I'm guessing you're trying to drive some AC motor that was originally designed to run on 50 Hz AC.
I'm assuming you're using two IRS2113 chips, the "left" chip controlling the two FETs connected to the "left" wire going to the motor, and the "right" chip controlling the two FETs connected to the other wire going to the motor. You probably want to check out the IRS2113 datasheet and the FET datasheet(s).
It is your responsibility to never, ever drive both HIN and LIN of the left IRS2113 chip HI at the same time -- the "self-destruct" state.
As JustJeff points out, if both HIN and LIN of the left FET driver chip are ever high at the same time, the FETs connected to it will, in effect, short the rails ("shoot through", "fuse test", etc.), and much unwanted excitement will follow.
(Never, ever drive both HIN and LIN of the right IRS2113 chip HI at the same time, either).
A few people don't use a translation circuit at all; they directly connect the left HIN and LIN and the right HIN and LIN directly to 4 independent output pins of the microcontroller, and hope that the software doesn't have any bugs that would set the FETs to the self-destruct state.
I prefer using a digital logic gate, such as the external NOT gate you mention, to enforce this "avoid the self-destruct state" condition in the motor driver hardware, rather than software.
It looks to me that the IRS2113 (like many other "high-side drivers") uses a boost circuit that can't really produce a 100% "ON" in the high-side transistor; it assumes that software will periodically turn that transistor OFF, so your software might need to limit the maximum PWM duty cycle to a maximum of something like 255 ticks HI + 1 tick LO.
bipolar
The simplest (in hardware) approach is to drive the entire H bridge/motor system in one of two basic states of the H bridge: either
State 0:
The microcontroller PWM output is LOW.
This drives the left LIN and the right HIN, turning the corresponding FETs off.
That output also drives a NOT gate which drives HI the left HIN and the right LIN,
turning those corresponding FETs ON.
This in effect connects the motor's left wire to HI and the motor's right wire to LO.
If things stay in this state long enough, the motor turns in the direction I call "forward".
State 1:
The microcontroller PWM output is HI.
This drives the left LIN and the right HIN, turning the corresponding FETs ON.
...
This in effect connects the motor's left wire to LO and the motor's right wire to HI.
If things stay in this state long enough, the motor turns in the direction I call "reverse".
With a PWM running at reasonable speeds, setting the duty cycle to 50% HI, 50% LO, you end up with the motor stationary (possibly humming a little).
This 2-state (bipolar) system is sometimes called "Locked-Antiphase". a b
trilevel
Trilevel, aka "Sign-Magnitude Drive"
Adding a third state makes things a little more complicated and difficult to debug, but it typically improves (reduces) harmonics and makes the system more power efficient.
State 3:
Some people use a third state turns off all the FETs, letting the motor freewheel.
(If you manage to turn off any 3 of the 4 transistors and leave the other one ON, that also works just as well as a freewheel state).
If things stay in this state long enough,
the motor "stops".
(In some cases, external forces spin the motor in one direction; the direction is not under the control of the electronics).
Alternate State 3:
Some people use a third state that turns on both the lower FETs, turning off both upper FETs, which slows down the motor ("brake").
(Others turn on both the upper FETs, turning off both lower FETs; that also works just as well as a "brake" state).
If things stay in this state long enough,
the motor stops.
(Even when external forces push the motor in one direction,
this "brake" generally causes the motor to stop).
There are a variety of ways to implement digital logic between the microcontroller and the FET driver that supports trilevel drive, but also avoids the self-destructive states.
(Some motor driver chips such as the IXYS IXDN404 include this anti-shoot-through direction/PWM translation circuit, but the IRS2113 does not).
Such translation circuits generally require a "direction" and a "PWM" line from the microcontroller.
The software sets the "direction" to "forward", and then adjusts the duty cycle of the PWM to control the speed from "idle" to "full speed forward".
(The two states of the PWM, in effect, are translated to the two states "State 0 forward" and "State 3 idle").
Much later, the software sets the "direction" to "reverse", and then adjusts the duty cycle of the PWM to control the speed from "idle" to "full speed reverse".
(The two states of the PWM, in effect, are translated to the two states "State 0 forward" and "State 3 idle").
Translation circuits that support tristate motor control:
a b c d e
You might think that these circuits would have 100% LO, 0% HI PWM duty cycle give "idle" (in either direction), and 0% LO, 100% HI give "full speed" (in whatever direction the "direction" line indicates).
But lots of people use a direction/PWM translation circuit that does something more confusing, and then fix it up in software.
Perhaps the simplest such circuit translation uses the PWM to drive the left LIN and a NOT gate that in turn drives the left HIN.
Another general-purpose output on the microcontroller drives the "direction" signal, which drives right LIN and a NOT gate that in turn drives the right HIN.
Best Answer
I looked at the code reference and, as the comments say, it's going to generate a kind of sawtooth - not a sinewave.
What you are expecting is the blue sine-wave. What that Arduino code will give is a short sawtooth (ramping up to 255 and back to -1) followed by 20ms of nothing. It won't even give 50 Hz correctly.
If you did get a sine wave out it would give you a 2.5 V DC output with a 5 V peak-to-peak sinewave on top of it. The DC will bias the transformer which expects the input voltage to be symetrical about 0 V (and not 2.5 V).
5 V p-p = 1.7 V RMS so even if you got rid of your DC you still have a very low voltage to work with.
(You think) you have \$5V \times 1A = 5 W\$ to work with but \$5 V_{p-p} = \frac {5}{2 \cdot \sqrt 2} = 1.7 V_{rms}\$ and \$1 A \cdot 1.7 V_{rms} \to 1.7 W\$. Be aware that even if you got this to work efficiently and stepped up the voltage to 120 V the absolute maximum output would still be 1.7 W which is about 14 mA at 120 V.
I suspect that you hope to use your Arduino to power something that would normally run at 50 Hz. I think the answer is "no".
What you are trying to make is an inverter. Have a look on the web for inverter designs. You will find some very cheap designs but they will have terrible non-sine waveforms and poor voltage regulation. Anything better will require specialist and more expensive components.
Edit after Arduino code update.
Your updated code will give an output like this.
The updated code result.
Now your code now steps up by 1 step and waits 20 ms before doing the next step. It will take \$255 \times 20ms = 5.1s\$ to get to 255 and another 5.1 s to come back down to zero. That's one full cycle in 10.2 s or < 0.1 Hz.
It still has a DC offset in it.