A general-purpose vector signal analyzer offers a low-cost, flexible option for measuring the frequency settling time of PLLs.
By Douglas Olney, Keithley Instruments Inc.
The frequency-settling time of a phase-locked loop (PLL) is a critical specification for many RF circuits. There are various techniques for measuring both the amplitude and frequency-settling time of a PLL, but using a general-purpose vector signal analyzer (VSA) is one of the more attractive methods for several reasons:
• It is a piece of test equipment that most RF labs already own.
• It provides relatively low-cost spectrum-analyzer functions.
• The techniques employed can be accomplished with any VSA that allows the transfer of complex IQ time-domain samples to a PC.
• A wide variety of phase-locked loops can be measured, as long as a TTL trigger can be derived at or near the start of tune.
• 10-Hz resolution is easily achieved.
The specific technique described in this article resulted from a Keithley development project on an RF synthesizer. To avoid a major expenditure for a high-end spectrum analyzer or specialized test equipment with limited application, other solutions were investigated. After some thought, the development team recognized that a tuned instrument with a fast digitizer could be used to measure frequency-settling time. An evaluation of existing equipment revealed that a VSA had the appropriate features, and a technique was quickly developed to measure both amplitude and frequency settling.
Other Alternatives Considered
There are three basic techniques to detect frequency as a function of time. Frequency counters measure the length of time between zero crossing, discriminators convert a frequency change into a power change, and VSAs directly measure frequency versus time for signals within their acquisition bandwidth.
Prior to the advent of high-speed digitizers, the preferred technique for measuring frequency versus time was to use an instrument that timed the waveform’s zero crossings. Most RF frequency counters employ this technique, and more advanced instruments log data versus time. Frequency-versus-time instruments called modulation domain analyzers are now largely obsolete, and most have been replaced by VSA-based equipment.
The lowest-cost approach to measuring frequency versus time is to run the signal through a device that has a roughly linear magnitude slope versus frequency, causing the frequency variation to turn into an amplitude variation. This can be as simple as the partial-cutoff region of a bandpass or low-pass filter. The power at the filter/discriminator output is then measured using either a spectrum analyzer or a power sensor.
Though inexpensive, this approach makes accuracy difficult to maintain, as amplitude variations in the signal can be confused with frequency variation, and it’s difficult to resolve fine-grain variations on the order of 10 Hz, which we want for PLL settling time measurements.
Another possibility is to use a fast sampler or sampling oscilloscope to capture the signal. However, this technique lacks the dynamic range typically required to measure down to 10 Hz.
The preferred approach for modern solutions is to use a VSA. This instrument employs an RF downconverter followed by a fast digitizer, which feeds a digital IQ detector. The solution can be as simple as a standard VSA, plus external software to process the trace. More complex solutions include real-time spectrum analyzers and signal source analyzers. These instruments may provide more features and allow more types of measurements, but at significantly higher cost. Unless these features and functions are necessary for other applications, their cost is excessive for the basic task of measuring when a synthesizer frequency has settled.
Measurement Theory for VSA Solutions
When first faced with measuring frequency, many engineers will acquire a time record and use progressively larger FFTs to get increasingly finer resolution in the frequency domain. This works well when dealing with a modulated carrier that has rich frequency content, but can be frustrating for measuring the frequency trajectory of an unmodulated carrier. For an unmodulated carrier, the main concern is the dominant frequency at a given instant in time. In that case, FFT calculations return lots of unneeded information and cause a tradeoff in time resolution, because longer FFTs are used to get better frequency resolution.
This was the case in Keithley’s development of an RF synthesizer that uses a PLL to generate carrier frequencies. Therefore, development engineers began to consider the fundamental definition of frequency—the rate of change of phase. In effect, this says that you can estimate frequency as often as you can make a phase measurement. The resolution of the frequency estimate is limited only by the noise in the measurement. Still, caution must be exercised. The instantaneous frequency estimated from the derivative of phase can only be interpreted as the carrier frequency when there’s a single carrier present, without any modulation other than the tuning of that carrier. Other signals present in the measurement bandwidth will cause the measured phase to vary wildly.
In the case of a PLL, when measuring its voltage-controlled-oscillator (VCO) tuning, the single carrier assumption generally holds true. Unless the VCO breaks out into spurious oscillations, the derivative of the phase is the frequency estimate you want.
The measurement methodology is to set the VSA to the expected ending frequency, trigger the VSA at the start of tune for the synthesizer under test, and log an IQ time record long enough to cover the settling time. Eventually, the signal falls within the IF bandwidth of the VSA, and we can calculate phase versus time. From that we calculate frequency versus time. At the same time, we also have the magnitude versus time.
To measure phase versus time, an IQ detector multiplies the IF signal, s(t)=I(t)+jQ(t), with a cosine to get the real part and with a sine to get the imaginary part. A filter follows the multiplication to turn it into convolution and to remove spurious frequency components. The phase, θ
i, is computed as tan
-1\[Q(t)/I(t)\]. An IQ detector is inherently band-limited, and the magnitude and phase flatness across frequency is a key consideration. The best solutions use a digital IQ detector with perfectly matched channels and well-behaved filters.
To convert phase to frequency, we approximate the derivative with a time difference. One approach would be to subtract adjacent samples. Unfortunately, that introduces a half-sample delay in the data.
To avoid a time shift, it’s better to subtract two samples that are equidistant from the current sample, subtracting the value N samples before the current sample from the value N samples after the current sample. If the phase samples are in an array θi, then the frequency is computed using:
F ≈ (P
i+
N - P
i-
N)/(2NΔt)
where Δ
t is the sampling interval, the inverse of the sampling rate.
It’s necessary to use unwrapped phase samples for this estimate
1, and that the incoming signal not progress more than 180° in between samples. In general, having a detector bandwidth that’s below the Nyquist bandwidth ensures the second criteria.
By using samples equidistant before and after the current sample, there’s no effective time delay. In the equation above, the quantity (2
N) is known as the aperture, denoting the separation between the phase samples. It’s important to note that we divide by the aperture, which means that we can trade noise for time resolution with a simple post-processing operation.
The measurement bandwidth is set by the IQ detection filter and is an important consideration because frequency resolution is limited by noise, causing us to want smaller bandwidths. At the same time, filter bandwidth limits the frequency deviation we can see, and filter ringing can dominate the time response. In addition to bandwidth, the filter shape is important because its frequency shape affects the magnitude response, and its impulse response can show up as ringing on both the frequency and amplitude response.
A good rule of thumb is to use a bandwidth about 1000 times larger than the desired frequency resolution. For example, to measure settling down to 100 Hz, use a 100-kHz bandwidth. When measuring large deviations, use a flat-top filter to capture as much information as possible. When measuring the final settling with high precision, use a Gaussian filter to avoid ringing.
Figure 1 shows the ringing due to a flat-top filter at 10-kHz bandwidth (around 10-Hz resolution) as compared to a Gaussian filter.
Time accuracy is also important in this measurement, which means paying attention to trigger accuracy. We need a trigger signal to start the VSA acquisition. If there’s any ambiguity in the relationship between the start trigger and the actual start-of-tune of the VCO, the measurement accuracy is reduced. It’s best if the control module that starts the synthesizer tuning also generates the start-of-tune trigger. It’s sometimes possible to derive a valid trigger from existing signals in the system, such as existing control lines and synchronization signals.
Finally, it’s important that the digital acquisition system be calibrated for accurate trigger time. Because of pipeline delays in the digital processing system, the next sample available after a trigger event will take place sometime before the trigger event. Most commercial VSAs calibrate the time axis so that the correct sample at the trigger instant is identified.
1In general, a sinusoidal function of time can be characterized with its argument expressed as a general angle, ϕ(t), that changes in time, i.e, x(t) = A cos(ϕt)), and the time derivative of that unwrapped angle, ϕ(t), is the instantaneous frequency of that sinusoid at any given time t . The angle ϕ(t) is said to be unwrapped if it is continuous everywhere, except at places where the absolute value of the jump discontinuity is less than π radians.Example Measurement Data
In this example, an RF synthesizer was programmed to hop between 1142 MHz and 998 MHz. A Keithley Model 2810 VSA was used for the settling-time measurements (
see “Model 2810 VSA With Spectrum Analyzer Capabilities Fits Many Applications”). It was tuned to 998 MHz and used a 35-MHz flat-top filter. The synthesizer triggered the VSA at the time the VCO started tuning from 1142 MHz to 998 MHz.
Figure 2 shows that the signal comes within the measurement bandwidth about 33 µs after the VCO begins tuning, The VCO frequency overshoots that 998-MHz target by about 8 MHz, but the PLL brings the frequency in until there’s lock-on at about 220 µs after start-of-tune. The VCO amplitude
(Fig. 3) settles much earlier—after about 70 µs.
This measurement was made by placing the VSA in zero span mode at 998 MHz. The receive filter was configured as a 35-MHz flat-top filter. Sweep time was 300 µs. The trigger was configured for external trigger (from the synthesizer), with a trigger delay of 33 µs. Data were collected using the
:MEAS:IQ? SCPI command, which acquires the data and transfers it to the host computer in a binary format. The post-processing software used an aperture of 900 ns when computing the frequency trajectory.
Figure 4 shows approximately the final 200 Hz of settling time for a synthesizer that was tuned from 1214.39011 MHz to 998 MHz using a 100-kHz Gaussian filter. As before, data were processed externally to calculate frequency trajectory. This figure shows that the VSA technique can easily resolve frequency features as small as 10 Hz.
To simplify the measurement setup, we now build in the ability to command the synthesizer to hop between two frequencies every few milliseconds, and provide a TTL trigger at the start of tune.
Conclusions
What started as an effort to preserve capital budget quickly became a preferred method for measurement of frequency settling time. The litmus test was when we realized we could resolve 10-Hz differences in output frequency on samples less than 100 µs apart. That’s significantly better than the old engineering rule that says you have to measure over a 0.1-second period to resolve a 10-Hz frequency difference. This methodology has a number of advantages:
• Ability to zoom out and measure the last few megahertz of the trajectory with high time resolution.
• Ability to zoom in and resolve around 10 Hz for the final settling.
• Ability to trade time resolution for noise on the frequency trace by adjusting the aperture of the phase-to-frequency derivative.
• Valid amplitude trajectory once signal is within IF bandwidth.
• If a VSA isn’t already owned, it’s relatively economical to acquire (about $20,000 for a Keithley Model 2810), compared to a high-end spectrum analyzer.
Doug Olney, senior DSP engineer, has over 20 years experience with RF measurements and is currently working on measurement algorithms for OFDM signals.References
Agilent Application Note AN 1275, "Automatic Frequency Settling Time Measurement Speeds Time-To-Market for RF Designs," available at
http://cp.literature.agilent.com