Network analysis is a powerful and well-established method of characterizing and optimizing a control system. Unfortunately, making a successful measurement can be difficult and frustrating without the proper instrumentation. Having a good network analyzer is not enough. There must be a means for injecting a test signal into a closed loop over the frequency range of interest.
A common method of signal injection is to insert a 100-W resistor in the control loop, typically between the error amp and the plant, which is everything between the control output and the feedback input. For example, a buck converter plant consists of the sawtooth generator and comparator, the power transistor, the catch diode and the LC filter. The injection point must be between a low-impedance output and a high-impedance input. A transformer is then used to generate an ac test signal across the resistor. The reference signal is then measured at the plant input and the response is measured at the error-amplifier output.
Fig. 1 shows a typical control loop and two common locations for signal injection and measurement. It is very difficult to design a transformer that will provide a flat signal both at very low frequencies (< 1 Hz) and at higher frequencies (200 kHz). An engineer can forego this design and purchase a transformer, but the commercial versions still have the same frequency limitations, typically only operating over one or two decades. Electronic injection circuits exist but are expensive, around $1000. At least one open-source design exists according to published literature, but it suffers from chassis grounding issues.
Fig. 2 shows a simple circuit that performs the forementioned tasks very well. An instrumentation amplifier (inamp) isolates the sine-wave test signal from the network analyzer ground. The reference input is driven by the low-impedance output of an error amp or buffer. The sine wave then rides on the reference node, or control point of the circuit. It is important to realize that the reference input to an inamp is not a high-impedance node. Therefore, make sure that this node is truly driven by a low-impedance source.
The output of the inamp is connected to the high-impedance input of the plant. A designer does not actually need the 100-W resistor with this design, but its presence prevents accidental open-loop conditions if the device under test (DUT) is powered up before the test fixture. Analog Device's AD620 is able to be run on ±18-V rails, allowing for injection at nearly any conceivable point in most loops. Lastly, note the 1-MW resistor from the inverting input to ground. The inamp needs nanoamps of input bias current, which isn't much, but it is ill-advised to float the inputs completely. The resistor allows a very small amount of bias current without changing the return current paths appreciably.
The injection circuit was tested using a Venable 3120 network analyzer. First, the open-circuit performance of the fixture was checked with a test voltage of 500 mV and the reference input biased at 1 V (a graph of open-circuit performance is shown as Fig. A.) The region of practical use is from 0 kHz to 200 kHz. The phase shift is unimportant because it affects neither the network measurement nor the DUT. However, gain is important because the injected signal will start to decrease in the region of gain roll-off, reducing signal-to-noise and resulting in an inferior measurement. A higher-performance inamp would extend the range of the circuit.
It is very easy to believe erroneous results from a network analyzer. There are innumerable ways to compromise the measurement, from incorrect test signal amplitude to forgetting to turn on the injection circuit. It is important to have an oscilloscope hooked up to both the signal injection and measurement points. The signals should be relatively clean — distortion and noise compromise a reading. Once the setup has passed an open-loop test like the one previously described, it is prudent to test a standard, such as a simple RC circuit. As a means of verifying signal integrity, a buffered 10-kW/16-nF low-pass filter was tested in the fixture with a dc bias of 1 V.
Results of this test (see Fig. B.) showed that the performance was very good, matching the expected 3-dB point of 1 kHz. There are many other logical test circuits, but making a standard that closely matches the application is best. The results from a 10-Hz high-pass filter won't yield much information about how the test circuit will behave at 100 kHz.
After verifying performance open loop and then in a test circuit, the test fixture was used to take practical data. The injection circuit was used to make actual network measurements of a 45-kV X-ray source's kilovolt-drive circuit. This particular drive is for a handheld X-ray fluorescence spectrometer used in RoHS compliance testing. Because it is battery powered, the spectrometer needs to be stable over a rather wide input range of 12 V to 18 V.
Load conditions also vary widely, from 10 kV to 45 kV and 0 mA to 50 mA. The initial “guess” compensation worked well over most line and load conditions, but the supply occasionally oscillated at high kilovolt and no load. To find out what was going wrong, the inamp test fixture was used to probe stability at several line and load combinations, including the problem region.
Even if the compensation is in place and working, it is usually very instructive to first look at the plant response. Run the system closed loop and place the test input at the input of the plant, but instead of placing the measurement input at the output of the error amp, place it at the output of the feedback generation circuit (or the output of the plant).
A 100-W resistor was inserted between the error amp and the plant, and the injector was connected as previously described. The reference input of the network analyzer was connected to the output of the inamp, and the signal input of the network analyzer was connected to the buffered feedback reference (1 V = 10 kV). The DUT was brought up to 45 kV and 40 mA of load current, and the injected sine-wave amplitude was adjusted to yield a good signal, but not so much as to disrupt the operating point (about 100 mVPK-PK). A sweep from 1 Hz to 20 kHz yielded the results in Fig. 3.
By knowing that a type-1 integrating error amplifier provides a constant 270 degrees of phase shift (90 degrees from the capacitor and 180 degrees from the logic inversion of the amp), the phase can be shifted to see where the 0 degree crossing point is. The criterion for stability is that the gain must be less than 0 dB when the phase crosses 0 degrees. Knowing this, the integrator roll-off can be set for optimal performance.
The measurement described previously was repeated for several line and load conditions. It was found that the dip in phase shown in Fig. 4 shifted to lower frequencies with decreasing load. This effect was so pronounced that the ample phase margin at 45 kV and 50 mA was reduced to zero at no load, leading to the oscillation described previously. H. Dean Venable wrote a landmark paper on optimizing feedback compensation networks. Through use of the techniques prescribed by Venable, the phase in that area was boosted to ensure stability over all line and load conditions.
Note that there are two practical ways to make this plant measurement. The loop usually can be stabilized by swamping the capacitance of the integrator. Even if the dynamic performance is abominable, it will keep the plant in the operating region so that the designer can perform the forementioned test. Once the test is complete, the designer can calculate the optimal compensation and tune the error amp to a stable, high-performance state.
A second way to perform this measurement is to run the plant open loop, using a variable voltage reference to set the operating point. This dc level is injected into the reference input of the inamp, and the output of the inamp drives the plant directly. On pulse-width modulation chips with internal amps, this node is often available as the COMP pin. Running the chip in this configuration can be a little tricky since ICs such as Linear Technology's LT3431 pack a bunch of features into the feedback node, for example frequency reduction and current limiting. Clamping the feedback input to the correct voltage will often solve these problems (e.g., in the case of the LT3431, a voltage of 1 V keeps the chip operating normally).
The Bode plot in Fig. 3 shows a plant gain of about 11 dB and a 90 degree phase shift at about 1.5 kHz. Using Venable's methods, the loop was compensated to have 60 degrees of phase margin (the phase at the 0-dB point). This results in a slightly overdamped response to step changes in program, which minimizes the voltage stress on the components. In other systems, fast settling time might be more important, which would indicate a lower phase margin, but resulting in more overshoot.
Fig. 4 shows the Bode plot of the compensated power supply at 45 kV and 50 mA. The results of the test match the predicted response of the measured plant gain combined with the integrator, indicating that the test fixture performs its task well.
Network analysis can be difficult not only because of the challenges of signal injection, but also due to the practical difficulties of maneuvering so many test leads and scope probes. It is very easy to short out a critical point and destroy equipment or the DUT. The test jig described next and pictured in the online version of this article overcomes many of these challenges.
Fig. 5 shows the full schematic of the signal injection circuit. In the upper left-hand corner, U1 handles the actual signal injection. Note that there is isolation from chassis ground at the signal input. R4 provides enough current for input bias current requirements, but shouldn't cause ground loop errors. Jumpers J2 and J3 provide 10x and 100x attenuation for fine control of the injected signal, which can be very helpful for analyzers with limited attenuation control. Two batteries provide low noise and isolated rails (this could be changed to extend the usable range to ±18 V).
In the lower right-hand corner, U2 (LT1057) provides buffering for a variable voltage reference (needed for making open-plant measurements) and an auxiliary buffer (for providing a low-impedance feedback point). To make the open-plant measurements, J7 is jumpered and the pot adjusted until the plant is at the desired operating point. The plant response is then measured from “high Z side A” to “low Z side B” (Fig. 1). The TL431 is used for a low-battery-detect circuit. The various connectors are used to interface between the DUT and the network analyzer.
The fixture's mechanical layout is nearly as important as its electrical functionality. (A mechanical layout of the pc board appears as Fig. C.) Three BNC connections at the back of the board go off to the network analyzer (sine, reference, measurement). This keeps all the large cables out of the way and under control. Two 9-V batteries (mounted on the underside of the board) provide power, eliminating another piece of test equipment from the designer's bench. Additionally, the batteries provide some mass, which is needed to keep the board from falling off the bench.
Three banana jacks are mounted at the front of the board. These connect to the DUT ground and the low-impedance and high-impedance sides of the resistor. The resistor should be on the DUT and should preferably be built in to the pc board. There is negligible cost or mass penalty to one extra resistor in the assembly. Test points on either side of the resistor are extremely helpful for clipping in both the banana test leads and the scope probes. A three-pin SIP header is very useful for connecting all three signals at once.
When making plant measurements, the banana leads stay as they are, but the measurement BNC is unplugged, and the measurement test node is probed directly from the network analyzer to the node of interest on the DUT. Other useful user features on the board include a power-on LED and a low-battery LED. Both are very helpful at preventing measurement errors.
Venable, H. Dean. “Optimum Feedback Amplifier Design for Control Systems,” Proceedings of Intersociety Energy Conversion Engineering Conference, August 1986.
Venable, H. Dean. “Practical Testing Techniques for Modern Control Loops,” Venable Industries Technical Paper #16, www.venable.biz.
Galinski, Martin. “Use Op-Amp Injection for Bode Analysis,” EDN, Sept. 16, 2004, pp. 90, 92.
“AD620: Low Drift, Low Power Instrumentation Amp with Set Gains of 1 to 10000,” Analog Devices data sheet, www.analog.com.
“LT3431: High Voltage, 3A, 500kHz Step-Down Switching Regulator,” Linear Technology Datasheet, www.linear.com.