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Here is a circuit using two 556 (dual 555) timers that I believe meets your needs. The 556 is available in a 14-pin DIP package.

The top half of the first 556 (IC1) acts as a Schmitt trigger to clean up any noise bounce from the pushbutton.

The bottom half of IC1 provides a 1/4 second output. It goes high when the pushbutton is pressed. It's output is fed to the solenoid K1. I have the solenoid connected to 5v, but you can connect the top end to 12v or whatever voltage is needed. When the output falls, it riggers the next timer.

The top half of the second 556 (IC2) provides a one-second delay between the release of the solenoid and the beginning of the pulsed output to the spark gap generator. The bottom half of IC2 provides a 1/4 second output. It goes high when the one second delay is over.

I didn't know what that interface to the spark gap generator looks like so I just show an output line ("To Spark Gap Gen"). You could add another NPN transistor interface if you like. The output to the spark gap generator is also set at 1/4 second.

The times are all easily adjustable by modifying either the resistor values or capacitor values or both. C1/R1 controls the duration of the output to the solenoid (currently 1/4 second); R2/C3 controls the delay between the ending of the pulse to the solenoid and the beginning of the pulse to the spark gap generator (currently 1 second); and R3/C4 controls the duration of the output to the spark gap generator (currently 1/4 second).

I used this calculator to figure out the values of the resistors and caps needed.

By making any of these fixed resistor trimpots, you can adjust the timing to whatever you need.

I suggest using tantalum caps instead of electrolytic since you can twice as good tolerance (5% vs 10%).

You will also want to add 0.1 µF bypass caps between the Vcc (+5v) and GND leads (14 and 7) of each IC.

Here is a circuit using two 556 (dual 555) timers that I believe meets your needs. The 556 is available in a 14-pin DIP package.

The top half of the first 556 (IC1) acts as a Schmitt trigger to clean up any noise bounce from the pushbutton.

The bottom half of IC1 provides a 1/4 second output. It goes high when the pushbutton is pressed. It's output is fed to the solenoid K1. I have the solenoid connected to 5v, but you can connect the top end to 12v or whatever voltage is needed. When the output falls, it riggers the next timer.

The top half of the second 556 (IC2) provides a one-second delay between the release of the solenoid and the beginning of the pulsed output to the spark gap generator. The bottom half of IC2 provides a 1/4 second output. It goes high when the one second delay is over.

I didn't know what that interface to the spark gap generator looks like so I just show an output line ("To Spark Gap Gen"). You could add another NPN transistor interface if you like. The output to the spark gap generator is also set at 1/4 second.

As shown in the timing diagram, the pushbutton can be pressed either for less time than the 1/4 second delay for the solenoid, or greater; in the lattter case the 556 timer will not re-trigger.

The times are all easily adjustable by modifying either the resistor values or capacitor values or both. C1/R1 controls the duration of the output to the solenoid (currently 1/4 second); R2/C3 controls the delay between the ending of the pulse to the solenoid and the beginning of the pulse to the spark gap generator (currently 1 second); and R3/C4 controls the duration of the output to the spark gap generator (currently 1/4 second).

I used this calculator to figure out the values of the resistors and caps needed.

By making any of these fixed resistor trimpots, you can adjust the timing to whatever you need.

I suggest using tantalum caps instead of electrolytic since you can twice as good tolerance (5% vs. 10%).

You will also want to add 0.1 µF bypass caps between the Vcc (+5v) and the GND leads (14 and 7) of each IC.

Here is a circuit using two 556 (dual 555) timers that I believe meets your needs. The 556 is available in a 14-pin DIP package.

The top half of the first 556 (IC1) acts as a Schmitt trigger to clean up any noise bounce from the pushbutton.

The bottom half of IC1 provides a 1/4 second output. It goes high when the pushbutton is pressed. It's output is fed to the solenoid K1. I have the solenoid connected to 5v, but you can connect the top end to 12v or whatever voltage is needed. When the output falls, it riggers the next timer.

The top half of the second 556 (IC2) provides a one-second delay between the release of the solenoid and the beginning of the pulsed output to the spark gap generator. The bottom half of IC2 provides a 1/4 second output. It goes high when the one second delay is over.

I didn't know what that interface to the spark gap generator looks like so I just show an output line ("To Spark Gap Gen"). You could add another NPN transistor interface if you like. The output to the spark gap generator is also set at 1/4 second.

The times are all easily adjustable by modifying either the resistor values or capacitor values or both. C1/R1 controls the duration of the output to the solenoid (currently 1/4 second); R2/C3 controls the delay between the ending of the pulse to the solenoid and the beginning of the pulse to the spark gap generator (currently 1 second); and R3/C4 controls the duration of the output to the spark gap generator (currently 1/4 second).

I used this calculator to figure out the values of the resistors and caps needed.

By making any of these fixed resistor trimpots, you can adjust the timing to whatever you need.

I suggest using tantalum caps instead of electrolytic since you can twice as good tolerance (5% vs 10%).

Here is a circuit using two 556 (dual 555) timers that I believe meets your needs. The 556 is available in a 14-pin DIP package.

The top half of the first 556 (IC1) acts as a Schmitt trigger to clean up any noise bounce from the pushbutton.

The bottom half of IC1 provides a 1/4 second output. It goes high when the pushbutton is pressed. It's output is fed to the solenoid K1. I have the solenoid connected to 5v, but you can connect the top end to 12v or whatever voltage is needed. When the output falls, it riggers the next timer.

The top half of the second 556 (IC2) provides a one-second delay between the release of the solenoid and the beginning of the pulsed output to the spark gap generator. The bottom half of IC2 provides a 1/4 second output. It goes high when the one second delay is over.

I didn't know what that interface to the spark gap generator looks like so I just show an output line ("To Spark Gap Gen"). You could add another NPN transistor interface if you like. The output to the spark gap generator is also set at 1/4 second.

The times are all easily adjustable by modifying either the resistor values or capacitor values or both. C1/R1 controls the duration of the output to the solenoid (currently 1/4 second); R2/C3 controls the delay between the ending of the pulse to the solenoid and the beginning of the pulse to the spark gap generator (currently 1 second); and R3/C4 controls the duration of the output to the spark gap generator (currently 1/4 second).

I used this calculator to figure out the values of the resistors and caps needed.

By making any of these fixed resistor trimpots, you can adjust the timing to whatever you need.

I suggest using tantalum caps instead of electrolytic since you can twice as good tolerance (5% vs 10%).

You will also want to add 0.1 µF bypass caps between the Vcc (+5v) and GND leads (14 and 7) of each IC.

Here is a circuit using two 556 (dual 555) timers that I believe meets your needs. The 556 is available in a 14-pin DIP package.

The top half of the first 556 (IC1) acts as a Schmitt trigger to clean up any noise bounce from the pushbutton.

The bottom half of IC1 provides a 1/4 second output. It goes high when the pushbutton is pressed. It's output is fed to the solenoid K1. I have the solenoid connected to 5v, but you can connect the top end to 12v or whatever voltage is needed. When the output falls, it riggers the next timer.

The top half of the second 556 (IC2) provides a one-second delay between the release of the solenoid and the beginning of the pulsed output to the spark gap generator. The bottom half of IC2 provides a 1/4 second output. It goes high when the one second delay is over.

I didn't know what that interface to the spark gap generator looks like so I just show an output line ("To Spark Gap Gen"). You could add another NPN transistor interface if you like. The output to the spark gap generator is also set at 1/4 second.

The times are all easily adjustable by modifying either the resistor values or capacitor values or both. C1/R1 controls the duration of the output to the solenoid (currently 1/4 second); R2/C3 controls the delay between the ending of the pulse to the solenoid and the beginning of the pulse to the spark gap generator (currently 1 second); and R3/C4 controls the duration of the output to the spark gap generator (currently 1/4 second).

I used this calculator to figure out the values of the resistors and caps needed.

By making any of these fixed resistor trimpots, you can adjust the timing to whatever you need.

I suggest using tantalum caps instead of electrolytic since you can twice as good tolerance (5% vs 10%).

Here is a circuit using two 556 (dual 555) timers that I believe meets your needs. The 556 is available in a 14-pin DIP package.

The top half of the first 556 (IC1) acts as a Schmitt trigger to clean up any noise bounce from the pushbutton.

The bottom half of IC1 provides a 1/4 second output. It goes high when the pushbutton is pressed. It's output is fed to the solenoid K1. I have the solenoid connected to 5v, but you can connect the top end to 12v or whatever voltage is needed. When the output falls, it riggers the next timer.

The top half of the second 556 (IC2) provides a one-second delay between the release of the solenoid and the beginning of the pulsed output to the spark gap generator. The bottom half of IC2 provides a 1/4 second output. It goes high when the one second delay is over.

I didn't know what that interface to the spark gap generator looks like so I just show an output line ("To Spark Gap Gen"). You could add another NPN transistor interface if you like. The output to the spark gap generator is also set at 1/4 second.

The times are all easily adjustable by modifying either the resistor values or capacitor values or both. C1/R1 controls the duration of the output to the solenoid (currently 1/4 second); R2/C3 controls the delay between the ending of the pulse to the solenoid and the beginning of the pulse to the spark gap generator (currently 1 second); and R3/C4 controls the duration of the output to the spark gap generator (currently 1/4 second).

I used this calculator to figure out the values of the resistors and caps needed.

By making any of these fixed resistor trimpots, you can adjust the timing to whatever you need.

I suggest using tantalum caps instead of electrolytic since you can twice as good tolerance (5% vs 10%).