Page 16 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL BUILDING BLOCKS Ronald E. Best 13
scaler. We will discuss the properties of phase detectors in Sec. 2.4, the loop filters in Sec. 2.5,
controlled oscillators in Sec. 2.6, and down scalers in Sec. 2.7.

Phase Detectors

A phase detector is a circuit capable of delivering an output signal that is proportional to the
phase difference between its two input signals u and u ′ (cf. Fig. 2.1). Many circuits could
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be applied. In mixed signal PLLs, mainly four types of phase detectors are used. The first
phase detector in the history of the PLL was the linear multiplier (also referred to as four-
quadrant multiplier). When the PLL moved into digital territory, digital phase detectors
become popular, such as the EXOR gate, the edge-triggered JK-flipflop and the so-called
phase-frequency detector (PFD). Let us start with the discussion of the multiplier phase
detector.

Type 1: Multiplier phase detectors

The multiplier phase detector is used exclusively in linear PLLs (LPLLs). In a LPLL, the input
signal u is mostly a sine wave and is given by
1

(2.10a)

where U is the amplitude of the signal, ω is its radian frequency, and θ its phase. The
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second input signal u ′ (cf. also Fig. 2.1) is usually a symmetrical square wave signal
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(sometimes also called Walsh function) and is given by
(2.10b)

where rect stands for “rectangular” (square wave), and U is the amplitude, ω ′ the
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radian frequency and θ ′ the phase. These signals are shown in Fig. 2.3. The dashed curve in
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Fig. 2.3a is a sine wave having a phase of θ = 0; the solid line has a nonzero phase θ . For
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1
simplicity, we assume here that the phase is constant over time. The dashed curve in Fig. 2.3b
shows a symmetrical square wave having a phase θ ′ = 0; the solid line has a nonzero phase.
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The output signal of the four-quadrant multiplier is obtained by multiplying the signals u and
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u ′. To simplify the analysis, the square wave signal is replaced by its Fourier series. For
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u ′(t), we then get
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(2.11)

The first term in square brackets is the fundamental component; the remaining terms are
odd harmonics. For the output signal u (t), therefore we get
d