ideal lossless lines: Apply now this formula to the circuit of Fig.

The amount of power dissipated in the line length AB is readily found by taking into account the energy conservation: The Smith chart is provided with a scale that allows the fast evaluation the attenuation increase due to the line mismatch.

The expressions 5. In this section we analyze it, by assuming that the primary constants L, C, R, G do not depend on frequency. This amounts to neglecting the dielectric dispersion and the frequency dependence of the skin eect see Chapter 4.

Considering the equations 5. To obtain the expressions for the low frequency range it is convenient to rst rewrite 5.

Note that is linear at both high and low frequency, but the slope of the two straight lines is dierent. As for the low frequency approximation of the characteristic admittance, from Eq. We notice that both at low and high frequency the real part of the characteristic admittance is essentially frequency independent, but the two constant values are dierent.

The imaginary part instead tends to zero in both regimes. In the intermediate frequency range no approximation is possible and the general expressions 5. The plot of is of log-log type, so that the dierent slopes in the two frequency ranges is represented as a vertical translation.

The other plots are instead of semi-log type. We note that the imaginary part of Y is maximum when the real part has the maximum slope. This is a general property Kramers Krnig o relations , related only to the fact that Z can be considered to be a transfer function, that is the Fourier transform of the impulse response, which is a causal time function.

Indeed, from Eq. Since in practice the line parameters do not fulll this condition, one can load the line with periodically spaced series inductors. Figure 5. Plots of , , G , B for a realistic transmission line. The values of the primary constants are specied in the text. In this chapter we address a subject with great practical importance in the eld of distributed parameter circuits, i. Actually there are two types of matching, one is matching to the line, the other is matching to the generator.

When a transmission line must be connected to a load with an impedance dierent from the characteristic impedance of the line, it is necessary to introduce a matching device, capable of eliminating the presence of reected waves on the line. The other type of matching, not specic of distributed parameter circuits, has the property of allowing a generator to deliver its available power.

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. Circuit consisting of a lossless transmission line, connected to a generator and a load. The power absorbed by the load can also be expressed in terms of the maximum voltage on the line. Depending on the values of the internal impedance of the generator ZG and of the load impedance, two dierent cases can be considered: Observe that, xing the active power delivered to the load PB , the maximum line voltage Vmax has the minimum value when the load is matched to the line.

Alternatively, we can say that xing the maximum voltage on the line, the power delivered to the load is maximum when the load is matched.

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. We can ask what is the optimum value of Zin that allows the maximum power to be extracted from the generator. Rewrite 6. It can be readily checked that it corresponds to a maximum. Note that when Zg is real, the Kurokawa reection coecient is coincident with the ordinary one, whereas it is a dierent concept when Zg is complex.

If we now set V. The center is in the point with coordinates , , lying on the segment joining the point coordinates a, b , to the origin. In other words, energy matching and line matching are independent. The optimum operating condition for the circuit of Fig. Indeed, in these conditions the generator delivers the maximum power. Moreover, because of the line matching, the voltage on the line is the minimum for that value of active power ow.

If the losses were not negligible, the line attenuation would be the minimum one and would be coincident with nominal one. Finally, as it will be discussed in Chapter 8, the line matching condition is essential to minimize distortions. In the rest of this chapter we will show how to design impedance transformers that allow the matching condition to be reached.

First of all, we observe that a matching network must be formed by at least two components, since two conditions must be enforced, one on the real and one on the imaginary part of the input impedance.

If the network contains more than two independent elements, multiple matching conditions can be enforced, i. First we address the simplest case of single frequency matching. 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. In the case of matching to the line, Zin is the charac-. In the case of conjugate matching, Zin is the complex conjugate of the generator internal impedance.

There are various solutions to this problem, all consisting of ideally lossless networks. Two structures are possible see Fig. Consider the rst conguration. The corresponding X values are found from the second of 6. Obviously the square root must be real: In this case the condition to.

Also in this case, the radicand of the square root must be positive. It is interesting to ascertain for which combinations of load and input impedance each form of the L circuit can be used. Suppose that ZL is specied.

From eq. Next, consider the circuit of Fig. 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. For other values of desired input impedance, both circuits can be used. In this way we have solved the matching problem in the most general case. It is interesting to note that the problem can also be solved graphically by means of the Smith chart. The susceptance B and the reactance X can be realized by lumped elements inductors and capacitors if the frequency is low enough.

This means that with present day technology this matching technique can be used up to some GHz vedi Pozar p. Alternatively, for frequencies in the microwave range, B e X can be realized with transmission line lengths, terminated in short circuit or open circuit which, as discussed in section 3. The matching networks of this type consist essentially of a a transmission line length and a reactance, that can be connected in shunt or in series to the line itself.

This reactance is realized by another length of transmission line, terminated with an open or short circuit, called stub. Suppose that a shunt stub matching network is to be designed, to match a load to the feeding transmission line, Fig. Figure 6. Realizability of L matching networks. For Zin in the circle, only network a can be used, for Zin to the right of the vertical line, only network b.

The matching network is an impedance transformer: Using again the Smith chart, the stub length is readily found as soon as its termination short or open circuit has been chosen. Example 1 Design a line matching network, having a shunt short circuited stub. The data are: From the intersection of the constant I circle eqB. The length of the stub is found from the Smith chart of Fig. In this case the length of AB becomes 0.

The relevant Smith charts are shown in Fig. The input impedance of the matching network would still be Z at the design frequency, but its bandwidth would be smaller. There is indeed a general rule: A similar remark holds for the stub. If the stub were to be connected in series to the main line, the design procedure would be only slightly modied. In this case we would have employed an impedance Smith chart: The procedure described above to design a line matching network can be generalized to solve the problem of designing a conjugate matching network.

In this case the arrival point on the Smith chart is not the origin but a generic point, corresponding to the complex conjugate of the generator internal impedance. Let us make reference to a shunt stub. The matching network structure is the same as before: The problem, now, is that the rst step is not always successful. Indeed, it is evident from Fig.

Incidentally, it is simple to recognize that whatever the value of yL , the origin belongs always to Rd , so that the line matching is always possible.

When the admittance yg to be reached lies outside of the region Rd , we can still use a stub matching network, provided the structure is reversed, as shown in Fig. Indeed, in this case, the values of yA that can be obtained from yL are those belonging to the region Rr see Fig. We see that the union of Rr and. Rd equals the entire Smith chart, hence every matching problem can be solved by a stub network either straight or reversed L.

Moreover the intersection of Rr e Rd is not empty, so that the solution for certain values of yA can be obtained with both types of networks.

Example 2 Design a conjugate matching network with an open circuit shunt stub. The points of the region Rd represent the input admittances of a matching network of the type of Fig.

Let us try a reversed L conguration. The length of AB is 0.

The relevant Smith chart is shown in Fig. We see easily that the constant I circle through yL and the constant conductance circle through yg have no intersections, hence only the reversed L conguration is possible. The reason for which only examples of shunt stubs have been discussed is that this type of connection is more common, because it is easier to realize, for example by the microstrip technology.

Even if the straight or reversed L matching networks can solve any practical problem, sometimes double stub networks are used. Because of their form, shown in Fig. It is clear that the stub susceptances and their separation can be chosen in an innite number of dierent ways.

Sometimes the distance AB is xed a priori.

Calculus 11th Edition by George Thomas

In this case the solution are at most two, but are not guaranteed to exist. The design can be carried out in two dierent ways,. The points belonging to the region Rr represent input admittances of a stub matching network loaded by yL. If we start from the load, see Fig.

The procedure for the design of the same matching network, but starting form the generator see Fig. Obviously, even if the diagrams on the Smith chart are dierent in the two cases, the values of b1 e b2 turn out to be the same. If the distance between the stubs is xed a priori, the solution for certain load and input admittances is not guaranteed to exist. This limitation is not present in the case of a triple stub matching network, even if the relative distances are xed a priori see Fig.

In this case, in fact, the desired matching can always be obtained, provided convenient stub lengths are selected. This device can be useful in the laboratory: If we want to design such a matching network, we can use the method described above. The detailed procedure is the following: Smith chart relative to the design of the conjugate matching network reversed L discussed in Example 2.

From these nontrivial examples we can appreciate the power of the Smith cart as a design tool. The wavelength is to be evaluated at the design frequency. The scheme is shown in Fig. The normalized input impedance is the inverse of the normalized load impedance see Eq. In conclusion, the characteristic impedance of the line must be the geometric mean of the two resistances to be matched. Their purpose is that of transforming the complex impedances into pure resistances, as.

The Scattering matrix In this Chapter we develop a convenient formalism to describe distributed parameter circuits containing multiport devices.

First we review the matrix characterization of multiport devices based on the use of total voltage and total current as state variables. This description is appropriate to the case of lumped networks. As discussed at length in the previous Chapters, in the case of distributed parameter circuits a change of basis is highly convenient: The simplest two-lead circuit element is characterized by its impedance ZL , or its inverse, i.

Suppose this element is linear, so that the impedance does not depend on the excitation I but only on frequency. Often a couple of leads of a device is called a port: As known in circuit theory, these concepts can be generalized to the case of devices with several ports, say N. We start this presentation by focusing our attention to the important case of two-port devices see Fig. In general, the two voltages V1 and V2 depend on both I1 and I2: The relation 7. Pearson offers special pricing when you package your text with other student resources.

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Thomas' Calculus, 11th Edition. George B. Thomas, Jr.

Giordano, Naval Postgraduate School. Availability This title is out of print. Description The new edition of Thomas is a return to what Thomas has always been: Series This product is part of the following series. MyMathLab Series. Carefully developed exercises — the benchmark by which all other books are measured.

Applications to the physical world — a Thomas trademark. Transparent integration of technology. Complete and careful multivariable calculus section.Read More. Lumped parameter circuit theory deals with the dynamics of systems made of elements of negligible electrical size. Note that for simplicity we have assumed that both the generator internal impedance and the load impedance are pure resistances, hence frequency independent. We see easily that the constant I circle through yL and the constant conductance circle through yg have no intersections, hence only the reversed L conguration is possible.

With a suitable choice of the reference planes, its S matrix can be written: