Real world use case for a battery with phase?

Question

Referencing the following exercise by Dan: $9\rm V$ Battery with $45^\circ$ phase, I'm curious on what this would be used for in real world applications?

Answer 1

This is a fun exercise for working with complex signal processing. I use this question to point out the confusion we get with conflating phase with time delay, and makes it clear with complex signal processing such as the Fourier Transform how any of the bins can have a magnitude and phase, including the "DC bin". Often baseband equivalent signals are used where the carrier becomes "DC" facilitating further analysis and simulation. It also helps to point out how we do implement hardware with such complex signals in practice, using two datapaths (just as we need two real numbers to represent a complex number on paper)!

A big take-away when working with complex signals is that phase is a rotation on the complex plane (I recommend not associating it with two sine-waves offset in time-- it's true that such a delay will cause the rotation I describe, but those stuck in that thinking will believe then that it is impossible to create in hardware a signal that is constant with time that also has phase). A time delay results in a linear phase in frequency, but that is not the only way to induce a phase rotation.

A "DC signal with phase" is used in practice anywhere we want to add a static phase offset (the "static" part makes it DC, as in not changing with time). One example of this is an analog RF phase shifter using a vector modulator such as the Analog Devices HMC630 shown in the block diagram below:

A vector modulator has two ports labeled "I" and "Q", where "I" represents "In - phase" and "Q" represents "Quadrature". Each of the ports can take a DC signal, and therefore together represent a complex DC signal that we would apply as $I+jQ$. The device will upconvert this complex DC signal to the RF carrier, which is the signal applied differentially to IN and /IN ports. Thus if we rotate our DC signal x degrees, x degrees will also be added the RF carrier's phase appearing at the RFOUT port. The way this works in practice is the multipliers shown serve as biphase attenuators based on the voltage applied to I or Q. It will control the amplitude of the incident RF signal, and if made negative, it will flip the RF signal 180° in addition to the amplitude control based on magnitude. The RF signal is split in quadrature, each leg is adjusted with the biphase attenuation, and then the two are combined resulting is a full 360° phase and amplitude control corresponding to $I + jQ$ applied. We can use one of these at each antenna input to an antenna array, and with a static phase and amplitude setting on each antenna, form a static beam pointing in a particular direction (no one would ever in practice refer to these as "DC batteries" controlling each antenna, but my point is they are indeed complex values that are constant with time-- so DC!).

The above implementation when shown in complex form is functionally equivalent to this block diagram below, which (once understood) demonstrates the significant simplifying convenience of the complex representations (the exponents simply sum in the product). This is also why I consider single frequency tones as spinning phasors of the form $e^{j\omega t}$ and not sinusoids which complicate everything as consisting of two such tones.

The "Hilbert" block creates the "Analytic Signal" and is implemented in practice with a 90 degree quadrature splitter, which produces two outputs $\cos(\omega_c t)$ and $\sin(\omega_c t)$ which I represent above with the thicker line as a single complex datapath as $\cos(\omega_c t)+j\sin(\omega_c t)= e^{j\omega_c t}$. The multiplier shown above is a full complex multiplier, which would require four multipliers and two adders generally as a product of the real and imaginary terms as $(I_1+Q_1)(I_2-jQ_2)= (I_1I_2 + Q_1Q_2) + j(I_2Q_1-I_1Q_2)$, where $I_1$, $Q_1$ are the real and imaginary ports from the output of the Hilbert, and $I_2$, $Q_2$ is our "DC phase". Since we in this case only need the real of the result (as a passband signal), we only need two of the multipliers representing the real terms of the complete product: $I_1I_2 + Q_1Q_2$, matching the implementation shown.

“DC offset” in demodulation is another example of DC with phase as demonstrated for QAM in this link:

https://dsp.stackexchange.com/a/40740/21048

Answer 2

In a stepper motor driver.

A stepper motor runs on two-phase alternating current. The phases are usually called A and B, but they can be interpreted as real and imaginary part of a complex rotation. The frequency is variable, including coming to a full stop and rotating backwards. If you want the motor stopped in a specific position, you need a DC voltage with a phase angle.