Comparison of Linear and Nonlinear Differentiating Devices for Forming Nanosecond Pulses

R. B. Bakshit

(Presented by the scientific seminar of the Scientific-Research Institute of High Voltages)

The dependence of the magnetic permeability \(\mu(H)\) for many ferrites and ferromagnetic materials has a sharply pronounced maximum, as a result of which, when \(H\) changes rapidly, a voltage pulse appears at the terminals of a coil with a core made of ferromagnetic material. This effect was used in works [1, 2] to obtain short high-voltage pulses; however, in the circuits proposed in [1, 2], the pulses obtained had a small amplitude \((2—8\ \text{kV})\). In work [3] a device was proposed for differentiating high-voltage nanosecond pulses that is free from the shortcomings of the circuits [1, 2]. It consists of a nonlinear ferrite element made in the form of a brass rod built into a coaxial chamber, with ferrite rings of grade F-1000 placed on it.

Fig. 1. a) differentiation circuit with a nonlinear ferrite element; b) schematic diagram of differentiation with a linear inductance.

It is of interest to compare the parameters of the pulse obtained according to the circuit of Fig. 1, a with the parameters of a pulse differentiated according to the usual circuit using a linear inductance (Fig. 1, b). The principal parameters of a high-voltage nanosecond pulse of peaked shape are its amplitude and its duration, measured at the 0.1 level. If the front of the input pulse has an exponential form, then for the image of the output signal we obtain

\[ U_2(p)=\frac{aU_0p}{(p+a)(p+b)}, \tag{1} \]

whence

\[ U_2(t)=\frac{aU_0}{a-b}\left(e^{-bt}-e^{-at}\right), \tag{2} \]

where

\[ b=\frac{z}{2L}, \]

\(Z\) is the characteristic impedance,
\(L\) is the differentiating inductance.

Equating the time derivative \(\dfrac{d u_2(t)}{dt}\) to zero, we obtain the time at which \(u_2(t)\) is maximal:

\[ t=\frac{\ln a-\ln b}{a-b}. \tag{3} \]

Using (3) and (2), for the amplitude value of the output signal differentiated according to the circuit in Fig. 1b, we have

\[ U_{a_2}=\frac{1}{1-\frac{1}{A}} \left(A^{-\frac{1}{A-1}}-A^{-\frac{1}{1-A}}\right), \tag{4} \]

where \(A=\dfrac{a}{b}\).

Thus, the amplitude of the output signal depends on the parameters of the differentiating circuit and on the rise time of the input signal. When the parameter \(b\) is decreased (i.e., when the inductance is increased), the amplitude of the output signal increases, approaching \(U_{a_1}\) exponentially; it is quite clear that, along with the increase in amplitude in this case, the duration of the output signal also increases. Figure 2 shows the dependence of the relative amplitude of the output signal on its duration, calculated at the level \(0.1\,U_{a_2}\); it also gives

Fig. 2. Dependence of the relative amplitude of the output pulse on its duration: 1 — for circuit 1a; 2 — for circuit 1b.

an analogous experimental dependence for a differentiating circuit with a nonlinear element. Both dependences are given for an input pulse with rise time \(t_{\phi}=20\cdot10^{-9}\) sec, since such a pulse can readily be obtained in a circuit with a spark gap operating at atmospheric pressure, or in a circuit with a thyratron [4]. As follows from the graphs in Fig. 2, the duration of the pulse obtained in the circuit with a nonlinear differentiating element, at the same amplitude of the input pulse, is less than when differentiation is performed in a circuit with a linear inductance. For example, for a brass rod of diameter

30 mm and 40 mm long, with 4 rings, 10 mm thick, placed on it, at an amplitude ratio \(\dfrac{U_{a2}}{U_{a1}} = 0.75\), the pulse obtained according to the circuit of Fig. 1a is 5 times shorter than the pulse obtained according to the circuit of Fig. 1b. For steeper input pulses \((t_{\phi1} = 2 \cdot 10^{-9}\ \mathrm{sec})\), obtained by means of a discharger with high nitrogen pressure, [4] experimentally obtained an output pulse of duration \(t_{i2} = 14\ \mu\mathrm{sec}\) at an amplitude of 15 kV and a ratio \(\dfrac{U_{a2}}{U_{a1}} = 0.6\); linear differentiation in this case, at the same duration, gives the ratio \(\dfrac{U_{a2}}{U_{a1}} = 0.265\). The foregoing indicates the substantial advantages of ferrite differentiating devices over ordinary ones for obtaining nanosecond pulses.

REFERENCES

1. K. Kult. Slabopr. Obz. 19, No. 5, p. 292, 1958.
2. S. I. Andreev, M. P. Vanyukov, V. A. Serebryakov. PTE, No. 3, 89, 1962.
3. R. B. Baksht, G. A. Mesyats. PTE, in press.
4. G. A. Vorob’ev, G. A. Mesyats. Technique for Forming High-Voltage Nanosecond Pulses, Gosatomizdat, 1963.