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KrisBlueNZ
Sadly passed away in 2015
Hi and welcome to the Electronics Point forums 
Note: It's not normal to connect IC pins together by drawing wires on top of the IC symbol. Besides, it looks a bit like a swastika on your 555, and the 555 was designed by a Swiss guy, not a German!
If you want a bipolar output, there are two ways:
(a) generate a negative rail - this can be done with a charge pump IC like the ICL7665 (very limited current is available from the rail) or an inverting switching converter;
(b) measure the voltage with reference to a voltage point between the supply rails.
What do you want to achieve? There may be a simpler way to do it.
Note: It's not normal to connect IC pins together by drawing wires on top of the IC symbol. Besides, it looks a bit like a swastika on your 555, and the 555 was designed by a Swiss guy, not a German!
If you want a bipolar output, there are two ways:
(a) generate a negative rail - this can be done with a charge pump IC like the ICL7665 (very limited current is available from the rail) or an inverting switching converter;
(b) measure the voltage with reference to a voltage point between the supply rails.
What do you want to achieve? There may be a simpler way to do it.
KrisBlueNZ
Sadly passed away in 2015
If you want to feed the primary of a step-up transformer with a bipolar square wave, the best way is to drive it using an H-bridge. This is an arrangement of four switching devices (transistors or MOSFETs), like two amplifier output stages, with the load connected between them, forming an "H" shape. The same idea as a bridge-tied load in audio applications. But each output stage is either high or low, i.e. digital, not analogue.
In your case, you want the two sides of the H-bridge to be driven to opposite states. When one side of the H-bridge is high, the other side is low, and voltage is applied across the transformer primary in one direction. On the other half-cycle, the first output is low and the other is high, and voltage is applied across the primary in the opposite direction. Running from a 12V supply you effectively get 24V peak-to-peak across the primary.
H-bridges are available as single ICs with current ratings up to around 5A. More powerful devices are available but they seem to be very expensive. Have a look at the Digikey selection guide: http://www.digikey.com/product-sear...mic-motor-and-fan-controllers-drivers/2556626
In the "Output configuration" column, select "Full H-Bridge (1) Single" and "Half Bridge (2) Dual". (Ctrl-click to select multiple lines.)
In the "Current - Output" column, select the current range you need.
In the "Mounting type" column, if you don't want to use SMT (surface-mount technology), select "Through Hole" only.
Then check the "In Stock" box and click "Apply Filters".
Each line in the selection table has a direct link to the IC's data sheet, indicated with the PDF symbol.
Of course, you can make an H-bridge with four discrete switching components - MOSFETs are best because they are easy to drive and they can have extremely low ON-resistance. But an integrated device is much easier to work with.
When there's an inductive load involved, it's normal to have a diode across each switching device. Many of these ICs have diodes built-in. Some of the older ones require external diodes.
Using an H-bridge you also have the option of driving the transformer with a so-called "modified sine wave". Google the term for more information.
In your case, you want the two sides of the H-bridge to be driven to opposite states. When one side of the H-bridge is high, the other side is low, and voltage is applied across the transformer primary in one direction. On the other half-cycle, the first output is low and the other is high, and voltage is applied across the primary in the opposite direction. Running from a 12V supply you effectively get 24V peak-to-peak across the primary.
H-bridges are available as single ICs with current ratings up to around 5A. More powerful devices are available but they seem to be very expensive. Have a look at the Digikey selection guide: http://www.digikey.com/product-sear...mic-motor-and-fan-controllers-drivers/2556626
In the "Output configuration" column, select "Full H-Bridge (1) Single" and "Half Bridge (2) Dual". (Ctrl-click to select multiple lines.)
In the "Current - Output" column, select the current range you need.
In the "Mounting type" column, if you don't want to use SMT (surface-mount technology), select "Through Hole" only.
Then check the "In Stock" box and click "Apply Filters".
Each line in the selection table has a direct link to the IC's data sheet, indicated with the PDF symbol.
Of course, you can make an H-bridge with four discrete switching components - MOSFETs are best because they are easy to drive and they can have extremely low ON-resistance. But an integrated device is much easier to work with.
When there's an inductive load involved, it's normal to have a diode across each switching device. Many of these ICs have diodes built-in. Some of the older ones require external diodes.
Using an H-bridge you also have the option of driving the transformer with a so-called "modified sine wave". Google the term for more information.
