What is the beam steering range of a phased array dependent on? For example, if I use an antenna array of Nx1, with a phase shifter resolution of \$\frac{2\pi}{k}\$ and the antenna elements spaced at \$\frac{\lambda}{2}\$, what is the beam steering range?
Also, the beam steering angle, \$\theta\$, for phase shifts of \$\phi, 2\phi, 3\phi,....\$ and antenna element spacing of \$\frac{\lambda}{2}\$ is \$\theta = sin^{-1}(\frac{\phi}{\pi})\$ right?
I am assuming, when you ask about beam steering range, you are asking if there are any limitations on the directions in which one can steer the main beam of the array.
First, we must have the number of elements in the array \$N>1\$ or else there will be no steerable beam.
If grating lobes are of no concern, then there is no limitation on the beam steering angle, \$\theta\$. The phase at the element, \$\phi_n\$ can then be set to:
$$\phi_n = 2\pi\frac{x_n}{\lambda}\sin\theta$$
where \$x_n\$ is the location of the nth element and \$\lambda\$ is the wavelength.
If the array is uniformly spaced with inter-element spacing \$d\$ this can be written as:
$$\phi_n = 2\pi\frac{nd}{\lambda}\sin\theta = n\Delta\phi$$
where \$\Delta\phi = 2\pi(d/\lambda)\sin\theta\$
Most of the time we do need to consider grating lobes however. Grating lobes occur when the element spacing is greater than half of the wavelength and we steer far enough from broadside that an "aliased" replica of the main beam appears in visible space.
The maximum angle that can be steered from broadside without introducing grating lobes is: $$\theta_{max} = \arcsin\left(\frac{\lambda}{d}-1\right)$$
If \$d/\lambda < 1/2\$ then all steers are grating lobe free. If \$d/\lambda \geq 1\$ then all steers have grating lobes. For all intermediate values there is a maximum grating lobe free steer given by the above expression.
As far as phase shifter resolution goes, in general, the finer the resolution the higher fidelity of the realized patter including steering direction. (See Graphical Investigation of Quantisation Effects of Phase Shifters on Array Patterns)
With a single bit of resolution, the only possible phase states are 0 and 180 degrees. This makes it essentially impossible to steer the beam and results in the inability to distinguish steering left and right. With increasing phase control resolution the steering accuracy is improved. Also, if dithering of phase states is used, steering accuracy will improve as the element count grows.
