From Tracking Filters to Demodulation: Exploring the Applications of PLLs

Phase-locked loops (PLLs) are incredibly versatile circuits. Their applications span communications, control systems, signal processing, and power systems. Here, we'll limit ourselves to discussing the first of these domains. Let's jump right in.

 

Tracking Filters

The PLL can serve as a dynamic filter that adapts to the input signal. Consider the basic PLL configuration shown in Figure 1.

 

Figure 1. The basic PLL.

 

Due to its feedback, the PLL tracks the phase and frequency of its input. This tracking works well when phase variations are gradual. If the phase variations are too quick, the PLL cannot track them due to the loop's finite bandwidth. As a result, the PLL acts as a lowpass filter for phase variations.

To understand the practical significance of this, think about a satellite communication system. As the satellite moves across the sky, the Doppler effect alters the carrier frequency detected by the ground receiver.

The PLL is capable of adjusting to these gradual frequency changes, allowing the system to maintain a stable connection. If a narrowband filter were employed instead of a PLL for recovering the input carrier, the signal would soon drift beyond the limits of the narrow passband, compromising the feasibility of communication.

The PLL follows slow changes like Doppler shifts or limited frequency variations caused by modulation schemes. At the same time, it filters out high-frequency noise and jitter.

The PLL isn't a linear filter. It also discards amplitude information of the input. However, it effectively suppresses rapid fluctuations in the input frequency. The loop bandwidth determines the PLL's tracking capability: a narrow bandwidth improves noise rejection but limits tracking speed, whereas a wide bandwidth allows faster tracking but admits more noise.

We can use the linear model of the loop to determine its transfer function and bandwidth. For instance, for a PLL using a simple lag filter, the 3-dB bandwidth can be approximated by:

$$BW ~\propto~ \sqrt{K_0 \omega_f}$$

Equation 1.

 

In other words, the bandwidth of the PLL-based tracking filter is proportional to the square root of the loop filter's cut-off frequency (ω_f) times the loop gain constant (K₀).

The PLL functions as a bandpass filter that is centered around the frequency of the VCO, which matches the frequency of the signal being tracked. However, some authors describe the PLL as a lowpass filter centered at the VCO frequency. This interpretation stems from the fact that we commonly depict the PLL transfer function (Figure 2) as a lowpass characteristic.

 

Figure 2. Typical magnitude response of a PLL with a lag filter for various values of the damping factor.

 

The above graph shows that the PLL efficiently tracks slow phase changes while suppressing rapid fluctuations. Since the PLL exhibits this behavior for both positive and negative phase variations, the actual response is bandpass rather than lowpass.

In this section, we viewed the PLL's filtering behavior as a distinct application. However, this is actually the fundamental property that supports nearly all PLL applications. For example, PLLs used for carrier recovery in receivers maintain lock on the carrier frequency despite gradual shifts caused by Doppler effects or oscillator drift. Other applications include FM demodulation and synchronization systems.

 

PM Generation Using a PLL

The PLL can be readily used for both frequency and phase modulation. The latter application is illustrated in Figure 3. Before we look at the math, let's first develop a qualitative understanding of how this circuit operates.

 

Figure 3. Using a PLL for phase modulation.

 

The PLL locks to the stable frequency source provided by the crystal oscillator. This sets the center frequency of the output signal. The tolerance of the output carrier frequency is determined by the crystal oscillator circuit. The message signal, m(t), is introduced into the loop as an error signal before the lowpass filter. This prompts the loop to adjust the VCO and create the message-dependent phase shift between its output and the reference signal.

In the steady state, the output phase is proportional to the modulating signal. Therefore, the above configuration serves as a phase modulator.

We'll now proceed to the mathematical analysis and derive a relationship between the output phase and the message signal. Assuming that a linear model is used to analyze the PLL operation, we can apply superposition to find the output in terms of ϕ_in(s) and M(s), which represent the two inputs in the Laplace domain. Using algebraic methods, we can express the output equation in the s-domain as follows:

$$\phi_{vco}(s) ~=~ \frac{k_d k_{v} \frac{F(s)}{s}}{1~+~ \frac{k_d k_v F(s)}{Ns}} ~\times~ \phi_{in}(s) ~+~ \frac{k_{v} \frac{F(s)}{s}}{1~+~ \frac{k_d k_v F(s)}{Ns}} ~\times~ M(s)$$

Equation 2.

 

where:

k_d is the phase detector gain in volts per radian

k_v is the VCO gain factor in radians per second per volt.

At low frequencies, we have:

$$\left|\frac{k_dk_vF(j \omega)}{N \omega}\right|~\gg~1$$

Equation 3.

 

Assuming Equation 3 holds true, Equation 2 simplifies to:

$$\phi_{vco}(s) ~=~ N \Big (\phi_{in}(s) ~+~ \frac{1}{k_d} ~\times~ M(s) \Big )$$

Equation 4.

 

If the input phase (ϕ_in) is constant, the instantaneous output phase (ϕ_vco) varies linearly with the message signal. We therefore know this is a phase-modulated wave.

It's worthwhile to mention that the relationship shown in Equation 3 holds at frequencies below the loop bandwidth. Therefore, the PLL's bandwidth should exceed the modulating signal's bandwidth to prevent the loop from distorting the signal.

In other words, when the bandwidth of the message signal is less than that of the PLL, the PLL can effectively track the phase variations introduced by the message signal to generate the PM wave. The linear operating range of this type of modulator depends primarily upon the phase detector.

 

FM Generation Using a PLL

Instead of using the message signal as an error signal before the lowpass filter, it can also be applied after the lowpass filter, as shown in Figure 4.

 

Figure 4. PLL-based modulator that applies the message signal after the loop filter.

 

Applying the integral of the desired modulating signal to the system results in the production of the corresponding FM wave at the output. This is illustrated in Figure 5.

 

Figure 5. Incorporating a PLL arrangement to generate wideband FM waves.

 

FM Demodulation Using a PLL

As well as generating FM signals, PLLs can also be used to demodulate them. Figure 6 shows such a demodulator.

 

Figure 6. Basic block diagram of a PLL used for FM demodulation.

 

To understand the operation of this circuit, we must first note the following:

• The negative feedback incorporated in the PLL forces the VCO output to follow the input FM wave.
• The VCO's frequency shift from its free-running frequency is proportional to its control voltage.

Since the VCO output reproduces the input FM wave, its control voltage must vary in accordance with the underlying message signal.

A PLL-based FM demodulator provides superior noise performance compared to other FM demodulators, such as a slope detector. It can also provide a greater level of linearity than many other FM detection techniques, provided that the frequency shift of the FM wave is within the loop's tracking range.

If the phase detector output varies between ±A, then the maximum frequency deviation of the VCO is:

$$\Delta \omega_{max} ~=~ \pm k_{f} k_{v}A$$

Equation 5.

 

where k_f is the DC gain of the loop filter, and k_v is the gain factor of the VCO.

Note that Equation 5 assumes that the VCO operates linearly across this frequency range. For the PLL-based demodulator to work effectively, the frequency deviation of the input FM wave must be within this range.

Numerous PLL-based FM demodulators incorporate an amplifier followed by an amplitude limiter prior to the PLL. In this way, the demodulator removes unwanted amplitude variations from the input FM wave. This approach helps to avoid any unintended sensitivity of the PLL to the amplitude variations of its input signal.

 

FSK Demodulation

There is a close relationship between FM and frequency shift keying (FSK). Therefore, it is unsurprising that PLLs also offer an effective approach for demodulating FSK signals. In the FSK scheme, the frequency shifts between discrete values to represent digital data. For instance, as the binary frequency shift keying (BFSK) signal transitions between two frequencies, the control voltage of the VCO will switch between two distinct voltage levels.

 

AM Demodulation

In order to demodulate an AM signal, we need to multiply it by a local oscillator of the same frequency. Figure 7 illustrates the demodulation process for a double-sideband suppressed-carrier (DSB-SC) AM signal.

 

Figure 7. Demodulation of a DSB-SC signal.

 

In the above figure, r(t) is the received DSB-SC signal and c_r(t) is the locally generated carrier in the receiver. There is a phase difference of ϕ between c_r(t) and r(t). At the output of the multiplier, we obtain a low-frequency replica of the message signal centered around DC, as well as a high-frequency replica of the message signal centered at twice the carrier frequency (2f_c).

The lowpass filter effectively removes the high-frequency components, allowing only the message signal to pass through to the final output. However, the output signal is influenced by the phase discrepancy (Φ) between the carrier wave used at the transmitter and that generated at the receiver. For a non-zero Φ, the amplitude of the output signal is reduced by a factor of cos(Φ).

To ensure maximum output voltage, the phase discrepancy should be eliminated (Φ = 0). Since the input phase angle is not known and may vary due to the channel variations, we need a PLL to lock the local oscillator signal to the incoming carrier signal. Figure 8 illustrates this concept.

 

Figure 8. Using the PLL to generate the local oscillator signal required for coherent AM demodulation.

 

The above block diagram assumes that the VCO output is in phase with the input. If the type of phase detector used causes a certain phase shift between the incoming signal and the VCO output, then we need to account for that phase shift. For instance, with a multiplier phase detector, we'll need to introduce a 90 degree phase shift to the VCO output before applying it to the multiplier.

 

Wrapping Up

In this article, we explored several key applications of PLLs. In fact, the versatility of PLLs extends well beyond these examples. In communication systems, they also play essential roles in clock and data recovery, frequency synthesis, and carrier recovery. PLLs are also integral to control systems, power electronics, and a variety of other engineering fields.

 

All images used courtesy of Steve Arar