A Pulse Modulator with a Nonuniform Artificial Line

N. I. Vorob'ev

(Presented at the scientific seminar of the Faculty of Physics and Technology)

A modulator with an output pulse transformer is considered. It is proposed to correct the shape of the voltage pulse on the load by replacing the shaping uniform artificial line with a nonuniform one. The possibility of reducing the weight of the pulse transformer is indicated. A method is given for calculating a nonuniform ladder line.

Introduction

The type of modulator considered (Fig. 1) is widely used in various pulse devices whose output is required to produce a rectangular voltage pulse. Ideal shaping two-terminal networks are long lines, but their technical use is impractical because of their large dimensions and very low characteristic impedance [1].

Fig. 1

Usually, uniform ladder lines are used as shaping two-terminal networks (Fig. 2). The use of a ladder line as a—

Fig. 2

… of the forming circuit expresses the desire to replace a line with distributed parameters by an artificial line.

In a number of cases the shape of the pulse across the load may differ substantially from rectangular (Fig. 3). The principal distortions are an increase in the front duration \(t_{\phi}\) and a decrease \(\Delta U\) of the flat portion of the pulse.

Fig. 3

The front duration is determined from the expression [1]:

\[ t_{\phi}=k_{\phi}\cdot \frac{\sqrt{L_{p}\cdot C_{n}}} {\sqrt{1+\dfrac{R_{g}}{R'_{n}}}}, \tag{1} \]

where

\(k_{\phi}\) — coefficient of the active duration of the pulse front,
\(L_{p}\) — leakage inductance of the transformer,
\(C_{n}\) — total value of the parasitic capacitance of the transformer and the load, referred to the primary winding,
\(R_{g}\) — internal resistance of the modulator,
\(R'_{n}\) — load resistance, referred to the primary winding of the transformer.

The decrease of the flat portion of the pulse for \(R'_{n}=R_{g}\) [1] is

\[ \Delta U=\frac{1}{2}I_{k}\cdot R_{g} =40\,\frac{l_{c}\cdot t_{u}\cdot U_{1}\cdot R_{g}} {w_{1}^{2}\cdot S_{c}\cdot \mu_{k}}, \tag{2} \]

where

\(I_{k}\) — apparent magnetizing current of the transformer at \(t=t_{u}\),
\(l_{c}\) — mean length of the magnetic circuit,
\(S_{c}\) — core cross section,
\(w_{1}\) — number of turns of the primary winding of the transformer,
\(\mu_{k}\) — apparent permeability.

From comparison of formulas (1 and 2) it is evident that the requirements of minimum front duration and minimum decrease of the flat portion are contradictory [2]. For one and the same voltage \(U_{1}\) and pulse duration \(t_{u}\), a transformer designed for good reproduction of the flat portion will considerably increase the front duration, while a transformer designed for good reproduction of the front will give a larger decrease of the flat portion of the pulse. As the product \(t_{u}U_{1}\) increases, it becomes increasingly difficult to obtain a good pulse shape.

For a number of applications (as, for example, for magnetron generators), pulses of good rectangular shape are required. The problem of obtaining a rectangular voltage pulse can be solved much more easily if, instead of a uniform artificial line, a nonuniform artificial line is used.

A nonuniform artificial line as a shaping two-terminal network

In recent years, nonuniform lines with distributed parameters have found ever wider application as broadband resistance transformers, delay circuits, filters, coaxial helical lines in centimeter-wave amplifiers, etc. They can also be used as lines that shape pulses of a special form.

Any nonuniform line is characterized fairly completely by three parameters: the series resistance per unit length \(z(x)\), the shunt conductance per unit length \(y(x)\), and the equivalent resistance of the line \(\rho(x)\) at the point \(x\), with allowance for the boundary conditions [3]. The equivalent resistance is defined as the ratio of voltage to current at the given point of the line,

\[ \rho(x)=\frac{u(x)}{i(x)}. \tag{3} \]

Let a nonuniform line (Fig. 4), whose equivalent resistance \(\rho(x)\) varies from point to point and decreases as the coordinate \(x\) increases, be precharged to the voltage \(E\). If the line is now connected to the resistance \(R=\rho(0)\), while at the same time disconnecting it from the voltage source, then the form of the voltage and current waves in the line will be analogous to those shown in Fig. 5.

Fig. 4

Consequently, if the equivalent resistance of the line monotonically decreases (increases) as the coordinate \(x\) increases, then the voltage across the load \(R\) monotonically increases (decreases). The character of the variation of the ordinate \(u(t)\) of the pulse (Fig. 6) will be determined by the character of the variation of \(\rho(x)\).

Instead of nonuniform lines with distributed parameters, because of the shortcomings noted above, it is more convenient in practice to use nonuniform lumped-constant lines (Fig. 7). The wave resistance \(W\) of such a line on the load side is, to a first approximation,

\[ W_1=\sqrt{\frac{L_1}{C_1}}. \tag{4} \]

For matching with the load,

\[ W_1=R'_н, \tag{5} \]

Fig. 5

Visible labels in the diagram: \(x\), \(u=E\), \(t=0\), \(i=0\), \(0<t<\ell/v\), \(t=2\ell/v\), \(i_0\), \(i_n\), \(u_n\).

Fig. 6

Visible labels in the diagram: \(U(t)\), \(t\), \(O\), \(t_u\), \(\rho(0)>\rho(x)\).

Fig. 7

Visible labels in the diagram: \(L_1, L_2, L_3, L_k, L_n\); \(C_1, C_2, C_3, C_k, C_n\).

\[ L_0=\sum_{1}^{n} L_k \]

\[ C_0=\sum_{1}^{n} C_k \]

Fig. 7

and the law of variation of the characteristic impedance \(W_k=f(k)\) (here \(k\) is the number of the line cell) is determined by the required pulse shape.

Owing to considerable mathematical difficulties, in the present article no attempt is made to obtain analytical expressions that would make it possible to select the necessary law of variation of the line parameters as a function of the required pulse shape.

The task was set of experimentally verifying the possibility of using a nonuniform lumped line in a modulator in order to obtain a rectangular pulse at the load whose shape would be better than when a uniform lumped line is used.

The experiments showed that, by using a nonuniform lumped line, it is possible to correct a considerable decrease (up to \(20\text{–}25\%\)) in the flat portion of the pulse, which would occur when a uniform line is used.

Fig. 8 gives an oscillogram of a voltage pulse of duration \(t_u=10\ \mu\text{sec}\) at the load when the pulse is formed by a uniform line, and Fig. 9 shows an oscillogram of the voltage pulse at the load when the pulse is formed by a nonuniform line. For comparison, Fig. 10 gives an oscillogram of a pulse formed by a nonuniform line.

Fig. 8. Relative decrease of the flat portion \(\dfrac{\Delta u}{u}=0.1\), calibration frequency 500 kc/s.

Fig. 9. Relative decrease of the flat portion \(\dfrac{\Delta u}{u}=0\), calibration frequency 500 kc/s.

Thus, the problem is posed in principle in a new way. By using a nonuniform line in the modulator, one can design a pulse transformer while allowing for a considerable decrease of the flat portion of the pulse that would occur with a forming uniform line. It follows from formula (2) that in this case the transformer may have either a smaller number of winding turns \(w\), or a smaller core cross section \(S_c\). It is easy to see that in this case the weight of the transformer is reduced and the steepness of the front of the transformed pulse is increased.

It should be noted that in some cases, when the thermal regime of the transformer is very severe, the use of a nonuniform line can

may prove impossible if there is no way to make the cooling of the transformer more intensive. However, for most cases, when the requirements on the pulse shape are high and the thermal duty of the transformer is not severe, the use of a nonuniform line is expedient.

Especially good results can be obtained by using nonuniform lines in high-voltage modulators, for example those used in the injector devices of a number of charged-particle accelerators, in pulse magnetron generators, etc. The dimensions of pulse transformers in such modulators are determined by the requirements of the electric strength of the insulation. The transformers turn out to be of very large dimensions, and obtaining the desired pulse shape is not always possible.

Fig. 10. Relative rise of the flat portion, \(\Delta u/u\); calibration frequency \(500\ \mathrm{kHz}\).

In conclusion, the design relations for a nonuniform chain line (Fig. 7) should be given:

1. The inductance \(L_1\) and capacitance \(C_1\), for the number of line cells \(n\), are taken the same as for a uniform line [1]

\[ L_1=\frac{R'_n}{2\cdot n}\,t_i \tag{6} \]

\[ C_1=\frac{1}{2\cdot n\cdot R'_n}\,t_i . \tag{7} \]

1. To obtain a pulse of specified duration \(t_i\), the product of the total inductance of the line \(L_0\) by the total capacitance of the line \(C_0\) must be equal to the same product for the case of a uniform line forming a pulse of the same duration.

2. The flattest pulse top is obtained when

\[ W_k=\sqrt{W_{k-1}\cdot W_{k+1}}, \tag{8} \]

where

\[ W_k=\sqrt{\frac{L_k}{C_k}}, \]

and \(k=1, 2, 3,\ldots,n\) is the number of line cells.

The quantities \(W_1\), \(W_k\), and \(W_{k+1}\) are connected by the relation

\[ W_1=\frac{W_k^k}{W_{k+1}^{\,k-1}} . \tag{9} \]

1. The ratio

\[ \frac{W_1-W_n}{W_1} \]

should, in a first approximation, be taken equal to twice the relative decrease of the flat portion of the pulse,

\[ \frac{\Delta W_n}{W_1}=2\,\frac{\Delta U}{U}, \tag{10} \]

where \(\Delta W_n=W_1-W_n>0\).

If the decrease of the flat portion proves to be undercorrected or overcorrected, then \(W_n\) should be taken somewhat smaller or larger than the value calculated from (10).

1. The quantities \(L_k\) and \(C_k\) (taking item 2 into account) are determined from the formulas:

\[ L_k=L_1\frac{W_k}{W_1} \tag{11} \]

\[ C_k=C_1\frac{W_1}{W_k} \tag{12} \]

From formulas (6–12) the magnitudes of all line elements are determined.

Appendix

Example of an electrical calculation of a nonuniform chain line

A calculated and manufactured pulse transformer gives, when transforming a rectangular pulse of duration \(10\ \mu\text{sec}\), formed by a uniform line, a relative drop

\[ \frac{\Delta U}{U}=5\%. \]

It is required to calculate a nonuniform line that would correct this drop of the top while keeping all the other pulse parameters unchanged. The load resistance is \(R'_н=100\ \Omega\), and the number of line cells is \(n=10\).

1. Inductance of the first cell

\[ L_1=\frac{R'_н}{2\cdot n}\cdot t_и =\frac{100\cdot 10^{-5}}{2\cdot 10} =50\ \mu\text{H}. \]

Capacitance of the first cell

\[ C_1=\frac{1}{2\cdot n\cdot R'_н}\cdot t_и =\frac{10^{-5}}{2\cdot 100\cdot 10} =5000\ \text{pF}. \]

1. Wave resistance of the last cell of the line

\[ W_{10}=W_1\left(1-2\cdot\frac{\Delta U}{U}\right) =100(1-2\cdot 0.05)=90\ \Omega. \]

1. The wave resistances of the remaining cells are determined from formula (9), and the inductances and capacitances from formulas (11 and 12). The results of the calculation are collected in Table 1.

Table 1

We check:

\[ L_0=474.5\ \mu\text{H}, \qquad C_0=52760\ \text{pF} \]

\[ L_0\cdot C_0=474.5\times 52760=2.5\cdot 10^7\ \mu\text{H}\cdot\text{pF}. \]

For a uniform line we obtain

\[ L_0\cdot C_0=n^2\cdot L\cdot C =10^2\times 50\times 5000 =2.5\cdot 10^7\ \mu\text{H}\cdot\text{pF}. \]

1. The energy stored in a uniform line,

\[ \frac{C_{oo}\cdot U^{2}}{2}, \]

and the energy stored in a nonuniform line,

\[ \frac{C_{on}\cdot U^{2}}{2}. \]

Consequently, in the nonuniform line the stored energy is greater by a factor of \(\frac{C_{on}}{C_{oo}}\). In our example,

\[ \frac{C_{on}}{C_{oo}}=\frac{52760}{50000}=1.06, \]

i.e., 6% greater. This energy is mainly dissipated in the pulse transformer.

Literature

1. Itskhoki, Ya. S. Pulse Engineering. Publishing House “Soviet Radio,” 1949.
2. Matkhanov, P. N. “On the Engineering Calculation of High-Power Pulse Transformers.” Radio Engineering, No. 5, 1951.
3. Ilyin, V. A. “Long Lines with Parameters Varying along the Length.” Electricity, No. 2, 1950.