User:Graeme Bartlett/phase-shift network

A phase-shift network is an electronic circuit designed to shift the phase of an electronic signal by a desired quantity.

Theory
Causing a phase shift is equivalent to adding a delay:

shifted signal (t) = signal(t + Δ)

equivalent to multiplying

SS(t) = signal(t) cis(Δ) using complex numbers

=signal(t) eiΔ

Application
Audio signals can be shifted by 90° to generate a single sideband modulation.

Electronic signals can be combined in order to produce a difference or sum frequency without producing the other by using 90° shifts.

Implementation
Ladder network

In a simple RC network the input goes into a capacitor and then a resistor. The output is measured across the resistor. The phase shift is given by

ΔΦ = tan−1($1 2πfRC$)

There can be multiple stages cascaded. This can increase stability.

Lattice phase equaliser
A lattice phase equaliser uses capacitors and inductors to achieve a constant impedance but delayed shift.

tan $Φ/2$ = −$ω/ω_{m}$ ; ωm = $1/√LC$

or Φ = 2 tan−1 −2πf√LC

Lattice delay network
A lattice delay network attempts to achieve a constant delay over a frequency range. These also use inductors and capacitors in a ladder arrangement. hey can be designed to have maximally flat group delay, or with controlled passband phase ripple.

Practical limits
Frequency cannot go to zero, otherwise delay becomes infinite.