4.3 FEEDBACK DEMODULATORS: THE PHASE-LOCKED LOOP
4.3.1 Phase-Locked Loops for FM and PM Demodulation
A block diagram of a PLL is shown in Figure 4.18. The basic PLL contains four basic elements.
These are 1. Phase detector 2. Loop filter
3. Loop amplifier (assume𝜇 = 1) 4. Voltage-controlled oscillator (VCO).
In order to understand the operation of the PLL, assume that the input signal is given by 𝑥𝑟(𝑡) = 𝐴𝑐cos[2𝜋𝑓𝑐𝑡 + 𝜙(𝑡)] (4.95) and that the VCO output signal is given by
𝑒0(𝑡) = 𝐴𝑣sin[2𝜋𝑓𝑐𝑡 + 𝜃(𝑡)] (4.96) (Note that these are in phase quadrature.) There are many different types of phase detectors, all having different operating properties. For our application, we assume that the phase detector is a multiplier followed by a lowpass filter to remove the second harmonic of the carrier. We also assume that an inverter is present to remove the minus sign resulting from the multiplication.
With these assumptions, the output of the phase detector becomes 𝑒𝑑(𝑡) = 1
2 𝐴𝑐𝐴𝑣𝐾𝑑sin[𝜙(𝑡) − 𝜃(𝑡)] = 1
2 𝐴𝑐𝐴𝑣𝐾𝑑sin[𝜓(𝑡)] (4.97)
Phase detector
Loop f ilter
VCO
Loop amplif ier
xr(t) ed(t)
ev(t) e0(t)
Demodulated output
Figure 4.18
Phase-locked loop for demodulation of FM.
where𝐾𝑑 is the phase detector constant and𝜓(𝑡) = 𝜙(𝑡) − 𝜃(𝑡)is the phase error. Note that for small phase error the two inputs to the multiplier are approximately orthogonal so that the result of the multiplication is an odd function of the phase error𝜙(𝑡) − 𝜃(𝑡). This is a necessary requirement so that the phase detector can distinguish between positive and negative phase errors. This illustrates why the PLL input and VCO output must be in phase quadrature.
The output of the phase detector is filtered, amplified, and applied to the VCO. A VCO is essentially a frequency modulator in which the frequency deviation of the output,𝑑𝜃∕𝑑𝑡, is proportional to the VCO input signal. In other words,
𝑑𝜃
𝑑𝑡 = 𝐾𝑣𝑒𝑣(𝑡)rad∕𝑠 (4.98)
which yields
𝜃(𝑡) = 𝐾𝑣∫
𝑡𝑒𝑣(𝛼)𝑑𝛼 (4.99)
The parameter𝐾𝑣is known as theVCO constant and is measured in radians per second per unit of input.
From the block diagram of the PLL it is clear that
𝐸𝑣(𝑠) = 𝐹 (𝑠)𝐸𝑑(𝑠) (4.100)
where 𝐹 (𝑠) is the transfer function of the loop filter. In the time domain the preceding expression is
𝑒𝑣(𝛼) =∫
𝑡𝑒𝑑(𝜆)𝑓(𝛼 − 𝜆)𝑑𝜆 (4.101)
which follows by simply recognizing that multiplication in the frequency domain is con- volution in the time domain. Substitution of (4.97) into (4.101) and this result into (4.99) gives
𝜃(𝑡) = 𝐾𝑡∫
𝑡
∫
𝛼sin[𝜙(𝜆) − 𝜃(𝜆)]𝑓(𝛼 − 𝜆)𝑑𝜆𝑑𝛼 (4.102)
where𝐾𝑡is the total loop gain defined by 𝐾𝑡= 1
2 𝐴𝑣𝐴𝑐𝐾𝑑𝐾𝑣 (4.103)
Equation (4.102) is the general expression relating the VCO phase𝜃(𝑡)to the input phase𝜙(𝑡).
The system designer must select the loop filter transfer function𝐹 (𝑠), thereby defining the filter impulse response𝑓(𝑡), and the loop gain𝐾𝑡. We see from (4.103) that the loop gain is a function of the input signal amplitude𝐴𝑣. Thus, PLL design requires knowledge of the input signal level, which is often unknown and time varying. This dependency on the input signal level is typically removed by placing a hard limiter at the loop input. If a limiter is used, the loop gain𝐾𝑡is selected by appropriately choosing𝐴𝑣,𝐾𝑑, and𝐾𝑣, which are all parameters of the PLL. The individual values of these parameters are arbitrary so long as their product gives the desired loop gain. However, hardware considerations typically place constraints on these parameters.
Loop f ilter
Amplif ier +
– (t)
θ(t)
sin ( ) Σ
Phase detector
Demodulated output Kv t( )dt
AvAcKd 1 2
ed(t)
ev(t) ϕ
Figure 4.19
Nonlinear PLL model.
Loop f ilter
Loop amplif ier +
−
(t) Σ
Phase detector
Demodulated output Kv t( )dt
AvAcKd 1 2 ϕ
θ(t)
Figure 4.20 Linear PLL model.
Equation (4.102) defines the nonlinear model of the PLL, having a sinusoidal nonlin- earity.2This model is illustrated in Figure 4.19. Since (4.102) is nonlinear, analysis of the PLL using (4.102) is difficult and often involves a number of approximations. In practice, we typically have interest in PLL operation in either the tracking mode or in the acquisition mode. In the acquisition mode the PLL is attempting to acquire a signal by synchronizing the frequency and phase of the VCO with the input signal. In the acquisition mode of operation, the phase errors are typically large, and the nonlinear model is required for analysis.
In the tracking mode, however, the phase error𝜙(𝑡) − 𝜃(𝑡)is typically small the linear model for PLL design and analysis in the tracking mode can be used. For small phase errors the sinusoidal nonlinearity may be neglected and the PLL becomes a linear feedback system.
Equation (4.102) simplifies to the linear model defined by 𝜃(𝑡) = 𝐾𝑡∫
𝑡
∫
𝛼[𝜙(𝜆) − 𝜃(𝜆)]𝑓(𝛼 − 𝜆)𝑑𝜆𝑑𝛼 (4.104)
The linear model that results is illustrated in Figure 4.20. Both the nonlinear and linear models involve𝜃(𝑡)and𝜙(𝑡)rather than𝑥𝑟(𝑡)and𝑒0(𝑡). However, note that if we know𝑓𝑐, knowledge of𝜃(𝑡)and𝜙(𝑡)fully determine𝑥𝑟(𝑡)and𝑒0(𝑡), as can be seen from (4.95) and (4.96). If the
2Many nonlinearities are possible and used for various purposes.
Table 4.3 Loop Filter Transfer Functions PLL order Loop filter transfer function,F(s)
1 1
2 1 + 𝑎
𝑠 = (𝑠 + 𝑎)∕𝑠
3 1 + 𝑎
𝑠 + 𝑏
𝑠2 = (𝑠2+ 𝑎𝑠 + 𝑏)∕𝑠2
PLL is in phase lock,𝜃(𝑡) ≅ 𝜙(𝑡), and it follows that, assuming FM, 𝑑𝜃(𝑡)
𝑑𝑡 ≅ 𝑑𝜙(𝑡)
𝑑𝑡 = 2𝜋𝑓𝑑𝑚(𝑡) (4.105)
and the VCO frequency deviation is a good estimate of the input frequency deviation, which is proportional to the message signal. Since the VCO frequency deviation is proportional to the VCO input𝑒𝑣(𝑡), it follows that the input is proportional to𝑚(𝑡)if (4.105) is satisfied. Thus, the VCO input,𝑒𝑣(𝑡), is the demodulated output for FM systems.
The form of the loop filter transfer function𝐹 (𝑠)has a profound effect on both the tracking and acquisition behavior of the PLL. In the work to follow we will have interest in first-order, second-order, and third-order PLLs. The loop filter transfer functions for these three cases are given in Table 4.3. Note that the order of the PLL exceeds the order of the loop filter by one.
The extra integration results from the VCO as we will see in the next section. We now consider the PLL in both the tracking and acquisition mode. Tracking mode operation is considered first since the model is linear and, therefore, more straightforward.