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A phase shifter changes the phase angle of a signal. Consider a sinusoidal signal given by

v v= asin(wt+f (2.123))

It has the following three representative parameters:

Amplitude, v_a Frequency, ω Phase angle, ϕ

Input

v_i Output

v_o +

A – R

R_f Resistance switching

circuit

FIGURE 2.49 A curve-shaping circuit.

The phase angle represents the time reference (starting point) of the signal. It is an important consid-eration when two or more signal components are compared and also when different time instants of a signal (generally not sinusoidal) are compared. The Fourier spectrum of a signal is presented as its amplitude (magnitude) and the phase angle with respect to frequency.

2.10.1.1 Applications

Phase shifting circuits have many applications. The applications can be classified into two types:

1. Detecting the phase angle of a signal (typically by shifting the phase angle and comparing with a reference signal)

2. Shifting the phase angle of a signal for subsequent use in the application

Phase lead or lag of two quadrature signals generated by a digital transducer determines the direction of motion. In this context, phase determination is used in determining the direction of motion.

Another application of phase angle determination is in system identification where the goal is to obtain an experimental model of a system. When a signal passes through a system, the phase angle of the signal changes due to the system dynamics. Consequently, the phase change provides very useful information not only about the output signal but also about the dynamic characteristics of the system.

Specifically, for a linear constant-parameter system, this phase shift is equal to the phase angle of the fre-quency-response function (i.e., frequency-transfer function) of the system at that particular frequency.

This phase shifting behavior is, of course, not limited to electrical systems and is equally exhibited by other types of systems, including mechanical systems and mixed systems. The phase shift between two signals can be determined by converting the signals into the electrical form (using suitable transducers) and shifting the phase angle of one signal through known amounts using a phase-shifting circuit until the two signals are in phase.

Another application of phase shifters is in signal demodulation. For example, as noted earlier in this chapter, one method of amplitude demodulation involves processing the modulated signal together with the carrier signal. This, however, requires the modulated signal and the carrier signal to be in phase. But, usually, since the modulated signal has already transmitted through hardware with imped-ance characteristics, its phase angle would have changed. Then, it is necessary to shift the phase angle of the carrier until the two signals are in phase, so that demodulation can be performed accurately.

Hence, phase shifters are used in demodulating, for example, the outputs of LVDT displacement sensors.

Phase shifters are used in signal communication (e.g., PM in digital communication and modems) and transmission (e.g., antennas that do not require re-orientation).

2.10.1.2 Analog Phase Shift Hardware

A phase shifter circuit, ideally, should not change the signal amplitude while changing the phase angle by a required amount. Practical analog phase shifters could introduce some degree of amplitude distor-tion (with respect to frequency) as well. A simple phase shifter circuit can be constructed using resistor (R) and capacitor (C) elements. A resistor or a capacitor of such an RC circuit is made fine-adjustable so as to realize variable phase shifting.

An op-amp–based phase shifter circuit is shown in Figure 2.50. We can show that this circuit provides a phase shift without distorting the signal amplitude. The circuit equation is obtained by writing the current balance equations at nodes A and B, as usual, noting that the current through the op-amp leads can be neglected due to high input impedance. Thus,

v v

On simplifying and introducing the Laplace variable s, we get

vi =(t 1s+ )vA (2.124)

and

vB=1 vi+vo

2( ) (2.125)

where, the circuit time constant τ is given by τ = RcC. Since v_A = v_B, as a result of very high gain in the op-amp, we have by substituting Equation 2.125 into Equation 2.124, v_i = (1/2)(τs + 1)(vi + v_o). It follows that the transfer function G(s) of the circuit is given by

v

v G s s

s

o i

= =

-( ) ( + )

( )

1 1

t

t (2.126)

It is seen that the magnitude of the frequency-response function G(jω) is G j( ) (w = 1+t w^{2 2})/( 1+t w^{2 2}), or

G j( )w = 1 (2.127)

and the phase angle of G(jω) is ∠G(jω) = −tan⁻¹τω−tan⁻¹τω, or

ÐG j( )w = -2tan^-1tw= -2tan^-1R Cc w (2.128) As needed, the transfer function magnitude is unity, indicating that the circuit does not distort the signal amplitude over the entire bandwidth. Equation 2.128 gives the phase lead of the output v_o with respect to the input v_i. Note that this angle is negative, indicating that actually a phase lag is introduced by the circuit, as expected. The phase shift can be adjusted by varying the resistance R_c.

2.10.1.3 Digital Phase Shifter

In a digital phase shifter, a digital hardware processor is used to phase shift an incoming sequence of data bits. Digital phase shifters in the form of monolithic IC chips (e.g., a 4 mm package of GaAs 6-bit digital

Input

v_i Output

v_o +

B – R

R

Rc A

C

FIGURE 2.50 A phase shifter circuit.

phase shifter with an integrated CMOS driver, operating frequency range 3.5 GHz, phase shift range 360°, max error 1°, phase shift step 6°, supply voltage ± 8 VDC) for frequency shifting of radio-frequency (RF) signals are commercially available. Their applications include satellite communication, antennas, and active phased array radars. Digital signal sequences and transmitted are received. The phase change of the received signal is used to determine the distance (or the time of flight of the data). Measurement of object deformation using laser holographic interferometry and phase shifting of holographic data frames (software-based) has been reported. Another application is in three-dimensional measurement that uses stereo images and phase shifted fringe patterns.