Microwaves

E. U. Condon*)

Chapter II. Transmission Lines

In the region of low frequencies, energy is transmitted from one point in space to another (for example, from a generator to an antenna) by means of a transmission line, usually consisting of two conductors—for example, two parallel wires or a coaxial cable. Such transmission lines also play an important role in microwave radio. However, alongside these transmission lines, in microwave practice hollow tubes, called waveguides, are also used for the transmission of energy.

The theory of waveguides will be the subject of the next, third chapter of the present review; here we shall confine ourselves to transmission lines made of two conductors.

11. Transmission Lines Made of Two Conductors

The most general type of transmission line of this kind is the coaxial cable, consisting of two circular conductors of inner radius \(r=a\) and outer radius \(r=b\). The theory of such a cable is closely connected with the theory of the coaxial hollow resonator considered in § 5. Let us direct the \(z\)-axis along the transmission line and suppose that the cross-section in an arbitrary plane \(z=\mathrm{const}\) is bounded by two curves: the inner \(C_1\) and the outer \(C_2\) (cf. Fig. 4). We shall seek solutions of equation (5.1) in which the dependence on the coordinate \(z\) has the form \(e^{-ik_3 z}\), i.e., represents traveling waves propagating in the positive direction of the \(z\)-axis. Their phase velocity is

\[ u_p=\frac{\omega}{k_3}. \tag{11.1} \]

If we denote by \(k\) the ratio \(\omega/C\) and put \(E_z=H_z=0\), then equations (5.1) lead to the following relations, describing—

*) Continuation; see UFN, vol. 27, p. 213, 1945. Translated by V. G. Levich. The editors ask readers to correct misprints that crept into the first part of Condon’s article. A list of these misprints is given in the present issue, p. 258.

dependence of the field components on the coordinates \(x\) and \(y\):

\[ \begin{gathered} kE_x=-k_3H_y,\qquad kE_y=-k_3H_x,\\ 0=\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y},\\ -kH_x=k_3E_y,\qquad -kH_y=-k_3E_x,\\ 0=\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}. \end{gathered} \]

The last \(z\)-components of these equations show that the vectors \(\mathbf E\) and \(\mathbf H\) can be represented through the gradient of a certain scalar function \(u(x,y)\). Analogously to how this was done in § 5, let us put:

\[ \left. \begin{aligned} \mathbf E_s&=-\operatorname{grad} u(x,y),\\ -k\mathbf H_s&=[k_3\mathbf k,\operatorname{grad} u(x,y)]. \end{aligned} \right\} \tag{11.2} \]

The last relations show that \(k_3=\pm k\), so that the phase velocity of the waves in the cable is equal to \(\pm C\). The \(Z\)-component of the equation for \(\operatorname{rot}\mathbf H\) requires that the function \(u(x,y)\) satisfy Laplace’s equation:

\[ \Delta u(x,y)=0. \tag{11.3} \]

Since the boundary conditions satisfied by the function \(u(x,y)\) are the same as for the analogous function in the theory of the resonator, namely \(u=\mathrm{const}\) on the curves \(C_1\) and \(C_2\), all the results obtained in § 5 for a hollow resonator with double walls can be transferred to a coaxial cable.

Let \(u(x,y)\) be such a solution of the boundary-value problem that the coordinates \(x\) and \(y\) are periodic functions with period \(2\pi\) of the conjugate to \(u\) function \(v(x,y)\), as was the case in (5.26). Let, further, the values of \(u\) on the inner and outer conductors be respectively \(u_1\) and \(u_2\). Then for waves propagating in the positive direction of the \(z\) axis we have:

\[ \left. \begin{aligned} \mathbf E_s&=-\operatorname{grad} u(x,y)\cos(\omega t-kz),\\ \mathbf H_s&=[-\mathbf k,\operatorname{grad} u(x,y)\cos(\omega t-kz)], \end{aligned} \right\} \tag{11.4} \]

and for waves propagating in the opposite direction:

\[ \left. \begin{aligned} \mathbf E_s&=-\operatorname{grad} u(x,y)\cos(\omega t+kz),\\ \mathbf H_s&=[+\mathbf k,\operatorname{grad} u(x,y)\cos(\omega t+kz)]. \end{aligned} \right\} \tag{11.5} \]

Characteristic impedance. As already indicated in § 5, solutions in which the coordinates \((x,y)\) are periodic functions of \(v\) with period \(2\pi\) correspond to current amplitudes on the inner and outer conductors equal to \(1/2\) abs. A and ampli-

where \((u_2-u_1)\) statvolts is the line integral of the field strength \(\mathbf{E}\), taken from one conductor to the other along a curve lying in the plane \(z=\mathrm{const}\). Consequently, the ratio of the amplitudes of the potential difference and the current is

\[ Z=60(u_2-u_1)\ \text{ohms}. \tag{11.6} \]

This quantity is called the characteristic, or wave, impedance of the transmission line.

The wave impedance of a transmission line, defined in this way, coincides with the impedance of a hollow resonator with double walls found in § 5.

The wave impedance of a transmission line can also be determined in a somewhat different way, which makes clearer the physical meaning of this quantity. Let \(C_1\) denote the capacitance per unit length of the capacitor formed by the two conductors of the transmission line. Suppose that the potential of one of them is equal to zero and that of the other to \(V\) statvolts. Then the charge per unit length is \(C_1V\) esu. If the transmission line is fed in such a way that waves are sent into it, propagating from left to right with velocity \(c\), then at its input end a current must be excited which maintains the proper value of the charge on both conductors. The corresponding current is equal to \(cC_1V\) absolute units, or \(C_1V\) absolute amperes. Consequently, if the input resistance of the transmission line is defined as the ratio of the potential difference to the current, we have

\[ Z=\frac{300V}{10C_1V}=\frac{30}{C_1}\ \text{ohms}. \tag{11.7} \]

If the space between the two conductors is filled with a medium having dielectric constant \(\varepsilon\) and magnetic permeability \(\mu\), then it is easy to see that the input resistance of the line will be equal to

\[ Z=\left(\frac{\mu}{\varepsilon}\right)^{1/2}\frac{30}{C_1}\ \text{ohms}, \tag{11.8} \]

where \(C_1\) is the capacitance per unit length for \(\varepsilon\) and \(\mu\) equal to unity.

The capacitance per unit length of a circular coaxial cable is equal to

\[ C_1=\frac{1}{2\lg\frac{b}{a}}, \tag{11.9} \]

and, consequently, the wave impedance of such a transmission line is equal to

\[ Z=\left(\frac{\mu}{\varepsilon}\right)^{1/2}60\lg\frac{b}{a}. \tag{11.10} \]

Similarly, in the case of coaxial confocal elliptic cylinders with focal distance \(f\) and semiaxes, respectively,

\(a\) and \(b\), for the wave impedance one obtains [cf. (5.27)] the expression:

\[ Z=\left(\frac{\mu}{\varepsilon}\right)^{1/2}60\left[\operatorname{Arch}\left(\frac{b}{f}\right)-\operatorname{Arch}\left(\frac{a}{f}\right)\right]. \tag{11.11} \]

Transmission lines in which one of the conductors is completely surrounded by the other are preferable to “open” lines, for example a pair of parallel wires, since the closed system is self-shielding and does not interact with surrounding conductors. However, the transmission theory set forth in this paragraph is equally applicable also to open transmission lines, where the curves \(C_1\) and \(C_2\) bound two conductors, one lying outside the other.

An important special case of an “open” transmission line is the case of two round wires of radius \(r\), whose centers are at a distance \(d\) from one another. In this case the capacitance of the system per unit length is

\[ C_1=\frac{1}{4\operatorname{Arch}\dfrac{d}{2r}}, \]

so that the wave impedance is

\[ Z=\left(\frac{\mu}{\varepsilon}\right)^{1/2}120\operatorname{Arch}\frac{d}{2r}. \tag{11.12} \]

Another important case is a system of two plates of width \(b\), situated at a distance \(d\) from one another, with \(d\ll b\). For such a line

\[ \left. \begin{aligned} C_1&=\frac{d}{4\pi b},\\ Z&=\left(\frac{\mu}{\varepsilon}\right)^{1/2}\frac{120\pi d}{b}. \end{aligned} \right\} \tag{11.13} \]

Example. How large must the distance between the surfaces of the parallel wires of a transmission line be in order that the wave impedance of the line be equal to \(73\ \text{ohms}\).

Answer. About \(19\%\) of the diameter of the wires.

What must be the ratio of the radii of the inner and outer cylinders of a coaxial cable in order that the wave impedance be equal to \(73\ \text{ohms}\).

Answer: \(b/a=3.36\).

Transmission-line equations. Up to now we have considered the theory of transmission lines from the point of view of field theory. In the technical literature* however, transmission lines are considered on the basis of

* As a good elementary introduction one may point to the manual by Everitt\(^9\), and also to several important articles, see \(^ {10}\).

of the method of circuits with lumped constants. In order to compare both methods of calculation, we shall briefly set forth this latter one as well.

A transmission line is regarded as equivalent to the limiting case of the circuit shown in Fig. 7, in which each of the lines is considered to consist of small parameters and a large number of lines is included so that, for example, the inductance of each of the lines, multiplied by their number, approaches the limit \(L\), the inductance per unit length. The same applies also to the resistance \(R\), conductance \(G\), and capacitance \(C\), referred to unit length.

Fig. 7. Circuit with lumped constants, equivalent to a transmission line.

If \(V(z,t)\) is the potential difference of the upper line with respect to the points of the lower line with the same coordinate \(z\), and \(I(z,t)\) is the current flowing from left to right in the upper line and in the opposite direction in the lower line, we must have:

\[ \frac{\partial V}{\partial z}=-RI-L\frac{\partial I}{\partial t}, \tag{11.14} \]

where \(R\) and \(L\) are the resistance in ohms and the inductance in henrys, referred to unit length.

Similarly, if \(G\) is the conductance in mhos and \(C\) is the capacitance in farads per unit length, then

\[ \frac{\partial I}{\partial z}=-GV-C\frac{\partial V}{\partial t}. \tag{11.15} \]

These two equations constitute the basis of the theory of circuits with lumped constants. The results obtained earlier by means of the equations of field theory correspond to the ideal case of \(R\) and \(G\) tending to zero. Before proceeding to discuss the solution of equations (11.14) and (11.15), it is desirable to relate the quantities appearing in them to the concepts that we used in field theory. First of all, in the theory of circuits with lumped constants we speak of the potential difference between two lines. On the other hand, we know that a rapidly varying electric field cannot be described by a single scalar potential. We can remove this contradiction by identifying \(V(z,t)\) with the line integral of the field strength \(\mathbf{E}(x,y,z,t)\) along a path joining the two lines and lying in the plane \(z=\mathrm{const}\). Since, as we have seen, in the plane \(z=\mathrm{const}\) the component of the field strength \(\mathbf{E}_s\) can be represented as the gradient of a scalar potential, the result of the integration is then independent of the choice of the path of integration.

The concept of current strength can be related to the quantities appearing in field theory in the following way. Let us calculate the line integral of

of the magnetic-field intensity $\mathbf H$ along a contour lying in the plane $z=\mathrm{const}$, infinitely close to one of the conductors. From the equation

$\operatorname{rot}\mathbf H=4\pi\mathbf j+\dfrac{1}{c}\dfrac{\partial\mathbf D}{\partial t}$

it follows that

\[ \oint \mathbf H\,dl=\int \operatorname{rot}\mathbf H\,ds=4\pi\int \mathbf j\,ds, \]

where the surface integral is taken in the plane $z=\mathrm{const}$ over the region enclosed within the contour of integration in the integral on the left.

The displacement current does not contribute to the line integral, since the vector $\partial\mathbf D/\partial t$ is everywhere directed perpendicular to the surface of the conductor. Consequently, the total current flowing in the conductor, $I(z,t)$, is equal to

\[ I(z,t)=\frac{1}{4\pi}\oint \mathbf H\,dl. \tag{11.16} \]

As for the inductance per unit length, it must be understood in the following way: at high frequencies the spatial distribution of the magnetic field between the conductors is the same as for steady currents. The magnetic energy enclosed between $z$ and $z+dz$ is expressed by the integral $\int (H^2/8\pi)\,dV$, taken over the volume enclosed between these planes. Equating this integral to $(L\,dz)I^2/2$, we obtain a satisfactory definition of the inductance $L$ per unit length. If the current $I$ is expressed in abamperes and the energy in ergs, $L$ will be a dimensionless number.

Similarly, the capacitance per unit length must be related to the energy of the electric field enclosed between the planes $z$ and $z+dz$ by the relation

\[ C\,dz\,\frac{v^2}{2}=\int \frac{\mathbf E^2}{8\pi}\,dv. \]

If $v$ is expressed in abvolts and the energy in ergs, $C$ is a dimensionless number.

It can be shown that $LC=1$ at such a frequency at which the magnetic flux passing through the conductor may be regarded as negligibly small. The resistance per unit length must be defined as the sum of the resistances of both lines, with proper allowance for the skin effect.

The conductance per unit length is determined through the dissipative characteristics of the dielectric, as will be shown below (see § 15).

Distribution of potential and current. We shall now consider the solution of equations (11.4) and (11.15) corresponding to propaga-

... to a simple harmonic traveling wave, i.e., let

\[ V=Ve^{i(\omega t-kz)},\qquad I=Ie^{i(\omega t-kz)}. \tag{11.17} \]

Substituting this value of \(V\) and \(I\) into (11.14) and (11.15), we find for the amplitudes \(V\) and \(I\):

\[ ikV=(R+i\omega L)I,\qquad ikI=(G+i\omega C)V. \tag{11.18} \]

This system of equations leads to nonzero values of the amplitudes \(V\) and \(I\) only if the constant \(k\) has the value

\[ k^2=\omega^2\left(L-\frac{iR}{\omega}\right)\left(C-\frac{iG}{\omega}\right)=-(R+i\omega L)(G+i\omega C). \tag{11.19} \]

In the case of a lossless line, in which \(R=0\) and \(G=0\), this leads to the condition

\[ k=\omega(LC)^{1/2}. \tag{11.20} \]

In such a circuit the waves propagate in both directions without attenuation, with phase velocity \(1/(LC)^{1/2}\). From the definition of \(L\) and \(C\) it follows that this velocity is precisely equal to the velocity of light in vacuum, \(c\). In the general case of a transmission line with losses, equation (11.19) leads to complex values of \(k\). This means that attenuation of the waves occurs. The magnitude of the attenuation (determined by the imaginary part of \(k\)) depends on the frequency, and therefore in the transmission line distortion occurs in the transmission of any nonmonochromatic signal. It is easy to see, however, that in the special case when \(LG=RC\), the attenuation does not depend on \(\omega\), while the real part of \(k\) is proportional to \(\omega\), so that the phase velocity turns out to be the same for all frequencies. Such a transmission line is called a distortionless line. The development of the theory of transmission lines of this kind plays a very important role in the transmission of electrical energy over long distances and in telephony.

For this reason the theory of circuits with lumped constants has received very broad development, and a large number of special manuals are devoted to its exposition; we refer the reader to them for details.

§ 12. Transmission line with a load

Let us consider a transmission line terminating at the point \(z=L\) in some load characterized by an impedance \(Z_L\). Generally speaking, in the line there will exist waves traveling both toward the load and in the opposite direction. Interfering with one another, they form a system of standing waves superposed on the traveling waves. Because of this, the ratio of voltage to current at different points of the circuit turns out to be different.

Suppose that the characteristic impedance of the circuit is equal to \(Z_0\). If the amplitudes of the voltage waves propagating in the positive and negative directions of the \(z\)-axis are respectively \(V_1\)

and \(V_2\), the field strength at any point \(z\) will be expressed by the real part of the quantity

\[ V=(V_1e^{-ikz}+V_2e^{ikz})e^{i\omega t}, \tag{12.1} \]

and the total current flowing in the circuit at the point \(z\),

\[ I=\frac{1}{Z_0}(V_1e^{-ikz}-V_2e^{ikz})e^{i\omega t}. \tag{12.2} \]

Let us note that the amplitudes \(V_1\) and \(V_2\) are, generally speaking, complex numbers.

At the point \(z=L\), at which the line terminates in an impedance \(Z_L\), we have \(V=Z_L I\), so that

\[ Z_L=\frac{Z_0(V_1e^{-ikL}+V_2e^{ikL})}{V_1e^{-ikL}-V_2e^{ikL}}. \tag{12.3} \]

At the external end of the line, for \(z=0\), the input impedance \(Z\) is equal to the ratio \(V/I\), i.e.

\[ Z_0\frac{V_1+V_2}{V_1-V_2}=Z. \tag{12.4} \]

Equation (12.3) determines the ratio of the amplitudes of the reflected and incident waves, \(V_2/V_1\). Solving it, we find:

\[ \frac{V_2e^{ikL}}{V_1e^{-ikL}}=\frac{Z_L-Z_0}{Z_L+Z_0}. \tag{12.5} \]

Here \(V_2e^{ikL}\) represents the amplitude of the reflected wave and \(V_1e^{-ikL}\) the amplitude of the incident wave at the point \(z=L\), i.e. at the load. From (12.5) we see that \(V_2=0\) only if \(Z_L=Z_0\), i.e. that reflection does not occur if the load impedance is equal to the characteristic impedance of the transmission line. It is convenient to introduce an auxiliary quantity \(\psi\), defined by the relation

\[ \frac{V_2}{V_1}=-e^{-2\psi}, \tag{12.6} \]

with the aid of which formulas (12.3) and (12.4) can be represented in the following form:

\[ Z_L=Z_0\,\operatorname{th}(\psi-ikL), \tag{12.7} \]

\[ Z=Z_0\,\operatorname{th}\psi. \tag{12.8} \]

If we write

\[ Z=Z_0\,\operatorname{th}(u+iv), \]

\[ Z_L=Z_0\,\operatorname{th}(u_L+iv_L), \]

where \(u+iv=\psi\) and \(u_L+iv_L=\psi-ikL\), then

\[ u=u_L \quad \text{and} \quad V=v_L+kL. \tag{12.9} \]

Let us now suppose that in the impedance plane, i.e. in the plane

\(Z=R+ix\), we draw a coordinate net from the families of mutually orthogonal curves \(u=\operatorname{const}\) and \(v=\operatorname{const}\). The load impedance \(Z_L\) corresponds to the pair of values \(u_L, v_L\). From (12.9) we see that a transmission line with “electrical length” \(kL\) corresponds to a transformation in the impedance plane, corresponding to motion along the curve \(u=u_L\) from the point \(v=v_L\) to the point \(v=v_L+kL\). Therefore it is of great interest to study in more detail the curves defined by the transformation

\[ Z=Z_0\,\operatorname{th}(u+iv)= Z_0\,\frac{\operatorname{th}u+i\,\operatorname{tg}v} {1+i\,\operatorname{th}u\,\operatorname{tg}v}. \]

For \(u=0\) this relation gives \(Z=iZ_0\operatorname{tg}v\), i.e. it corresponds to the variation of \(Z\) along the entire imaginary axis as \(v\) increases from zero to \(\pi\). For infinitely large values of \(u\), \(Z=Z_0\), and for any values of \(v\) the curve degenerates into a point. For \(v=0\), \(Z=Z_0\operatorname{th}u\), so that \(Z\) runs through all values lying on the real axis between \(Z=0\) and \(Z=Z_0\) as \(u\) varies from zero to infinity. For \(v=\pi/2\) we have \(Z=Z_0\operatorname{cth}u\), so that \(Z\) varies from infinity to \(Z_0\) as \(u\) increases from zero to infinity. Thus the curve \(u=\operatorname{const}\) intersects the real axis at two points: namely at the points \((Z_0\operatorname{th}u,0)\) and \((Z_0\operatorname{cth}u,0)\). The curves \(u=\operatorname{const}\) turn out to be circles whose centers lie at the points \((Z_0\operatorname{cth}2u,0)\) and whose radii are equal to \(Z_0/\operatorname{sh}2u\). Similarly, the curves \(v=\operatorname{const}\) also turn out to be circles whose centers lie at the points \((0,-Z_0\operatorname{ctg}2v)\) and whose radii are equal to \(Z_0/(\sin 2v)\). Suppose that we are given \(Z_L\) and the input resistance \(Z\), and we wish to find what the line length \(kL\) and its characteristic impedance \(Z_0\) must be for transforming \(Z\) into \(Z_L\).

Fig. 8a. Geometric construction for determining \(Z_0\) and \(kL\). Detailed construction for one case is shown in the upper drawing.

In Fig. 8a we draw the bisector perpendicular to the line \(Z_LZ\). Its intersection with the real axis determines the center of the circle \(u=u_L\), along which the transformation from \(Z\) to \(Z_L\) is carried out. This circle-

ness intersects the real axis at two points. The product of the abscissas of these points is equal to the square of the quantity \(Z_0\). This makes it possible to compute \(Z_0\), whose value can be entered on the diagram. Knowing \(Z_0\), we can carry out the construction, indicated in detail in the upper part of Fig. 8a, which makes it possible to find the points \(C(v_L)\) and \(C(v_L+kL)\), the centers of the circles \(v=v_L\) and \(v=v_L+kL\). Finally, the angle at which the line joining the points \(C(v_L)\) and \(C(v_L+kL)\) is seen from the point \(Z_0\) is equal to \(2kL\).

It is clear that the dependence of \(Z\) on frequency arises partly because of the frequency dependence inherent in \(Z_L\), and partly because of the change in the electrical length of the circuit \(R_L\), associated with the change in \(k\). If \(Z_L(k)\) is given, one can construct points corresponding to the values

Fig. 8b. Special grid for impedance calculations.
Circles \(u=\text{const}\) and \(v=\text{const}\) are plotted on the plane.

\(Z(k)\), i.e., find the frequency dependence of the line and the load. In this connection it is useful to note that if the geometrical length of the line \(L\) is large in comparison with the wavelength, even a slight change of \(k\) leads to changes of \(kL\) exceeding \(2\pi\) several times, which in itself leads to several revolutions about the point \(Z_0\).

The dependence of \(Z_L\) on \(k\) is usually slow, except in those cases where resonance frequencies of \(Z_L\) occur in the given frequency range.

If many calculations of this kind have to be made, it is convenient to prepare the special chart shown in Fig. 8b, on which the ordinary Cartesian coordinates \(R\) and \(X\) and the circles \(u=\mathrm{const}\) and \(v=\mathrm{const}\) have been plotted simultaneously for several particular values of \(Z_0\). The values of \(u\) and \(v\) corresponding to given \(R\) and \(X\) can be

are found at once with a degree of accuracy sufficient for a number of cases. The difficulty of these geometrical constructions is connected with the fact that the motion of \(Z\) along the circle \(u=u_2\), as \(kL\) increases at a constant rate, is nonuniform.

Another complication is connected with the fact that, in order to compute all possible impedances, the construction must be carried out over the entire infinite half-plane. The latter suggests that it is more convenient to use diagrams on which the circles \(u=u_2\) are concentric, and the lines \(v=\mathrm{const}\) are rays issuing from the center, as in the ordinary polar coordinate system. We shall now see that such a construction is possible with the aid of a stereographic projection of the plane onto a sphere.

Let (Fig. 8c) a sphere of diameter \(Z_0\) be tangent to the plane \(RX\) at the origin of coordinates, and let \(Q\) be the end of the diameter passing through the origin of coordinates. To each point \(Z\) on the plane \(RX\) there corresponds a point \(Z'\) on the sphere at which the line \(QZ\) intersects the sphere. The fundamental property of stereographic projection is that every circle in the plane is transformed into a circle on the sphere, and conversely.

If on the sphere one draws a system of parallels and meridians with an axis parallel to the \(R\)-axis, then, as can be shown, the system of circles

Fig. 8c. Stereographic projection of the plane \(R\cdot X\) onto a sphere of diameter \(Z_0\).

Fig. 8d. Section through the imaginary axis and \(Qu=0\).

\(u=\mathrm{const}\) in the \(Z\)-plane corresponds to lines of constant latitude, while the system of circles \(v=\mathrm{const}\) corresponds to lines of constant longitude on the sphere.

In Fig. 8d is shown a section of the sphere by the plane \(u=0\), passing through the imaginary axis. Since for \(u=0\) we have \(Z=iZ_0\operatorname{tg}v\), the angle \(OQZ\) is equal to \(v\), and, consequently, the angle \(OCZ'\) is equal to \(2v\). An increase of \(v\) by \(\pi\) corresponds to a change of \(2v\) by a full period \(2\pi\). Therefore the lines \(v=\mathrm{const}\) are meridian circles with longitude \(2v\).

In Fig. 8e is shown a section through the real axis and \(Q\). The arc \(OP\) represents the geometrical locus \(v=0\), and the arc \(PQ\) the geometrical locus \(v=\pi/2\). The value \(u\) corresponds to a certain line of latitude on the sphere \(u=\operatorname{th} R/Z_0\), the latitude of the point \(Z'\) on the sphere being connected with the point \((R,0)\) on the plane. In particular, the equator represents

the image of the circle \(u=0\), while the pole corresponds to the infinitely distant circle \(u\to\infty\).

We can, further, project onto the sphere the Cartesian coordinates on the planes \(R=\mathrm{const}\) and \(X=\mathrm{const}\). Obviously, the geometric locus of points lying on the straight line \(R=\mathrm{const}\) will be small circles, which are the geometric locus of the points of intersection of the sphere with a plane passing through \(Q\) and the line \(R=\mathrm{const}\). This family of circles will have a common tangent at the point \(Q\).

Fig. 8e. Section through the real axis and \(Q\cdot v=0\).

Similarly, the lines \(X=\mathrm{const}\) are represented by a family of circles orthogonal to the first and also having a common tangent at the point \(Q\). Therefore, near the point \(Q\), the net of lines representing the straight lines \(R=\mathrm{const}\) and \(Q=\mathrm{const}\) will have the form shown in Fig. 8f.

Having a system of circles \(R=\mathrm{const}\) and \(X=\mathrm{const}\) drawn on the sphere, one need not use the impedance plane at all. On the sphere we obtain two systems of mutually orthogonal circles, one of which is the representation of the \((R, X)\) coordinates of a point, and the other the representation of its \((u, v)\) coordinates. If we are given the values \(R_L\) and \(X_L\), we must represent them on the sphere by means of the \((R, X)\) grid. Then the change of impedance caused by a line of length \(kL\) is obtained by rotation along the arc of the small circle \(u=u_L\) until the longitude has increased by the amount \(2kL\). The graphical methods described above are very instructive, but in practice it is very inconvenient to use curves drawn on a sphere.

Fig. 8f. Projections onto the sphere of the coordinate lines \(R=\mathrm{const}\) and \(X=\mathrm{const}\).

We can, however, make use of the fact that every point \(Q\) on the sphere can again be projected onto a plane tangent to the sphere at the opposite end of the diameter passing through \(Q'\). Of all the plane diagrams that can be constructed in this way, one is especially convenient. Namely, in constructing this

of the diagram the pole \(Q'\) and the point \(Z=Z_0\) are laid off at opposite ends of a diameter.

Fig. 8g. Construction of the projection of a sphere onto a tangent plane: each point of the sphere \(Q'\) is projected onto a plane tangent to the opposite end of the diameter drawn through this point.

It is clear from Fig. 8g that the hemisphere corresponding to positive resistances is projected onto a circle of radius \(Z_0\), which represents the projection of the equator \(u=0\) and whose center corresponds to the pole \(u\to\infty\). Other values of \(u\) will be represented by a system of concentric circles.

Similarly, the meridians \(v=\mathrm{const}\) are projected onto radial rays in this plane. The circles \(R=\mathrm{const}\) and \(X=\mathrm{const}\) on the sphere are projected onto an analogous, appropriately arranged system of circles in the plane, as is shown in Fig. 8g.

Thus we achieve the stated goal and obtain a diagram on which the circles \(u=\mathrm{const}\) are concentric circles, while the circles \(v=\mathrm{const}\) become rays equally spaced from one another for equal intervals of \(v\). Figure 8h shows a diagram of this type. In practical work with these diagrams they should be drawn as large as possible—

Fig. 8h. Grid for calculating impedances, in which the circles \(u=\mathrm{const}\) are concentric.

... on a scale that would ensure sufficient precision and accuracy of the construction.

Example. Find, for given \(Z_L\) and \(Z_0\), the input impedance for any line length \(kL\).

In Fig. 8i the center of the circle \(v=v_L\) lies on the imaginary axis, and the circle itself passes through the points \(Z_L\) and \(Z_0\). Having constructed the perpendicular bisector to the line \(Z_0 Z_L\) and extending it to its intersection with the imaginary axis, we find the center of the circle \(v=v_L\). To find the center of the circle \(u=u_L\), draw a perpendicular to the line \(C Z_L\) at the point \(Z_L\). Then the point of intersection of this perpendicular with the real axis will be the desired center. Next draw the circle \(u=u_L\). We obtain the impedance transformation of a line of length \(kL\) by adding the segment \(kL\) to \(v_L\); we thereby find the new center \(C'\), and, having drawn the new circle \(v=v_L+kL\), we find the point of intersection of the latter circle with the circle \(u=u_L\).

Fig. 8i. Construction for determining \(Z\) for any \(kL\).

13. Transformers with variable impedance

Since losses exist in a transmission line, and since insulation defects may also exist in the radiating system and in the network, it is desirable to bring the energy to the load in such a way that no reflected waves arise in the line. For this it is necessary that the load impedance \(Z_L\) be matched to the line impedance \(Z_0\). Therefore the question arises of the need to introduce, into the circuit between the transmission line and the load, suitable transformers that would make it possible to effect such matching.

Let us first see what can be obtained by connecting to the transmission line, in parallel with the load, a “matching section” of length \(L_1\). Suppose that the characteristic impedance of the parallel section is equal to the characteristic impedance of the transmission line and, according to (12.9), has the form \(i Z_0 \tg k h_1\). Since it is connected in parallel with the load, it is more convenient for further calculations to introduce, instead of impedance, the reciprocal quantity—the total conductance. Let \(Y_L=G_L-iB_L\) and \(-iY_0\) be the conductances of the load and of the parallel section of the circuit (here \(Y_0=1/Z_0\) is the characteristic conductance of the transmission line). Then the total conductance of the system...

of the load and of the “matching” section connected in parallel will be

\[ Y_1 = Y_L - iY_0 \operatorname{ctg} kL_1 . \tag{13.1} \]

When \(L\) is varied from zero to a value equal to half the wavelength, the second term in (13.1) runs through an infinite series of values, and consequently the total admittance may take any value on the vertical line \(Y_L\) in the complex admittance plane. Therefore, if the real part of \(Y_L\) proves to be equal to the characteristic admittance of the transmission line, then by choosing the corresponding value of \(L_1\) one can achieve ideal matching of the transmission line with the load. Since complete matching requires equality of two complex numbers, a universal transformer suitable for matching in all cases must possess at least two different “matching elements.” Let us therefore see what can be achieved by inserting into the transmission line a second “matching” section of length \(L_2\), located at a distance \(L_3\) from the load. From equation (13.1) it follows that, when a section \(L_3\) is added to the line, the total admittance will be

\[ Y = Y_0 \frac{1-w}{1+w}, \]

where

\[ w = \frac{Y_0 - Y_1}{Y_0 + Y_1} e^{-2i\theta} \quad \text{and} \quad \theta = kL_3 . \]

The effect of the section \(L_2\) is that the term \(-iY_0 \operatorname{ctg} kh_2\) must be added to the expression just written. Therefore a section connected in parallel with the load can be used to compensate any reactive term in \(Y\). The problem is thus reduced to studying the range of values that can be assigned to \(Y\) for different \(L_1\) and \(L_3\). Putting \(Y_1/Y_0 = g - ib\), we find:

\[ \frac{Y}{Y_0} \]

\[ = \frac{g}{(\cos\theta - b\sin\theta)^2 + g^2\sin^2\theta} -i \frac{\sin\theta\cos\theta(1-g^2-b^2)+b(\cos^2\theta-\sin^2\theta)} {(\cos\theta - b\sin\theta)^2 + g^2\sin^2\theta}. \tag{13.2} \]

It is evident from this that, by changing the magnitude \(b\), we can make the real part of \(Y/Y_0\) run through the whole range of values from zero (for an infinitely large value of \(b\)) to \(1/g\sin^2\theta\) (when \(\cos\theta - b\sin\theta = 0\)). Therefore any transmission line and load can be brought into correspondence, provided only that \(g\sin^2\theta\) is less than unity. Since the real part of \(Y_1\) is equal to the real part of \(Y_L\), it follows directly from this that, with the aid of such a transformer with two “matching sections,” one can match to the transmission line any

... load, the total conductance of which satisfies the condition

\[ G_L \sin^2 \theta < Y_0 . \tag{13.3} \]

At first glance it may seem that this limitation can be removed if \(L_3\) is chosen so that \(\theta = n\pi\), i.e. \(\sin \theta = 0\). If, however, \(L_3\) is selected in this way, we lose the possibility of finding the correct position of the wire \(L_2\), since it appears in the real part in (13.2) in the combination \(b \sin \theta\). It is therefore necessary to adopt the following compromise. In order to make the restriction (13.3) as little severe as possible, the transformer should be designed so that the value of \(\sin \theta\) is sufficiently small. At the same time, however, it is necessary to choose very accurately the position of the matching section \(L_2\).

It is convenient to choose \(\theta\) so that \(L_3=\lambda/8\) or \(3\lambda/8\), so that \(\sin \theta\) and \(\cos \theta\) are equal to \(1/\sqrt{2}\). It then becomes possible to match loads to transmission lines for all impedances for which the condition \(G_L<2Y_0\) is satisfied, without any substantial loss in the accuracy of determining the position of the section \(L_2\).

If \(Z_L=Ae^{i\alpha}\), so that \(G_L=1/A\cos\alpha\), then condition (13.3) reads \(1/A\cos\alpha \leq 2/Z_0\). In the impedance plane \(Z_L\) this means that the point \(Z_L\) must lie outside a circle of radius \(Z_0/4\) with center at the point \((Z_0/4,0)\). Another common type of transformer is a section of cable one quarter wavelength long, whose characteristic impedance can be varied continuously from a maximum to a minimum value. The corresponding construction of such a transformer is shown in Fig. 9. The inner conductor is mounted on an eccentric axis so that, when it is rotated through \(180^\circ\), it occupies positions ranging from coaxial with the outer conductor (dotted circle in Fig. 9) to very close to it (solid circle in Fig. 9). In the coaxial position of the inner cable the characteristic impedance has its maximum value; in the opposite position, when the center of the inner conductor occupies the position nearest to the outer wall, the characteristic impedance has its minimum.

Fig. 9. A quarter-wavelength line section with variable characteristic impedance.

If \(Z_L\) is the impedance of a load matched with such a transformer, then, according to (12.6), the input impedance is \(Z=Z_a^2/Z_L\). Therefore one may use a single “matching section” parallel to \(Z_L\) to cancel the reactive terms of \(Z_L\), and, connected after it, the transformer just described, which transforms the value \(R_L\) so that it becomes matched to the line impedance \(Z_0\).

Conversely, one may first connect the quarter-wavelength transformer and then, after it, a section of cable compensating the reactive components of the impedance.

§ 14. Losses in transmission lines

Losses in transmission lines arise as a result of the nonideality of the conductors and of the dielectric filling the space between them. The introduction of the latter into a cable is usually connected with the necessity of giving it the proper mechanical properties. Thus, for example, if a cable is flexible, then it is almost always necessary to fill it throughout with a solid dielectric, so that when the cable is bent its central conductor remains in place. Losses in conductors were considered in § 8. As was established there, the loss of energy per unit length is expressed by the formula

\[ \frac{c\delta_\mu}{\rho\lambda}\int H^2\,dS, \tag{14.1} \]

where the integration is carried out over the surface of both conductors. If \(I\) is the amplitude of the current in each of the conductors of a coaxial cable \([i(z,t)=I\cos(\omega t-kz)]\), the amplitude of the magnetic field at the inner conductor is \(2I/a\cos(\omega t-kz)\), and at the outer conductor is \(2I/b\cos(\omega t-kz)\). Therefore the average (over a period) loss of power per unit length is equal to

\[ \pi c\delta_\mu/2\lambda\,(1/a+1/b)I^2. \]

This loss of power corresponds to the effective resistance per unit length of the cable:

\[ R_1=15\left(\frac{\rho\mu}{\lambda}\right)^{1/2}\left(\frac{1}{a}+\frac{1}{b}\right). \tag{14.2} \]

If the current flowing in the line is \(I\) amperes, and the wave resistance is \(Z_0\) ohms, the average energy flux will be equal to \(Z_0I^2/2\) watts, and the power loss to \(R_1I^2/2\). Therefore the decrease of power along the cable is expressed by the law

\[ \frac{dP}{dz}=-\frac{R_1}{Z_0}P \]

and

\[ P(z)=P(0)e^{-(R_1/Z_0)z}. \tag{14.3} \]

Consequently, the quantity \(R_1/Z_0\) determines the distance over which the power in the cable decreases by a factor of \(e\). This quantity is analogous to the “quality factor” \(Q\) of a cavity resonator. We shall denote this quantity by \(L\). For a coaxial cable

\[ L=4b\left(\frac{\lambda}{\rho\mu}\right)^{1/2}\frac{\lg \dfrac{b}{a}}{1+\dfrac{b}{a}}\,\text{cm}. \tag{14.4} \]

In radio engineering, the ratio of powers is usually expressed in decibels (db). \(1\ db\) corresponds to the power ratio \(10^{0.1}=1.258\). Since \(\lg_{10} e=0.434\), the factor \(e^{-1}\) corresponds to an energy loss of \(4.34\ db\). Since the power losses in commonly used copper wires are small, it is convenient to express \(L\) in meters per decibel. Substituting for the electrical conductivity of copper \(\rho\), \(\rho=5.7\cdot 10^{-6}\ \mathrm{cm}\), we obtain the following practical formula:

\[ L=10,\ 72\, b\sqrt{\lambda}\, \frac{\lg \dfrac{b}{a}}{0.279\left(1+\dfrac{b}{a}\right)} \ \mathrm{m}/db, \tag{14.5} \]

where \(\lambda\), \(a\), and \(b\) must be expressed in centimeters. The factor in (14.5), which depends on the ratio \(b/a\), has a flat maximum at \(b/a=3.58\).

Table 6

\(x\)	\(\dfrac{\lg x}{0.279(1+x)}\)
1,5	0,58
2,0	0,83
2,5	0,94
3,0	0,98
3,5	1,00
4,0	0,99
4,5	0,98
5,0	0,96

In this case it is approximately equal to unity. Since this maximum is very flat, in order to obtain a good line there is no particular need to adhere exactly to the position of the maximum, as is evident from Table 6.

As a typical example, let us consider a cable with an outer conductor diameter of \(5/8\) inch in a line designed for the transmission of 15-centimeter waves. The maximum value of \(L\) is obtained when the diameter of the inner conductor is \(0.175\) inch and is equal to \(L=33\ \mathrm{m}/db\).

Example. Show that if the cable is filled with an ideal dielectric with dielectric constant \(\varepsilon\), and if the inner and outer conductors of the cable are made of different metals having resistivities and permeabilities respectively \(\rho_a,\mu_a\) and \(\rho_b,\mu_b\), formula (14.4) can be generalized as

\[ L=4b\left(\frac{\lambda}{\varepsilon}\right) \left(\frac{1}{\rho_b\mu_b}\right) \frac{\lg \dfrac{b}{a}} {1+\left(\dfrac{\mu_a\rho_a}{\mu_b\rho_b}\right)^{1/2}\dfrac{b}{a}}. \tag{14.4a} \]

In § 5 we saw how the theory of functions of a complex variable can be used to find the field in a line consisting of two conductors in the most general case. Let us now consider, with the aid of the theory of functions, the losses in a circuit for an arbitrary form of the conductors forming it. If \(z=f(w)\) and the inverse function is \(w=g(z)\), then

\[ \operatorname{grad}^{2} v = \left(\frac{\partial v}{\partial x}\right)^{2} + \left(\frac{\partial v}{\partial y}\right)^{2} = |g'(z)|^{2}. \]

On the curve \(u=\mathrm{const}\)

\[ ds=\left[\left(\frac{\partial x}{\partial v}\right)^2+ \left(\frac{\partial y}{\partial v}\right)^2\right]^{1/2} =\left|f'(w)\right|\,dv . \]

Since \(\left|g'(z)\right|=1/\left|f'(w)\right|\), the integral appearing in (14.1) can be represented in the form

\[ \int H^2\,ds=\int_0^{2\pi}\frac{dv}{\left|f'(w)\right|}; \]

the latter integral must be evaluated for the values \(u=u_1\) on the inner conductor and \(u=u_2\) on the outer conductors. Therefore, analogously to (14.2), we can write for the effective resistance per unit length of the line

\[ R_1=15\left(\frac{\rho\mu}{\lambda}\right)^{1/2}\frac{1}{2\pi} \left[ \left.\int_0^{2\pi}\frac{dv}{\left|f'(w)\right|}\right|_{u_1} + \left.\int_0^{2\pi}\frac{dv}{\left|f'(w)\right|}\right|_{u_2} \right], \tag{14.6} \]

and the quantity \(L\) can easily be found from its definition \(L=Z_0/R_1\), where for \(Z_0\) one must use formula (5.28).

As a typical example of computations of this kind, let us find the value of \(R_1\) for a line consisting of two confocal elliptic cylinders, whose wave resistance was computed in (5.27). For such a circuit we have \(z=f\operatorname{ch} w\), hence \(f'(w)=f\operatorname{sh} w\), and the integrals in (14.6) reduce to

\[ \frac{1}{2\pi}\int_0^{2\pi}\frac{dv}{f'(w)} = \frac{1}{2\pi f\,\operatorname{ch}u} \int_0^{2\pi}\frac{dv}{(1-k^2\sin^2 v)^{1/2}}, \]

where \(k^2=1/\operatorname{ch}^2 u\). The last integral is the complete elliptic integral of the first kind.

Since \(\operatorname{ch}u_1=a/f\) and \(\operatorname{ch}u_2=b/f\), we finally obtain:

\[ R_1=15\left(\frac{\rho\mu}{\lambda}\right)^{1/2} \left[ \frac{1}{a}\frac{2}{\pi}K\left(\frac{f}{a}\right) + \frac{1}{b}\frac{2}{\pi}K\left(\frac{f}{b}\right) \right]. \tag{14.7} \]

If the quantities \(f/a\) and \(f/b\) are small compared with unity, both elliptic integrals tend to the value \(\pi/2\), and formula (14.7) becomes the previously obtained formula for the resistance of a circular coaxial cable. To find the correction to \(R_1\) for small values of \(f\), in the first approximation one may use the known power-series expansion in the argument \(k\) for elliptic integrals and write

\[ R_1=15\left(\frac{\rho\mu}{\lambda}\right)^{1/2} \left[ \frac{1}{a}+\frac{1}{b} +\frac{f^2}{4}\left(\frac{1}{a^2}+\frac{1}{b^2}\right) +\ldots \right]. \tag{14.8} \]

Since the correction already in the first approximation depends only on \(f^2\), it is clear that the change in the magnitude of the losses in the line, for not too large deviations of the form of its constituent conductors from circular, is insignificant.

§ 15. Dielectric losses

Let us now suppose that the space between the conductors is filled with a dielectric with dielectric constant \(\varepsilon\). Then, according to (11.8), the wave impedance of the circuit will change, in comparison with its value in the absence of filling, in the ratio \(1/\sqrt{\varepsilon}\). If power losses occur in the dielectric, it must be described by a complex dielectric constant. To determine the magnitude of the dielectric losses it is necessary to return to equation (1.1) for \(\operatorname{rot} \mathbf H\). Assuming that the time dependence of the field is determined by the factor \(e^{i\omega t}\), we find

\[ \operatorname{rot}\mathbf H = 4\pi\mathbf j + ik\mathbf D . \]

If the substance is characterized by a resistivity \(\rho\) and a dielectric constant \(\varepsilon\), the right-hand side of the equation may be represented in the form

\[ ik\mathbf E\left(\varepsilon-\frac{2i\lambda}{\rho}\right), \]

analogously to the way in which we wrote it in § 8, devoted to the skin effect in metals. In metals \(\rho\) is so small that the second term in the parentheses proves to be considerably larger than the first. In dielectrics, however, the opposite relation obtains. The phenomena taking place in real dielectrics are much more complicated than is assumed in a formal approach to field theory[^11]. In reality, the dissipation of energy in a dielectric occurs not only as a consequence of the presence in it of finite conductivity. Thus, for example, energy is dissipated when molecules with a permanent dipole moment are oriented by the field in a viscous medium.

However, all dissipative processes have this in common: they all lead to the appearance in the dielectric of a certain nonzero current density, in phase with the field strength \(\mathbf E\), which formally can be described by introducing an imaginary term into the dielectric constant. In particular, dielectric losses may also be associated with ordinary conductivity, described by ohmic resistance. These losses also depend on frequency, and there is no unambiguous method for experimentally separating ohmic losses from losses due to other mechanisms.

Therefore it seems more expedient to us to abandon attempts to distinguish between “true” ohmic losses and other

dissipative mechanisms and to describe phenomenologically the nonideality of a dielectric by introducing a complex dielectric constant

\[ \varepsilon=\varepsilon'-i\varepsilon'', \tag{15.1} \]

where the quantities \(\varepsilon'\) and \(\varepsilon''\), which depend on frequency, are characteristic of the given material. Sometimes the losses are described by a complex dielectric constant written in the form

\[ \varepsilon=\varepsilon_0 e^{-i\delta}. \tag{15.2} \]

Before proceeding to the consideration of losses in a circuit caused by the nonideality of dielectrics, it is useful to review the results of § 3 and to ascertain how the properties of a hollow resonator change when it is filled with a nonideal dielectric.

The introduction into equation (3.1) of a complex dielectric constant leads to the fact that the refractive index \(n=\sqrt{\varepsilon\mu}\), which appears in the wave numbers \(k=n\omega/c\), also becomes complex. The entire theory set forth in Chapter 1 was basically reduced to finding the allowed values of \(k\) for which the field equations in the resonator would be satisfied with the corresponding boundary conditions.

Since we now regard \(n\) as complex, and all the allowed values of \(k\) as real, the frequency \(\nu\) must assume complex values.

Suppose that, for some type of oscillation, we have found an allowed (proper) value of the wave number equal to \(k_a\). In vacuum the field of the resonator is described by free undamped oscillations with frequency \(ck_a/2\pi\). If, however, the resonator is filled with a nonideal dielectric, the frequency of the oscillations will be

\[ \nu_a=\nu'_a+i\nu''_a=\frac{k_a c}{2\pi n}=\frac{k_a c}{2\pi}\sqrt{\varepsilon'-i\varepsilon''}, \tag{15.3} \]

so that the time dependence of the field oscillations is expressed by the factor \(e^{i2\pi\nu'_a t}\cdot e^{-2\pi\nu''_a t}\).

Thus the physical meaning of the imaginary part \(\nu''_a\) consists in the fact that it determines the magnitude of the damping of the free oscillations caused by losses in the dielectric.

The situation here proves to be quite similar to that which occurs for finite conductivity of the resonator walls (cf. § 9). We may introduce the concept of the “quality factor” of the dielectric \(Q'\), characterizing the magnitude of losses in exactly the same way as the quality factor of the conductor \(Q\) introduced in § 9.

For a resonator with ideally conducting walls, filled with a nonideal dielectric, the losses will be described by a decaying

with the time factor \(e^{-\omega t/2Q'}\), where

\[ Q'=\frac{\gamma'_a}{2\gamma''_a}=\frac{1}{2}\operatorname{ctg}\frac{\delta}{2}. \tag{15.4} \]

Here \(\delta\) is the phase of the dielectric constant.

If, along with dielectric losses, losses due to the finite conductivity of the walls occur in the resonator, the total damping may be characterized by the quantity \(Q_c\), equal to

\[ \frac{1}{Q_c}=\frac{1}{Q}+\frac{1}{Q'}. \tag{15.5} \]

We see that, even when the losses in the dielectric amount to only \(1\%\) (in other words, even when \(\operatorname{tg}\delta=0.01\)), the value of the quality factor \(Q\) of a resonator filled with this dielectric cannot reach 100. Moreover, in this case the dielectric losses will already be large in comparison with the losses in the walls, at least for ordinary values of conductivity.

Let us now consider the influence of the nonideality of a dielectric introduced into the cable of a transmission line. Maxwell’s equations will be satisfied if, instead of \(\mathbf E\), we substitute \(\sqrt{\tilde{\varepsilon}}\,\mathbf E\) and set

\[ k=\frac{n\omega}{c}, \]

where \(n\) is the complex refractive index.

The complex nature of the quantity \(n\) leads to the wave number \(k\) also becoming complex, i.e.

\[ k=k'-ik''=\frac{\omega}{c}\sqrt{\varepsilon'-i\varepsilon''}. \tag{15.6} \]

This, in turn, leads to the damping of waves propagating along the transmission line. For example, the current will now be equal to

\[ I=I_0 e^{-k''z}\cos(\omega t-k'z) \tag{15.7} \]

and, consequently, the power in the circuit will decrease according to the law \(e^{-2k''z}\). Therefore the “loss length” \(L'\) in a line with a nonideal dielectric will be equal to

\[ L'=\frac{1}{2k''}=\frac{\lambda}{4\pi\sqrt{\varepsilon}}\operatorname{cosec}\frac{\delta}{2}, \tag{15.8} \]

where \(L'\) and \(\lambda\) are expressed in centimeters. Note that in (15.8) the wavelength \(\lambda\) denotes the wavelength in vacuum.

Since the losses in the conducting walls are additive with the losses in the dielectric, the total effective length \(L_c\) is determined by the equality

\[ \frac{1}{L_c}=\frac{1}{L}+\frac{1}{L'}, \tag{15.9} \]

where \(L\) and \(L'\) are the loss lengths caused, respectively, by losses in the dielectric and in the conductors of the cable.

§ 16. Reflection from Fastenings

For the mechanical fastening of a cable one may use thin plugs of dielectric inserted inside the cable. At the surfaces of these plugs reflection of waves will inevitably occur; however, with a suitable choice of the distances between them these reflections can be reduced to zero. Moreover, a proper accounting of all the phenomena associated with the insertion of such dielectric plugs into the cable makes it possible to construct filters for microwaves, analogous to filters in circuits with lumped constants, which are used at low frequencies.

At each point \(z\) of a cable with dielectric plugs there exist an incident wave (from left to right, in the positive direction of the \(z\)-axis) and a reflected wave. Let \(n=\sqrt{\varepsilon}\) be the refractive index of the dielectric, \(A=nV_a\) and \(B=nV_b\), where \(V_a\) and \(V_b\) are the amplitudes of the incident and reflected waves. Then the electric field at each point of the cable will be described by the two-component quantity \(\binom{A}{B}\), which in the course of the further calculations we shall regard as a two-row matrix with one column. We shall assume, as always, that the time dependence of all quantities is expressed by the law \(e^{i\omega t}\); the coordinate dependence of \(A\) is determined by the factor \(e^{-ikz}\), where \(k=n\omega/c\), and, analogously, the coordinate dependence of \(B\) by the factor \(e^{+ikz}\). Therefore the amplitude \(\binom{A}{B}\) at a given point can be expressed in terms of the amplitude \(\binom{A_1}{B_1}\) at a point lying in the same medium at a distance \(z\) to the right, by the matrix relation

\[ \binom{A}{B} = \begin{pmatrix} e^{ikz} & 0\\ 0 & e^{-ikz} \end{pmatrix} \binom{A_1}{B_1}. \tag{16.1} \]

For readers not familiar with matrix algebra, let us note that the matrix equation

\[ \binom{a}{b} = \begin{pmatrix} c & d\\ e & f \end{pmatrix} \binom{g}{h} \]

is nothing other than a concise notation for the set of two linear equations:

\[ a=cg+dh,\qquad b=eg+fh. \]

In particular, (16.1) represents the combined notation of the pair of equations:

\[ A=e^{ikz}A_1,\qquad B=e^{-ikz}B_1. \]

The circumstance that in the transformation matrix entering into (16.1),

are zero, expresses mathematically the fact that \(A\) depends only on \(A_1\) and \(B\) only on \(B_1\). Physically this means that in a homogeneous cable no reflection of waves occurs. Let us now see how relation (16.1) changes if, moving from left to right, we cross the boundary between regions with dielectric constants equal, respectively, to unity and \(n\). At the interface the usual boundary conditions are satisfied—the continuity of the radial component of the electric field and of the circular component of the magnetic field.

Let

\[ \begin{pmatrix} A_1\\ B_1 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} A_n\\ B_n \end{pmatrix} \]

be the amplitudes on the two sides of the interface. Then the boundary conditions may be written in the form

\[ A_1+B_1=n^{-1}(A_n+B_n),\qquad A_1-B_1=A_n-B_n. \]

Expressing \(A_1\) and \(B_1\) in terms of \(A_n\) and \(B_n\), we can write, in matrix notation,

\[ \begin{pmatrix} A_1\\ B_1 \end{pmatrix} = \frac{1}{2} \begin{pmatrix} n^{-1}+1 & n^{-1}-1\\ n^{-1}-1 & n^{-1}+1 \end{pmatrix} \begin{pmatrix} A_n\\ B_n \end{pmatrix}. \tag{16.2} \]

To clarify the physical meaning of the result obtained, suppose that the dielectric completely fills the cable to the right of the interface, while to the left of it there is no dielectric at all. Suppose, moreover, that the cable is properly terminated so that \(B_n=0\). Then we have:

\[ A_1=\frac{1}{2}(n^{-1}+1)A_n,\qquad B_1=\frac{1}{2}(n^{-1}-1)A_n. \]

The energy flux in the incident wave is proportional to \(A_1^2\), or to \(\frac{1}{4}(n^{-1}+\)

\[ +1)^2 A_n^2, \]

and in the reflected wave to \(B_1^2=\frac{1}{4}(n^{-1}-1)^2A_n^2\). For simplicity we assume that \(n\) is real. Then the ratio of the reflected energy to the incident energy will be

\[ R=\frac{(n^{-1}-1)^2}{(n^{-1}+1)^2}=\frac{(n-1)^2}{(n+1)^2}. \tag{16.3} \]

The equivalence of the two expressions in (16.3) expresses the fact that, upon reflection from a simple interface, the reflected power is the same when passing from a medium with refractive index 1 to a medium with refractive index \(n\), or, conversely, when passing from a medium \(n\) to a medium 1.

Let us give a numerical example. If, for example, the cable is filled with polystyrene with \(\varepsilon=2.7\), we have \(n=1.65\) and \(R=5.8\%\). With such a large value of \(R\) for reflection already from a single boundary, it is obviously very important to take the appropriate measures to obtain such interference as would lead to mutual

to the attenuation of waves reflected from the various surfaces of discontinuity present inside the cable.

On a surface where the refractive index changes from \(n\) to 1, we find, analogously to (16.2):

\[ \binom{A_n}{B_n} = \frac{1}{2} \begin{pmatrix} n+1 & n-1\\ n-1 & n+1 \end{pmatrix} \binom{A_1}{B_1}. \tag{16.4} \]

As was to be expected, the transformation matrix in (16.4) is the inverse of the matrix in (16.2).

Let us remind the reader of the rule for multiplying matrices, which we shall need below. It consists in the fact that if

\[ \begin{pmatrix} a & b\\ c & d \end{pmatrix} = \begin{pmatrix} e & f\\ g & h \end{pmatrix} \begin{pmatrix} i & j\\ k & l \end{pmatrix}, \]

then

\[ a=ei+fk,\qquad b=ej+fl, \]

\[ c=gi+hk,\qquad d=gj+hl. \]

Fig. 10. Fastening in a coaxial cable.

Let us now consider all the effects associated with introducing into the cable a dielectric plug of thickness \(L\) (Fig. 10). With the aid of (16.4) we can express \(\binom{A_3}{B_3}\) in terms of \(\binom{A_4}{B_4}\), then \(\binom{A_2}{B_2}\) in terms of \(\binom{A_3}{B_3}\) by means of (16.1), and, finally, \(\binom{A_1}{B_1}\) in terms of \(\binom{A_2}{B_2}\) from (16.2). As a result, we obtain the following expression for \(\binom{A_1}{B_1}\) in terms of \(\binom{A_4}{B_4}\):

\[ \binom{A_1}{B_1} = \frac{1}{2} \begin{pmatrix} n^{-1}+1 & n^{-1}-1\\ n^{-1}-1 & n^{-1}+1 \end{pmatrix} \begin{pmatrix} e^{ia} & 0\\ 0 & e^{-ia} \end{pmatrix} \times \frac{1}{2} \begin{pmatrix} n+1 & n-1\\ n-1 & n+1 \end{pmatrix} \binom{A_4}{B_4}, \]

where \(a=\omega nL/c\). Multiplying all three matrices (remembering here that the order of multiplication is essential), we find the matrix describing the influence of a single plug:

\[ \frac{1}{4n} \begin{pmatrix} (n+1)^2 e^{ia}-(n-1)^2 e^{-ia} & -2i\,(n^2-1)\sin a\\ 2i\,(n^2-1)\sin a & (n+1)^2 e^{-ia}-(n-1)^2 e^{ia} \end{pmatrix}. \tag{16.5} \]

Relation (16.5) is often more conveniently written in the form

\[ \binom{A_1}{B_1} = \begin{pmatrix} P_1 & Q_1^{*}\\ Q_1 & P_1^{*} \end{pmatrix} \binom{A_4}{B_4}, \]

where

\[ P_1=\cos a+i\,\frac{n^2+1}{2n}\sin a, \tag{16.6} \]

\[ Q_1=i\,\frac{n^2-1}{2n}\sin a. \]

Hence we find that the power reflected from one plug is equal to

\[ R_1=\frac{(n^2-1)^2\sin^2\alpha}{(n^2+1)^2\sin^2\alpha+4n^2\cos^2\alpha}. \tag{16.7} \]

For polystyrene, for example, \(n=1.65\), and

\[ R_1=\frac{2.89\sin^2\alpha}{13.7\sin^2\alpha+10.8\cos^2\alpha}. \]

From the last relations it is clear that the reflected power becomes zero when \(\alpha=m\pi\), i.e., when \(L\) is equal to an even number of half-waves in the given dielectric material. Conversely, the greatest reflection occurs when \(L\) is equal to an odd number of half-waves. For polystyrene the maximum value of the reflected power is \(21\%\) of the incident power.

From the point of view of the mechanical properties of the cable, a plug thickness \(L\) of \(1/8\) inch is usually sufficient. For polystyrene plugs at a wavelength in vacuum equal to \(15\) cm, this corresponds to the value \(\alpha=12^\circ\) and \(R\) about \(1.1\%\). At the same time, in the region of longer waves the very same plug produces an entirely negligible reflection. This is why the question of reflection from plugs introduced into a cable is of such great importance precisely in the region of microwaves.

§ 17. Choke and Shunt Capacitor

Suppose that we wish the high-frequency currents in a transmission line to reach only a certain point, and that beyond this point the transmission line should turn into a circuit in which only low-frequency currents could flow. The corresponding device introduced into the line is called a choke. Let a cup-shaped fitting be made on the inner conductor of a coaxial cable, as shown in Fig. 11. Let, further, the impedance of the load located behind the fitting be equal to \(Z_L\). If the length of the fitting is \(L\), then the impedance at its open end is, according to (12.5),

Fig. 11. High-frequency choke (voltage node at the far end of the cup).

\[ Z' = Z_2 \frac{Z_L\cos kL+iZ_2\sin kL}{iZ_L\sin kL+Z_2\cos kL}, \tag{17.1} \]

where \(Z_2\) is the characteristic impedance of the circuit element formed by the outer conductor and the outer wall of the fitting.

Similarly, the input impedance at the open end of the cup is equal to

\[ Z''=iZ_1\operatorname{tg} kL, \]

where \(Z_1\) is the characteristic impedance of the section formed by the inner conductor and the inner wall of the sleeve.

For currents flowing as shown by the arrows in the diagram, the two impedances are connected in series, so that the total input impedance proves to be

\[ Z=Z' + Z''. \tag{17.2} \]

If the length of the sleeve is equal to one quarter of the wavelength,

\[ Z=iZ_1\infty+\frac{Z_2^2}{Z_L}=\infty, \tag{17.3} \]

i.e., the impedance from the open side of the sleeve becomes infinite.

Therefore complete reflection of the high-frequency waves from the sleeve takes place. At the input aperture of the sleeve there is then an antinode in the voltage wave and a node in the current wave, just as though the line ended at this point in an open circuit.

Consider now the same kind of sleeve placed inside a cable, as shown in Fig. 12. In this case the impedance of the sleeve is simply \(iZ_1\operatorname{tg} kL\), and the impedance is connected in series with the load impedance \(Z_L\), so that the total impedance is equal to \((Z_L+iZ_1\operatorname{tg} kL)\).

Fig. 12. High-frequency choke (voltage node at the near end of the sleeve).

If the length of the sleeve is equal to one quarter of the wavelength, the impedance again becomes infinite. Therefore the waves arriving from the left are completely reflected, and at the bottom of the sleeve a node of the voltage waves is obtained.

Thus, quarter-wavelength sleeves block high-frequency currents and do not pass them into the right-hand part of the cable.

Sometimes, on the contrary, it is desirable to block low-frequency currents so that this does not affect the high-frequency currents. For this purpose the transmission line is cut as shown in Fig. 13. If the length of the overlapping sections of the outer conductors is equal to one quarter of the wavelength of the high-frequency radiation in the cable, the infinite impedance at the open gap between the outer conductors is transformed into zero impedance. Therefore the high-frequency currents flowing in the larger (left-hand) conductor pass

Fig. 13. Low-frequency choke (convenient for assembled cable sections).

inside the right-hand conductor, bypassing the discontinuity without a voltage drop. Low-frequency currents, however, prove to be blocked because of the absence of contact between the two outer conductors.

Another cable design that makes it possible to block low-frequency currents is shown in Fig. 14. In this case the outer conductor of the cable is likewise made split, and both sections terminate in flanges. The radii of both flanges, for which such a design satisfies the stated requirements, can be found as follows. Let the radius of the outer conductor be equal to \(a\). Then we must have \(E_z=0\) at \(r=a\), so that no voltage drop occurs in the gap between the flanges, as though the outer conductor were continuous.

Fig. 14. Low-frequency choke (for a rigid cable).

In general, we may put

\[ E_z=A J_0(kr)+B N_0(kr), \]

where \(J_0\) and \(N_0\) are Bessel functions of zero order. The requirement \(E_z=0\) at \(r=a\) therefore reduces to the equation

\[ A J_0(ka)+B N_0(ka)=0, \]

which determines the ratio \(B/A\).

At the outer radius of the flange, at \(r=b\), the radial current must vanish. For this it is necessary that \(H_\varphi=0\), whence, in turn, it follows that the equality \(\partial E_z/\partial r=0\) must hold at \(r=b\). Therefore

\[ A J'_0(kb)+B N'_0(kb)=0. \]

The last equation makes it possible, for a given ratio \(B/A\), to determine the flange radius \(b\).

Let, for example, \(a=0.5\ \text{cm}\) and the wavelength \(\lambda=3\ \text{cm}\), so that \(k=2.08\) and \(ka=1.04\). In this case \(J_0(ka)=0.7473\) and \(N_0(ka)=0.1188\), so that if we put

\[ E_z=0.1188J_0(kr)-0.7473N_0(kr), \]

then the first condition at \(r=a\) will be satisfied. The value \(kb\) satisfying the condition \(\partial E_z/\partial r=0\) can most simply be found from tables of Bessel functions. With the aid of the tables we find that \(E_z\) has a maximum value at \(kb=2.4\), or \(b=1.15\ \text{cm}\).

§ 18. The transmission line as a resonator

Any finite section of a transmission line can be used by itself, or in connection with lumped capacitances or inductances, to obtain a resonant circuit.

First let us consider a section of transmission line of length \(z\), open at one end and closed at the other. The impedance at the open end will be, according to formula (12.7),

\[ Z=iZ_0 \operatorname{tg} kz. \]

The natural oscillations in the line must be such that the current flowing is equal to zero even for a finite value of the voltage amplitude. Consequently, the frequencies of the natural oscillations must be such that the impedance at the open end of the circuit is equal to infinity. Therefore the resonant values of \(k\) can be found from the condition

\[ kz=\left(n+\frac{1}{2}\right)\pi, \]

where \(n\) are integers. This condition may be rewritten in the form

\[ z=\left(\frac{n}{2}+\frac{1}{4}\right)\lambda . \tag{18.1} \]

Thus the lowest resonant frequency must be such that \(z\) is equal to one quarter of the wavelength.

If the resonator is closed at both ends, the resonant frequencies must be such that the impedance at both ends of the line is equal to zero. This gives \(kz=n\pi\), i.e., in this case the length of the section must be equal to an integral number of half-waves. The latter statement is in agreement with the results obtained in § 5 by means of arguments based on field theory.

Suppose now that at the open end of the line shown in Fig. 15 there is a capacitor of capacitance \(C\). The input impedance at terminals 1, 2 will be equal to the sum of the separate impedances, i.e.

\[ Z=iZ_0 \operatorname{tg} kz+\frac{1}{i\omega C}. \tag{18.2} \]

Fig. 15. Resonator composed of two sections of line.

All frequencies for which \(Z=0\) will be resonant frequencies, since in the present resonator, in which terminals 1, 2 are connected, the current at the boundary must flow without a voltage drop. This gives for \(k\) the equation

\[ kz \operatorname{tg} kz=\frac{z}{cCZ_0}. \tag{18.3} \]

If \(C\) is small, the roots of equation (18.3) differ little from the roots of (18.1). In the general case, for given \(L\), \(C\), and \(Z_0\), the resonant values—

\(k\) can most simply be found graphically, by plotting \(x \tg x\) as a function of \(x\). From such a graph one can directly conclude that increasing \(C\) leads to a decrease of all resonant frequencies. It is useful to compare these considerations with the calculations of § 7, based on the application of field theory.

The resonator may be constructed, as shown in Fig. 15, from two lines connected to one another, of lengths \(z_1\) and \(z_2\), with impedances \(Z_1\) and \(Z_2\), respectively. The input impedance in the plane of the section, equal to

\[ i\left(Z_1 \tg kz_1 + Z_2 \tg kz_2\right), \]

must vanish at the resonant frequencies. The latter can be found graphically. For this it is necessary to plot the graphs of the functions \(Z_1 \tg kz_1\) and \(-Z_2 \tg kz_2\) as functions of \(k\) and find the points of intersection of both curves.

Fig. 16. The same resonator (dimensions are given for the exercise).

Example. Calculate the lowest resonant frequency of the resonator shown in Fig. 16 (surface of rotation about the horizontal axis). The dimensions indicated in the drawing are equal to

\[ a=2,\quad b=3,\quad c=1,\quad d=\frac{1}{2}\quad \text{and}\quad e=\frac{1}{4}. \]

Answer: 468 megacycles/sec.

§ 19. Tapered transmission lines\(^ {12}\)

By a tapered line we shall mean a cable with geometrical dimensions varying along the line, for example a coaxial cable with a variable ratio of the radii of the outer and inner conductors.

In what follows we shall assume that all transverse dimensions of the cable are small in comparison with the wavelength. The initial relations of the theory are equations (11.14) and (11.15). Since in practice tapered lines are used only for short transition sections of the main line, we shall neglect losses and take \(R=0\) and \(G=0\). The necessary generalization of the equations written earlier as applied to tapered lines consists in now considering \(L\) and \(C\) as functions of \(z\). Therefore the equations of the transmission line take the form

\[ \left. \begin{aligned} \frac{\partial V}{\partial z} &= -L\,\frac{\partial I}{\partial t},\\ \frac{\partial I}{\partial z} &= -C\,\frac{\partial V}{\partial t}. \end{aligned} \right\} \tag{19.1} \]

Assuming that the time dependence of the quantities is expressed by the factor \(e^{i\omega t}\), for the amplitudes \(V\) and \(I\) we find:

\[ \left. \begin{aligned} V''-\frac{d\lg L}{dz}V'+\omega^2LCV&=0,\\ I''-\frac{d\lg C}{dz}I'+\omega^2LCI&=0. \end{aligned} \right\} \tag{19.2} \]

If \(\varepsilon=\mu=1\), then \(LC=\dfrac{1}{c^2}\), and the characteristic impedance of the circuit \(Z\) is related to \(L\) and \(C\) by the relations

\[ Z=cL=\frac{1}{cC}. \]

Therefore the logarithmic derivatives appearing in (19.2) can be expressed in terms of the logarithmic derivative of \(Z\). Putting \(k=\omega/c\), we have:

\[ \left. \begin{aligned} V''-\frac{d\lg Z}{dz}V'+k^2V&=0,\\ I''+\frac{d\lg Z}{dz}I'+k^2I&=0. \end{aligned} \right\} \tag{19.3} \]

We need obtain the solution of only one of these equations, since if, for example, the solution for \(V(z)\) is known, the current \(I(z)\) can be found with the aid of the first of equations (19.1) in the form

\[ I=\frac{i}{\omega L}\frac{\partial V}{\partial z} =\frac{i}{kZ}\frac{\partial V}{\partial z}. \tag{19.4} \]

We shall therefore consider the solution of the first of equations (19.3), which determines the variation of the potential difference between the two conductors of the cable along the transmission line.

For lines that remain homogeneous along their length, \(d\lg Z/dz=0\), the equation has a solution of the form \(e^{ikz}\) or \(e^{-ikz}\), i.e. it represents ordinary unperturbed harmonic waves propagating along the cable with the speed of light. If, however, the properties of the cable vary along its length and \(\dfrac{d\lg Z}{dz}\ne0\), it is convenient to seek the solution of (19.3) in the form

\[ V=\sqrt{Z}\,U, \tag{19.5} \]

where the function \(U\) satisfies the equation

\[ U''+\left[k^2+\left(\frac{Z''}{2Z}\right)-\left(\frac{3Z'^2}{4Z^2}\right)\right]U=0. \tag{19.6} \]

The solution of the last equation is expressed in terms of elementary functions in two cases.

Exponential line. The first of these is the case when the cable is tapered in such a way that its characteristic impedance decreases exponentially along the line, i.e.

\[ Z(z)=Z_0 e^{2k_0 z}. \tag{19.7} \]

In this case the differential equation (19.6) for \(U(z)\) is simplified and takes the form

\[ U''+(k^2-k_0^2)U=0. \tag{19.8} \]

The character of the solutions obtained in this case depends on the sign of the quantity

\[ k'^2=k^2-k_0^2. \]

If \(k'^2\) is positive, the function \(U\) is purely periodic, so that undamped waves propagate along the line, while the amplitude of the waves \(V\) increases exponentially in the direction of increasing impedance \(Z\). If, however, \(k'^2\) is negative, \(V\) represents a real exponential function, and the waves attenuate along the line. Therefore such an exponentially tapering line behaves as a pass filter—only waves with wave number \(k\) greater than \(k_0\) pass through it. The cut-off frequencies begin at the higher frequency, the more rapidly the cable tapers.

Let us consider a wave propagating in the positive direction of the \(z\)-axis. The potential difference \(V\) will have the form

\[ V=V_0 e^{k_0 z} e^{i(\omega t-k'z)} \]

and, consequently, the current

\[ I=\frac{V_0}{Z_0}\frac{k'+ik_0}{k}\,e^{-k_0 z}e^{i(\omega t-k'z)}. \]

The ratio of the potential difference to the current at any point gives the value of the load impedance with which the line may be terminated at that point without the formation of reflected waves. This impedance is, evidently,

\[ Z_i=\frac{k}{k'+ik_0}Z_0 e^{2k_0 z}. \tag{19.9} \]

Thus this impedance is to some extent reactive, although its phase angle tends to zero if the transmitted frequency is large in comparison with the limiting cut-off frequency, i.e. if \(k'\) is large in comparison with \(k_0\).

Let us consider a concrete example. Suppose we wish to investigate the properties of a transition section of a line connecting two cables with characteristic impedances of 50 and 100 ohms. If the internal

the conductor in both cables has one and the same diameter—0.125 inch, the diameters of the outer conductors must be, respectively, 0.288 and 0.660 inch. If the transition section has a length of \(1\ \text{m}\), we have \(200 k_0 = x \ln 2\), or \(k_0 = 3.47 \cdot 10^{-3}\ \text{cm}^{-1}\). Consequently, the limiting clipped wavelength is equal to \(\dfrac{2\pi}{k_0} = 1810\ \text{cm}\).

If radiation with wavelength \(15\ \text{cm}\), or \(k = 0.418\ \text{cm}^{-1}\), is transmitted in this line, it is easy to calculate that the phase angle of the impedance of the transition section is less than one degree.

A line in which \(Z\) varies with \(z\) according to a power law. Another case in which equation (19.6) can be solved in elementary functions is the case when \(Z\) varies proportionally to some power of the distance \(z\) from the beginning of the tapered cable, i.e.

\[ Z(z)=Z_1 z^n, \tag{19.10} \]

where \(Z_1\) is the characteristic impedance at a point located at a distance equal to unity from the place where \(z\) becomes zero. In practice one has to deal with a finite section of tapered cable, for example with a section from \(z=+a\) to \(z=+b\). Therefore no difficulties arise associated with negative values of \(Z\) or its going to zero, which, as may seem at first glance, arise when using the law (19.10).

In this case \(\dfrac{d\lg Z}{dz}=\dfrac{n}{z}\), and (19.10) is transformed into

\[ V''-\frac{h}{z}V' + k^2 V = 0. \tag{19.11} \]

The solution of equation (19.11) is expressed in Bessel functions. Namely,

\[ V(z)=z^m Z_m(kz), \]

where \(m=\dfrac{1-n}{2}\) and \(Z_m(kz)\) denotes a Bessel function of order \(m\).

Therefore the properties of tapered lines of this type can also be completely investigated.

LITERATURE

1. Everitt, Communication Engineering, N. Y., 1937, Ch. 4 and 5; Guillemin, Communication Networks, N. Y., 1935, v. 2.
2. Nergaard, RCA Rev., 3, 156, 1938; Nergaard and Salzberg, Proc. I. R. E., 579, 1939; Reukema, Elec. Eng., 56, 1002, 1937; King, Proc. I. R. E., 23, 885, 1935; Mason and Sykes, Bell Sys. Techn. J., 16, 275, 1938.

1. Manning and, Rev. Mod. Phys., 12, 215, 1940; W. Kausman, Rev. Mod. Phys., 14, 12, 1942.
2. Eckart, Z. Hochfrequ., 55, 173, 1940; Ballantine, J. Frank. Inst., 203, 561, 1927; Wheeler and Murnaghan, Phil. Mag., 6, 146, 1928; Starr, Proc. I.R.E., 20, 1052 1932; Burrows, Bell. Sys. Tech. J., 17, 555, 1938; Wheeler, Proc. I.R.E., 27, 65, 1939.

The editors ask readers to correct the misprints that slipped into the first part of Condon’s article in UFN, vol. XXVII, pp. 213—264, 1945.

Page	Line	Printed	Should be
219	6 from top	$\mathbf{k}\cdot \mathbf{E}$	$[\mathbf{k}\mathbf{E}]$
»	6 from top	$\mathbf{k}\cdot \mathbf{H}$	$[\mathbf{k}\mathbf{H}]$
»	8 from top	$\mathbf{k}(\mathbf{k}\cdot \mathbf{E})$	$[\mathbf{k}[\mathbf{k}\mathbf{E}]]$
227	7 from bottom	$\operatorname{div}(\mathbf{a}\mathbf{b})$	$\operatorname{div}[\mathbf{a}\mathbf{b}]$
»	5 from bottom	$\operatorname{div}(\mathbf{A}_n\operatorname{rot}\mathbf{A}_m+\operatorname{rot}\mathbf{A}_n\mathbf{A}_m)$	$\operatorname{div}([\mathbf{A}_n,\operatorname{rot}\mathbf{A}_m]+[\operatorname{rot}\mathbf{A}_n,\mathbf{A}_m])$
229	7 from bottom	$\displaystyle \int \operatorname{div}(\mathbf{A}_n\,\operatorname{rd}\mathbf{A}_m)\,dV$	$\displaystyle \int \operatorname{div}[\mathbf{A}_n,\operatorname{rot}\mathbf{A}_m]\,dV$
235	5 from top	$\mathbf{k}\operatorname{grad}_s E_z$	$[\mathbf{k},\operatorname{grad}_s E_z]$
238	formula (5.20)	$\displaystyle \mathbf{k}\operatorname{grad}\frac{\partial U}{\partial z}$	$\displaystyle \left[\mathbf{k},\operatorname{grad}\frac{\partial U}{\partial z}\right]$
239	20 from top	$\mathbf{k}\operatorname{grad}u$	$[\mathbf{k},\operatorname{grad}u]$
241	15 from top	$z=f(\cos hu\cos v+i\sin hu\sin v)$	$z=f(\operatorname{ch}u\cos v+i\operatorname{sh}u\sin v)$
»	18 from top	$\displaystyle \left(\frac{x}{f\cos hu}\right)^2+\left(\frac{y}{f\sin hu}\right)^2=1$	$\displaystyle \left(\frac{x}{f\operatorname{ch}u}\right)^2+\left(\frac{y}{f\operatorname{sh}u}\right)^2=1$
»	11 from bottom	$\displaystyle \cos hu_1=\frac{a}{f}\ \text{and}\ \cos hu_2=\frac{b}{f}$	$\displaystyle \operatorname{ch}u_1=\frac{a}{f}\ \text{and}\ \operatorname{ch}u_2=\frac{b}{f}$
»	8 from bottom	$\displaystyle u_2-u_1=\cos h^{-1}\left(\frac{b}{f}\right)-\cos h^{-1}\left(\frac{a}{f}\right)$	$\displaystyle u_2-u_1=\operatorname{arch}\left(\frac{b}{f}\right)-\operatorname{arch}\left(\frac{a}{f}\right)$
»	formula (5.27), 1st from bottom	$\displaystyle \left[\cos h^{-1}\frac{b}{f}-\cos h^{-1}\frac{a}{f}\right]$	$\displaystyle \left[\operatorname{arch}\left(\frac{b}{f}\right)-\operatorname{arch}\left(\frac{a}{f}\right)\right]$
249	formula (7.18)	$\mathbf{C}\cdot\operatorname{grad}\psi$	$[\mathbf{C},\operatorname{grad}\psi]$
»	formula (7.20)	$\mathbf{k}\operatorname{grad}\psi$	$[\mathbf{k},\operatorname{grad}\psi]$