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The Forbidden Frequency Analysis: Mixed-Signal and Switching Circuits

If you simulate circuits that include not only analog, but also digital or switching/PWM aspects, then you know that traditional “AC” or frequency-domain analysis cannot be used for these designs. That’s because they have multiple operational states that change, often rapidly, over time. AC analysis depends on circuit linearization at one (singular) operating point.

For these circuits, time-domain frequency sweep* (TDFS) analysis provides an effective alternative. It requires neither circuit linearization nor “state-average” modeling, but rather simply the injection of a small sinusoidal disturbance signal into the circuit at steady-state in time-domain simulation. Sweeping the stimulus over a specified range of frequencies provides the desired frequency response.

The above example shows the use of TDFS to measure the frequency response of a mixed-signal band-pass filter (BPF) circuit. This includes both the analog anti-aliasing low-pass filter, as well as an 8-bit, second-order digital BPF. The TDFS instrument measures the transfer function of each stage of the filter, both separately and combined. It is not possible to perform a standard AC analysis of the entire filter, because of the discrete states used to describe the digital behavior.

The TDFS oscillator stimulus voltage is applied to the anti-alias filter’s input. The output of the anti-alias filter is monitored by channel 1 of the TDFS instrument. The BPF output is monitored by channel 2. In the upper-left waveform viewer, the dark blue waveform is the channel 1 to oscillator amplitude ratio (in dB), so it is the gain of the anti-alias filter alone. The magenta waveform is the channel 2 to oscillator ratio, which gives the gain of the entire mixed-signal filter.

(Note: If you click on a link to open the “live” version of any design, you can observe signals on any net or within any component and view component parameter values. You can also modify and re-simulate a copy of the design, to see the effect of the changes you make.)

TDFS Measurement Concept and Model Features

The operating concept of TDFS is quite simple. The circuit under test is biased to its steady-state operating point, and a time-domain simulation is performed while the sinusoidal input stimulus signal is applied. The response to that stimulus is observed at other points in the circuit. The in-phase and quadrature components are measured and used to compute the gain and phase of the transfer function at that frequency. The stimulus frequency is automatically stepped to discrete values over a user-specified range, to generate the complete frequency-response profile. Note that the unprocessed time-domain filter output signal (green waveform) is shown in the upper-right viewer window. The amplitude peaking at mid-range frequencies can be observed near the end of the simulation time.

The user can specify the number of frequency points, stimulus amplitude, and bias levels, and other calibration parameters that may be needed to accommodate circuit-specific requirements. The TDFS measurement model automatically stops the simulation run when the measurements are complete, to minimize run-time.

Loop-Stability Analysis

Stability is an essential quality of most practical circuits and systems. A traditional stability metric for closed-loop control systems is gain and phase margin, based on the open-loop transfer function (OLTF) or frequency response. A special physical measurement technique for obtaining the OLTF of an operating closed-loop system, pioneered by Dean Venable** in the 1980s, involves injecting a small sinusoidal stimulus in series with the loop and measuring the absolute signal levels at the injection site. This is to identify the effective open-loop output-to-input ratio, or the transfer function value, at that particular stimulus frequency. By sweeping the frequency value of the stimulus signal over time, the complete frequency response of the circuit or system under test can be obtained.

The “buck” or step-down DC/DC switching power converter above demonstrates the TDFS Loop Stability measurement method. Note that this circuit contains no “state-average” or continuous equivalent model of the PWM section of the design , as is normally used for AC analysis. Rather, the actual switching circuit component models (Power MOSFET, Diode, modulator ramp function, etc.) can be used directly.

This converter is operating at 200 kHz and is converting a 12V DC input to a regulated 5V output, while supplying a 5A current to the load resistor. The TDFS measurement shows the open-loop gain crossover frequency is at 26 kHz (green waveform), and the phase margin is approximately 60 degrees (light blue waveform). This verifies that the op-amp based, lead-lag compensator is providing adequate stability margin for this load condition.

Impedance vs. Frequency Measurement and Impedance Stability Analysis

The TDFS method can also be used to measure the input or output impedance vs. frequency of mixed-signal and switching circuits. This is particularly helpful for analyzing the “impedance stability” of distributed power systems. These systems may include many DC-to-DC converters, some used in sources and others in loads. When used in loads they often draw constant power from the bus, which makes them appear as destabilizing negative resistances. These power systems may also be distributed over significant distances, so that the impedance characteristic of the interconnecting cables (i.e., transmission lines!) can contribute to system destabilization.

The key stability metric is the ratio of the source-to-load impedance, Tm = Zsource/Zload, as defined in the paper STABILITY OF LARGE DC POWER SYSTEMS…*. This paper describes the impedance specification design approach used for the International Space Station’s electric power system.

In the example power system schematic shown below, the switching buck DC/DC converter used for loop stability assessment above is shown in a “down hole” LED lighting application for a drilling rig camera. The regulated output voltage drives three parallel 8.4 watt LEDs. The converter is supplied through a 400 meter power cable with 8 AWG wire. An 18V battery is used to accommodate the DC voltage drop of the cable.

But, a long cable has a more “complex” characteristic than DC resistance! To measure it, as well as the converter input impedance, the TDFS impedance stability measurement model injects a 0.2A sinusoidal stimulus current at the point where the cable connects to the converter. It monitors the voltage at that point, as well as the stimulus current splits toward the source and the load. The associated source and load impedances vs. frequency are computed from these three measured values, and the ratio of the source-to-load impedance (Tm) is computed. Like the open-loop frequency response requirement for closed-loop system stability, Tm must not have magnitude = 1.0 and phase = 180 degrees at any frequency.

The results show that the impedance ratio magnitude (green waveform) reaches unity, or 0 dB, at approximately 2 kHz. The phase (light blue waveform) is approximately 165 degrees at that point, which implies a phase margin of only 15 degrees.

To verify this stability margin, a transient test circuit similar to the TDFS configuration above was used. A switch was added in order to inject a load transient event. For the 400 meter cable configuration, underdamped ringing is observed in the buck converter’s input current (dark blue waveform, lower-left chart). The measured overshoot is 0.822A (70%), as expected for 15 degrees of phase margin.

The TDFS phase margin and the corresponding ringing are sensitive to cable length, as well as the value of the input filter capacitor of the converter. The reader is invited to copy this “live” circuit, change these design parameters, and run new simulations to see the impact of those changes. For example, if the cable length is increased to 800 meters, the power system becomes unstable, as shown by the red waveform in the chart on the lower right.

Summary

SystemVision Cloud from Mentor Graphics provides a family of measurement models that perform “Time Domain Frequency Sweep” (TDFS) analysis, to generate the frequency response characteristics of many circuits that defy traditional “AC” or frequency-domain simulation. These include:

  • Switching circuits
  • Mixed-signal (analog + digital) circuits
  • Sampled data control systems
  • Systems that include modulation or transformation functions

TDFS measurement models are available for traditional gain/phase (“Bode Plots”), closed-loop system stability, impedance vs. frequency, and impedance stability analyses.

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* Available in SystemVision Cloud, a free on-line circuit and system simulation platform

** D. Venable, “Testing Power Sources for Stability”, Venable technical paper #1, Venable Industries.

*** E.W. Gholdston (Rocketdyne), K. Karimi (Boeing), F.C. Lee, J. Rajagopalan, Y. Panov (Virginia Tech.) and B. Manners (NASA), “STABILITY OF LARGE DC POWER SYSTEMS USING SWITCHING CONVERTERS, WITH APPLICATION TO THE INTERNATIONAL SPACE STATION”, 31st Intersociety Energy Conversion Engineering Conference, 1996. IECEC 96-96079

1 comment on “The Forbidden Frequency Analysis: Mixed-Signal and Switching Circuits

  1. Dave E
    March 30, 2017

    Very interesting article.  How long has you web-based design service been offered?

    Regarding the abilitity to analyze switching circuits, specifically switched-mode power supplies, does your software create the equivalent of a hardware ac frequency response analyzer (FRA) for time domain loop-gain analisis in the face of a large switched mode carrier such as is done to varying degrees of success by the following software?

    Most notable of this breed are SIMPLIS (SiMetrix), PSIM (Powersim), PLECS (Plexim) and NL5 (Sidelinesoft).

    These engines feature very quick steady state solvers for switched circuits and switched mode power supplies, typically named POP or PSS (Periodic Operating Point or Periodic Steady State), a long, drawn out initial transient is avoided.

    Near instant POP/PSS and transient run speeds allows a time domain DFT ac analysisto be performed on switched circuits so that loop-gain of switching power supplies and class-d amplifiers may be quickly analyzed in the time domain.  Only a single nonlinear switched model is required – no need to resort to equivalent circuits with added “sampling effect” networks.  With this method, simulated results precisely match FRA lab measurements (Venable, HP4194A, etc.) to and beyond the switching frequency.

    Could you please link to a loop-gain example of a SMPS that covers low frequency up through several times the switching frequency (so that sinx/x looking aliasing effects may be observed)?

    Thank you.

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