1.3
Pulse forming lines
1.3.1 Single line
A distributed constant circuit has been frequently used to generate a pulsed power. Fig. 1 shows a schematic circuit of a single line using the distributed constant circuit. A high voltage source charges the single line, which behaves as a capacitor for a slow charging voltage. After completing the charging of the single line, the closing switch S is turned on.
Fig. 1 Single line
Fig. 2 shows the principle of a pulsed power generation from a single line. The charging voltage and the length of the single line are V0 and l, respectively. The closing switch is turned on at t=0. The waves with a voltage V0/2 and a velocity v propagate into two directions, to the load side and to the source side. If the load resistance R is the same as the characteristic impedance of the line Z0, that is, the matching resistor, all the wave energy propagating to the load side is absorbed by the load resistor. The wave propagating to the source side is reflected back to the load side, since the source impedance is usually larger than Z0. The voltage distributions on the single line at different times are easily obtained as shown in Fig. 2.
Fig. 2 Principle of pulsed power generation from a single line.
The waveform of the output voltage on the load is shown in Fig. 3. The voltage becomes half of the charging voltage of the single line, and the pulse width becomes two times of l/v. Since v=1/(me)0.5, the pulse width increases with increasing e.
Fig. 3 Waveform of the output voltage
The strip line in Fig 4 is also used instead of the coaxial line. Three plates are used to decrease the influence of the stray coupling capacitance. Though the parameters such as the inductance and the capacitance per unit length are easily changed by changing the width, the separation and the dielectric material, it is difficult to eliminate the influence of the edge effect.
Fig. 4 Strip line
1.3.2 Blumlein line
Fig. 5 Blumlein line
Blumlein proposed a pulse forming line shown in Fig, 5, which has three coaxial conductors. Since "c" is connected to "a" through an inductor, there is no potential difference between a and c during a slow charging voltage to V0. b becomes a high voltage. The closing switch S is turned on after completing the charging. The wave between a and b (or b and c) behaves like that which is shown in Fig. 2.
Fig. 6 shows the potential distribution on the Blumlein lines. The waves propagating to the load and to the source between a and b are reflected positively at both sides, since the impedances at the both ends are larger than the characteristic impedance of the line. The waves to the load and to the source between b and c are reflected with the same and the opposite polarities, respectively. This is because the impedance at the load and at the source sides are larger and smaller than the characteristic impedance, respectively. The gap switch G remains off till t=l/v, since there is no potential difference between the two electrodes of the gap. The potential difference becomes V0 at t=l/v, and then the gap switch turns on.
Fig. 6 Potential distribution on the Blumlein line. The left figures for a-b and the right figures for b-c
If a matching load is connected, the output voltage on the load is V0 between l/v and 3l/v, as shown in Fig. 7. The output voltage is the same as the charging voltage, on the other hand the output voltage of the single line is half of the charging voltage.
Fig. 7 Output voltage
1.3.3 Stacked line
Fig. 8 Stacked strip line
The high voltage pulse is produced by stacking the coaxial and strip lines. Fig. 8 shows the stacked line of three Blumlein lines. During charging the line at t<0, the direction of the electric field is shown by the arrows with the different directions.
The three closing switches are closed at t=0. The arrows show the same direction at l/v < t < 3l/v from the same consideration shown in Fig. 6. The pulse width of this stacked line is
(1)
Since three Blumlein lines have 6 strip lines, which each has a characteristic impedance of Z0, the total characteristic impedance is 6Z0. The total characteristic impedance Z for n Blumlein lines is
(2)
The output voltage between a and b is
(3)
for the matching load with a resistance of 2nZ0,
(4)
for the load with much higher impedance than Z, since the voltage wave is reflected positively at the load.
For a realistic operation of Fig. 8, the closing switches have to turn on at the same time, and to have a low impedance. Fig. 9 shows the stacked line using one closing switch, where coaxial cables are used. Though the line using one closing switch can avoid the problem of the synchronization of closing switch operation, the impedance of the circuit connecting the switch increases. One solution of the synchronization problem is to use laser triggered spark gaps.
Fig. 9 Stacked coaxial line
1.3.4 Spiral line
Fig. 10 shows the principle of a spiral line, which folds the strip line one time. After charging to V0, the closing switch is turned on at t=0. The electric field becomes at the same direction at l/v < t < 3l/v from the same consideration of Fig. 6. The half of the line length is l. The output voltage between a and b for the open circuit load is 4V0 due to the positive reflection of the voltage wave.
Fig. 10 Principle of spiral line
Fig. 11 Spiral line
The real spiral line is shown in Fig. 11. After charging, the directions of the arrows are shown in A. The output voltage between a and b is zero. The reflection waves at the closing switch propagate into both sides. The direction of the electric field at
t=l/v becomes like the arrows shown at B. After the voltage waves arriving at a and b are reflected with the same polarity, the reverse electric field appears. The direction of the electric field at t=2l/v is shown at C. The output voltage between a and b
increases like stairs to t=2l/v (V=2nV0), and then decreases to t=4l/v (V=0). If the triangle is used instead of the stairs, the waveform of the output voltage is shown as
(5)
(6)
The spiral line is sometimes used as a triggered circuit of the main gaps.