2. ANALYSIS OF THE SHAPER CIRCUIT

The schematic of the shaper circuit is given in Figure 1.

Fig. 1. Circuit of the shaper of high-power rectangular pulses with the controlled stabilization of the pulse peak. С₁ – storage capacitor, К₁ – active switch, С₂ – reference capacitor, D₁ – unilateral conductive switch, R₂ – active load, R₂ – limiting resistance, К₂ – protection switch.

Storage capacitor C ₁ and reference capacitor C ₂ is charging to U ₀₁ and U ₀₂, respectively, by independent suppliers, so that U ₀₂ < U ₀₁. A resistor R ₁ limits the charging current of a capacitor C ₂ and the discharge current of a capacitor C ₁ at a shorted load R ₂. A switch K ₂ plays a protective role for the load and is not compulsory. Reactive elements in the scheme are considered to be small and not considered in the analysis.

The circuit is described by the following system of equations:

(1) $$\left\{ \matrix{U_1 (t) - R_1 \cdot I_1 (t) - U_2 (t) = 0; \hfill \cr U_2 (t) = I_2 (t) \cdot R_2 ; \hfill \cr I_1 (t) = - C_1 \cdot \displaystyle{{dU_1 (t)} \over {dt}}; \hfill \cr I_1 (t) = I_2 + C_2 \cdot \displaystyle{{dU_2 (t)} \over {dt}}. \hfill} \right.$$

The type of solution for this system with respect to U ₁ and U ₂ depends on the discriminant value of the quadratic equation

(2) $$\eqalign{ D & = ({\rm \tau} _1 + {\rm \tau} _2 + {\rm \tau} _3 )^2 - 4 \cdot {\rm \tau} _1 \cdot {\rm \tau} _2 \cr & = ({\rm \tau} _2 + {\rm \tau} _3 - {\rm \tau} _1 )^2 + 4 \cdot {\rm \tau} _1 \cdot {\rm \tau} _3.} $$

For this circuit, at any values of elements D > 0, the solution of a system of Eq. (1) looks in the following way (Korn & Korn, Reference Korn and Korn1968):

(3) $$\left\{ \eqalign{U_{_1} (t) = & \;A_{_1} \cdot \exp ( - s_{_1} \cdot t) + A_{_2} \cdot \exp ( - s_{_2} \cdot t); \cr U_{_2} (t) = & \; A_{_1} \cdot (1 - {\rm \tau} _{_1} \cdot s_{_1} ) \cdot \exp ( - s_{_1} \cdot t) \cr & \quad + A_{_2} \cdot (1 - {\rm \tau} _{_1} \cdot s_{_2} ) \cdot \exp ( - s_{_2} \cdot t).} \right.$$

Here

$$\eqalign{& s_{1,2} = \displaystyle{{({\rm \tau} _2 + {\rm \tau} _3 + {\rm \tau} _1 ) \mp \sqrt D} \over {2 \cdot {\rm \tau} _1 \cdot {\rm \tau} _2}}, \cr & \quad {\rm \tau} _1 = R_1 \cdot C_1, \quad {\rm \tau} _2 = R_2 \cdot C_2, \quad {\rm \tau} _3 = R_2 \cdot C_1 ;} $$

and the integration constants A ₁ and A ₂ are determined from the initial conditions U ₁(0) = U ₀₁, U ₂(0) = U ₀₂:

$$\eqalign{& A_1 = \displaystyle{{U_{02} - U_{01} \cdot (1 - {\rm \tau} _{\rm 1} \cdot s_2 )} \over {{\rm \tau} _1 \cdot (s_2 - s_1 )}}, \cr & \quad A_2 = \displaystyle{{U_{01} \cdot (1 - {\rm \tau} _1 \cdot s_1 ) - U_{02}} \over {{\rm \tau} _1 \cdot (s_2 - s_1 )}}.}$$

If the initial conditions are chosen so that

(4) $$U_{01} - U_{02} \cdot \displaystyle{{R_1 + R_2} \over {R_2}} \gt 0,$$

then after switching K ₁ two process will take place:

• • Additional charging of capacity C ₂ from the capacity C ₁, followed by increasing of the voltage on C ₂ by the amount ΔU;

• • Discharge of capacity C ₁ on the load resistance R ₂; in this case dU ₂/dt ≥ 0.

In the limiting case of the idle running (R ₂ → ∞), the voltage maximum on capacity C ₂ is achieved at t ≫ R ₁ · (C ₁ · C ₂/C ₁ + C ₂) and makes U _max2 = (C ₁ · U ₀₁ + C ₂ · U ₀₂/C ₁ + C ₂). At the given voltage deviation k = (U _max2 − U ₀₂/U ₀₂) = ΔU ₊/U ₀₂, the ratio of C ₁ and C ₂ looks as the following:

$$\displaystyle{{C_2} \over {C_1}} = \displaystyle{1 \over k}\left[ {\displaystyle{{U_{01}} \over {U_{02}}} - 1 - k} \right].$$

Thus, the stabilization mode can be achieved at any ratio of C ₁/C _2, but to obtain significant in terms of duration pulses with small deviation of ∆U, the condition C ₂  $ \gg $  C ₁ should be satisfied.

Qualitatively, the process of the stabilization of the pulse peak occurs as follows. The slice of the pulse peak occurs at a capacitor C ₁ discharge at the level of the reference capacitor C ₂ voltage U ₀₂. The energy corresponding to the sliced pulse is supplied to the capacitor C ₂. The presence of the switcher D ₁ with unilateral conductivity prevents the capacitor discharge on a load R ₂ thereby retains the additional energy incoming to the capacitor C ₂. This energy can be used to preserve the efficiency of the process. In this paper, we do not concretize these opportunities.

3. DURATION AND ACCURACY OF THE PULSE PEAK STABILIZATION

This section discusses two important parameters characterizing the process of stabilization.

For the finite values of R ₂, the condition dU ₂/dt ≤ 0 is executed in the moment of time t _inc, let it be the time of the pulse peak growth:

(5) $$t_{{\rm inc}} = \displaystyle{1 \over {s_2 - s_1}} \cdot \ln \left[ {\displaystyle{{(1 - {\rm \tau}_1 \cdot s_1) - U_{02}/U_{01}} \over {(1 - {\rm \tau}_1 \cdot s_2) - U_{02}/U_{01}}} \cdot \displaystyle{{s_2} \over {s_1}} \cdot \displaystyle{{1 - {\rm \tau}_1 \cdot s_2} \over {1 - {\rm \tau}_1 \cdot s_1}}} \right].$$

The analysis of Eq. (5) shows that at any parameters of the circuit R ₁, R ₂, C ₁, and C ₂, the ratio U ₀₁/U ₀₂ can be selected so that the mode of the low-voltage growth on the load will be realized (stabilization mode).

At the specified excess of the voltage on the load ∆U ₊, the ratio of the circuit parameters is expressed implicitly by the equation:

(6) $$\eqalign{ \displaystyle{{\Delta U_ +} \over {U_{02}}} = & \left[ {1 - \displaystyle{{U_{01}} \over {U_{02}}} \cdot (1 - {\rm \tau} _1 \cdot s_2 )} \right] \cr & \cdot (1 - {\rm \tau} _1 \cdot s_1 ) \cdot \exp ( - s_1 \cdot t_{{\rm inc}} )/{\rm \tau} _1 \cdot s_2 - 1.} $$

The results of numerical calculations of the dependence of t _inc on the ratio R ₂/R ₁ and C ₂/C ₁ are shown in Figures 2 and 3, respectively.

Fig. 2. Dependences of the rise time of the pulse peak t _inc versus ratio R ₂/R ₁.

Fig. 3. Dependences of the rise time of the pulse peak t _inc versus ratio С ₁/С ₂ at ∆U ₊ = 1%.

It should be noted that at the given parameters of the shaper (C ₁) and load (U ₀₂, R ₂), three parameters – R ₁, C ₂, and U ₀₁ remain free. But the system of equations, describing this circuit, contains only two Eq. (2) [or one of the equations of (2) and the equation for t _inc (3)]. Therefore, to analyze the circuit, either one of free parameters, or one of the ratios U ₀₁/U ₀₂, C ₂/C ₁, or R ₂/R ₁ should be specified. The ratio C ₂/C ₁ was a fixed parameter for Figure 2, and R ₂/R ₁ was a fixed parameter for Figure 3.

As is seen from the behavior of the graphs shown in Figure 2, the rise time t _inc is determined by the time constant τ₃ = R ₂.C ₁ at any ratio of C ₂/C ₁, whereas, the ∆U ₊ rise results in the growth of the maximum possible pulse duration for the fixed parameters of the circuit.

The dependences shown in Figure 3 confirm also that the peak rise time is determined by the time constant τ₃. As for the last graph the R ₂.C ₁ is constant for each curve, the dependences have a tendency to saturation, however the capacity C ₂ can be changed.

When the moment of time t _inc is reached, the diode excludes capacity C ₂ from the process of additional charging and only the discharge of capacity C ₁ through resistances R ₁ and R ₂ takes place. When the voltage on capacitor C ₁ falls lower than the value of U ₀₂ − ∆U ₋, the discharger К₂ actuates and shunts the load. Let us indicate the time of the voltage drop from the level of U ₀₂ + ∆U ₊ to U ₀₂ − ∆U ₋ as the time of the pulse peak drop:

(7) $$t_{{\rm decr}} = C_1 \cdot (R_1 + R_2 )\ln \left( {\displaystyle{{U_{02} + \Delta U_ +} \over {U_{02} - \Delta U_ -}}} \right).$$

Thus, the pulse peak within the specified deviation ∆U ₊ ÷ ∆U ₋ makes t _p = t _inc + t _decr. It should be noted that in this approximation, the voltage pulse front time t _f and the drop time t _d are determined by the actuation time of the dischargers К₁ and К₂, respectively, and are negligibly small in comparison with the typical times of the circuit being considered.

Figure 4 shows the calculation curves of the pulse peak form behavior for the case of the constant load R ₂.

Fig. 4. The calculation forms of the pulse peak for the circuit in Figure 1 for various ratios U ₀₁/U ₀₂ (C ₁ = 16.7 μF, C ₂ = 276 µF, R ₁ = 2.5 Ω, R ₂ = 15 Ω). The gates are shown for the deviation of ∆U/U ₀₂ = ±1%. The ratio U ₀₁/U ₀₂, rise time t _inc and drop time t _decr are given in the comments.

As is seen from Figure 4, with the given value of ∆U/U ₀₂ the pulse duration adjustment in wide ranges is achieved by a simple change of U ₀₁/U ₀₂.

4. FEATURES OF THE STABILIZATION AT THE SHORT-TERM CHANGING OF A LOAD

In case of the changing resistance of the load, namely decreasing in time, three variants of the circuit operation are possible:

• • The mode of a complete stabilization failure, at which the load voltage drops lower than the level of U ₂ and does not recover to the required limits for stabilization after the retrieval of the load resistance. In such a case, the capacity C ₂ is excluded from the circuit, and the usual discharge of capacity C ₁ through the resistances R ₁ and R ₂ takes place.

• • The mode of a partial stabilization failure with the subsequent recovery.

• • The mode of continuous stabilization.

The criterion for withdrawal from the stabilization mode is the correlation R ₂<(R ₁/U ₁ (t)/U ₂ (t) − 1). It should be noted that in order to obtain the maximum pulse peak rise time t _inc one should know the nature of the load behavior as precisely as possible, otherwise, the rise time will decrease. In case of decreasing load resistance, it happens due to low charging of the capacity C ₂ to the level of <U ₂ + ∆U owing to deflection of a greater part of current to the load in comparison with the calculation one, and in case of the growing resistance it happens due to the capacity recharge above the specified level.

Figure 5 shows the calculation curves for the above listed modes.

Fig. 5. The calculation forms of the pulse peak for the graded load resistance (on the graph it is shown on the right top) at various ratios U ₀₁/U ₀₂. The values of the circuit elements are the same as for Figure 2. The gates are shown for the deviation ∆U/U ₀₂ = ±1%.

5. RESULTS OF EXPERIMENTAL TESTING OF THE PROTOTYPE SHAPER

This section presents the prototype shaper, designed for the low-energy (<20 kV) long-pulsed (pulse time ≤100 microseconds) ion beams generation. The shaper was realized according to Figure 1, with the following elements: the capacitive storage device C ₁ = 16.7 µF, the reference capacitor C ₂ = 276 µF, R ₂ = 15 Ω, R ₁ = 2.5 Ω. Ignitrons were used as switches K ₁ and K ₂. D₁ is a stack of fast diodes with a total voltage 20 kV. The maximum load current was limited by value 1.3 kA. Charging of capacitors C ₁ and C ₂ (Fig. 1) was carried out by independent rectifiers. Figure 6 shows waveforms U ₂(t) and U ₁(t) in the load and on the capacitor C ₁, respectively, with U ₀₁/U ₀₂ = 2, where U ₀₁ = 10 and U ₀₂ = 5 kV. The amplitude deviation was ΔU/U ₀₂ = 0.85% by the end of the pulse with duration 62.5 µs limited by the switch K ₂ triggering. Obtained data agree well with the calculated values given in Figure 4. After the switch K ₂ is triggered the capacity C ₁ discharges through the load R ₁. It should be noted that at the each pulse the pre-charge of the reference capacitor C ₂ occurs and its voltage becomes equal ΔU + U ₀₂. To restore the initial value of the reference voltage U ₀₂ on the capacitor C ₂ it is necessary to provide an energy recovery in the capacitor C ₁ for reasons of efficiency. This requires the appropriate circuit solutions requiring further consideration.

Fig. 6. The oscillograms of voltage pulses on the load [U ₁(t)] and on capacity [U ₂(t)] for the circuit in Figure 1 at the charging voltage U ₀₁ = 10 and U ₀₂ = 5 kV, time scale is 25 µs/div.