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Thomas' calculus 11th ed [solution]. 1. CHAPTER 1 PRELIMINARIES REAL NUMBERS AND THE REAL LINE 1. Executing long division. Download Thomas Calculus 11th Edition Solution Manual PDF eBook free. The “ Thomas' Calculus, 11th Edition Solution Manual” added. Thomas Calculus 11th [Textbook + Solutions] - Download as PDF File .pdf), Text File Contents 1 Transmission line equations and their solution Calculus by Thomas Finney 10th Edition Solution Manual Part I.

Contents Contents 1 Transmission line equations and their solution 1. Electromagnetism background. Circuit model of a transmission line. Lossless lines. Wave equations and their solutions. Review of Fourier transforms and phasors. Transmission line equations in the frequency domain.

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 [3].

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 wavelength is to be evaluated at the design frequency. Their purpose is that of transforming the complex impedances into pure resistances. The scheme is shown in Fig. In the case the two impedances to be matched are complex. The normalized input impedance is the inverse of the normalized load impedance see Eq.

RL By enforcing the condition that the input impedance ZA coincides with the desired input resistance Ri. Triple stub matching network. As discussed at length in the previous Chapters. I1 V1 I2 V2 Figure 7. This description is appropriate to the case of lumped networks. First we review the matrix characterization of multiport devices based on the use of total voltage and total current as state variables.

As known in circuit theory.

Chapter 7 The Scattering matrix In this Chapter we develop a convenient formalism to describe distributed parameter circuits containing multiport devices.

We start this presentation by focusing our attention to the important case of two-port devices see Fig. Suppose this element is linear.

The relation 7. The advantage of the matrix notation is that?? The diagonal elements are input impedances. It can be shown [4] that the matrices [Z] and [Y ] of reciprocal devices are symmetrical.

In fact it relates the electric state at the input to that at the output of the device. It can be shown that the ABCD matrix of a reciprocal device has unit determinant. Recall that a circuit made of resistors. The total power dissipated in the device is the sum of the powers entering through the various ports: I1 I2 that is. In this Chapter we are going to generalize this concept to N -port networks. Introduce Let us generalize these concepts to the case of a device with N ports.

If the two two-port devices are connected in parallel. To explain this name. To each port we assign a reference impedance Zri that can be interpreted as the characteristic impedance of a transmission line connected to the port.

Note also that the line characteristic impedance plays the role of a reference impedance. In the case of a two-port device.

In this way the access line is matched and only an outgoing wave is present on it. There are several reasons. There is also a noteworthy relation between the [S] matrix and the open circuit impedance matrix [Z]. Even if the characterization of a device by means of its scattering matrix [S] is. Let us start from the device characterization in terms of [Z] matrix: In general a wide band characterization of the devices is of interest and.

From 7.

Substitute into 7. This follows immediately from 7. Let us address the problem gradually. The proof can be found in [5]. Suppose that we want to compute the scattering matrix [Sn ] of the same device.

We want now to examine the transformation of the [S] matrix induced by a change of these planes. The complex amplitudes of these waves are function of the longitudinal coordinate. Consider a N -port device with scattering matrix S0 see Fig.

If [S0 ] is the scattering matrix of the original structure. Notice that these matrices are diagonal. Introduce now the column vectors [a]. N are indicated by a0i and b0i. Very often it is convenient to view a complex system as made out of interconnected simpler blocks. Assuming that their scattering matrices are known, the problem arises of computing the scattering matrix of the complete structure.

With reference to Fig. This operation implies a partition in blocks of the scattering matrices of the two structures. Implicit in this partition is the assumption that the ports to be connected are the last K ones. This condition can always be obtained by suitable exchanges of rows and columns. Suppose also that the ports to be connected have the same reference impedances. In order to determine the resulting [S] matrix, it is necessary to eliminate the variables [a ], [a ], [b ], [b ] from 7.

The steps to be performed are the following: Substitute 7. The second of 7. Substituting 7.

The expression to be rewritten is: This matrix is singular. Now substitute in the second of 7. Sostitute 7. For reference sake. If the constituents structures are passive. Another particular case is the one in which all the ports of the second structure are connected. The computation of the comprehensive scattering matrix requires the inversion of a matrix with dimension equal to the number of ports K that are connected.

An isolator contains a magnetic material 1 2 Figure 7. In the drawings we will use the common convention of using one line as the symbol of a port. Its scattering matrix is then: The phase shift is related to the physical size of the device.

Symbol of an ideal isolator. Its S matrix is then: Note that if port 3 is terminated with a matched load. The S matrix of the device is: It can be shown that a matched. The power incident on port 2 is completely dissipated in the device. Symbol of an ideal three port circulator. TX 1 3 2 RX Figure 7. S31 of practical devices is not small enough.

Circulator used as a diplexer. With a suitable choice of the reference planes. Symbol of directional coupler. The isolation I is related to the same concept. The concept of the measurement is illustrated in Fig.

Other parameters used to characterize a real directional coupler are the directivity D: The directivity measures the ability of the directional coupler to discriminate the incident waves at port 1 from those incident at port 2. From the ratio in amplitude and phase of the signals coming out of ports 3 and 4.

A directional coupler is the heart of the Network Analyzer. The coupling C denotes the fraction of the incident power at port 1 that is transferred to port 3. The equations of the structure are: To this end. Suppose that the S matrices of the two two-ports do not depend on frequency. The S matrix of the comprehensive structure can be obtained in two steps: In practice this is not strictly true.

Change of reference impedance. We obtain the result by passing through the impedance: We can make reference to Fig. The transmission matrix relates the electrical states at the two sides of the structure in the power wave basis. Referring to Fig. By means of 7. The second discontinuity is a one-port load. The characterization of the device by means of the transmission matrix [T ] is then the following: When two structures of the type of Fig.

Generalized two-port structure: To derive the relation between the [T ] and [S] matrices of the same device. Cascade connection of two generalized two-ports. For completeness we list also the inverse relations of 7. From the linear algebra point of view. N ports K ports N ports Figure 7. An example is the structure of Fig. Chapter 8 Time domain analysis of transmission lines 8.

In this chapter we consider instead signals with arbitrary time dependence. We want to compute the load voltage vB t that is produced by a generator with open circuit voltage e t. This problem can be conveniently described. It is well known that for linear time invariant systems LTI.

Also the load impedance ZL and the internal impedance of the generator are generic complex functions of frequency. Take the case of a simple circuit. By this technique. The generator waveform e t is the system input. The system is: This property explains the usefulness of the Fourier transform technique in the analysis of LTI systems. These functions are plotted in Fig.

Note that the envelope is slowly varying in comparison with the carrier of Fig. Example of an amplitude modulated signal. Hence the signal e t is indeed quasi-monochromatic. Observe that vB t is computed from 8.

This signal. Figure Also the spectrum of m t is shown. Spectrum of e t. Recalling 8. The higher order terms in the expansion 8. In general terms. It turns out to be always smaller than the speed of light in vacuum. The concepts of phase and group velocity lend themselves to a geometrical interpretation.

As a consequence of the constructive and destructive interference phenomena that here take place.

Consider a dispersion curve as the one sketched in Fig. We see also that. In this section we will discuss the distortions caused by a transfer function with constant magnitude and a phase curve that is non linear but can be approximated by a parabola. The simplest case for which an analytic expression can be obtained is that of a gaussian pulse. This assumption implies that the group velocity and the group delay are linear functions of frequency. Geometrical interpretation of phase and group velocity 8.

Rewrite 8. In normal applications the pulse duration is much larger of the carrier period. For the same reason. The integrand in 8. Distorted gaussian pulse at the line end.

Note that ld increases with the initial pulse duration T0. Even if the pulse retains the gaussian shape. Note that the transmission line is a symmetric device. Moreover it increases if the line dispersivity becomes smaller. If T0 is small. The general characteristics of the plot are easily explained. Received pulses are so distorted that the transmitted word is no longer recognizable The considerations made for the gaussian pulse hold qualitatively for any pulse waveform.

To quantify the phenomenon. Let BT be the bit rate. At the other extreme. T0 Plot of the maximum bit rate BT max on a digital link versus the pulse duration Note that T0opt does not depend on the receiver characteristics. Various technological solutions have been devised to unify at the same wavelength the properties of low losses and low dispersion. Why buy extra books when you can get all the homework help you need in one place?

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Textbook Solutions. Looking for the textbook? We have solutions for your book! Step-by-step solution:. JavaScript Not Detected. It is given that. Comment 0. Taking three numbers a , b , and c We know that So. Now, for number of variables, Thus, it is proved. View a full sample. Thomas' Calculus Early Transcendentals 11th Edition.

Maurice D. Weir , George B.