Ceramic Sensor Rate Networks: Bound an Analog Differentiator's Useful Frequency Range

System boundary

An inverting voltage-feedback amplifier with a series input capacitor and resistor and a parallel feedback resistor and capacitor, operated around a valid bias reference. It is an analog rate-extraction stage, not a complete motion controller or a digital PID derivative algorithm.

Integration interfaces

Integration risks

Define the rate quantity before selecting a circuit

If a sensor produces voltage v equal to a constant scale kx times displacement x around a valid operating region, its voltage derivative is kx times mechanical velocity. That relationship requires a known local sensor scale. A nonlinear resistive position curve has a position-dependent scale, so one fixed voltage-rate calibration does not automatically give correct velocity everywhere.

Identify the signal frequencies and transients needed for the actual decision. A slow positioning estimate and a fast contact-interruption detector are different applications. The circuit discussed here estimates the derivative of a band-limited analog voltage. It does not establish the sensor's mechanical accuracy, remove wiper noise or decide the derivative term of a heater controller.

Derive the two limiting poles from the impedances

Connect input capacitance Ci in series with resistance Ri to the inverting node. Connect feedback resistance Rf in parallel with capacitance Cf from output to that node. With an ideal amplifier maintaining a virtual reference, the small-signal voltage transfer is minus feedback impedance divided by input impedance. The numerator contains s Rf Ci, while the denominator contains both first-order RC factors.

At frequencies well below both pole frequencies, the denominator is approximately one and output is approximately minus Rf Ci times the input derivative. As frequency approaches either pole, that interpretation loses accuracy. The input resistor and feedback capacitor limit high-frequency behavior; they are not neutral additions that preserve ideal differentiation at every frequency.

Compare the actual result with the intended derivative

Let taui equal Ri Ci and tauf equal Rf Cf. For a sinusoid at angular frequency omega, the ratio of actual output magnitude to ideal derivative magnitude is one divided by the square root of [(1+(omega taui) squared)(1+(omega tauf) squared)]. The additional phase lag relative to the ideal inverting derivative is minus atan(omega taui) minus atan(omega tauf).

These are two different acceptance questions. A magnitude departure that looks small can coexist with a phase shift that matters to the receiving system. State whether the specification concerns peak rate, timing, phase alignment or all of them. A DC gain measurement cannot verify this transfer because an ideal series input capacitor blocks a constant input change after its transient has passed.

Calculate the usable band without borrowing an amplifier rating

For an independent example, take Ri equal to 10 kilohms, Ci equal to 10 nanofarads, Rf equal to 100 kilohms and Cf equal to 1 nanofarad. Both pole time constants are 100 microseconds, and the derivative scale Rf Ci is 1 millisecond. At 100 hertz, the magnitude relative to an ideal derivative is approximately 0.9961, with about 7.19 degrees of additional phase lag.

At 1 kilohertz the same relative magnitude is approximately 0.7170, with about 64.28 degrees of additional lag. A circuit can therefore produce a clean-looking output while understating and delaying the rate substantially. The example calculates the ideal passive-network effect only; the selected amplifier may impose a narrower usable range.

Distinguish a sustained ramp from its corners

With the example derivative scale of 1 millisecond, a sustained input ramp of 1 volt per second produces approximately minus 1 millivolt after the RC transients settle and while the amplifier remains linear. At the start and end of the ramp, the two poles shape the transition. Those corners cannot be interpreted using only the steady-ramp value.

An abrupt input step is different again. The ideal mathematical derivative contains an impulse, whereas the practical circuit has finite bandwidth and finite output swing. A clipped transient can hide both its actual peak and recovery time. Use physically representative ramps, sinusoids and disturbances rather than validating a velocity channel solely with an arbitrary square-wave edge.

Include the sensor impedance and unwanted fast signals

A resistive position sensor can have an output impedance that changes with position. In the simple series-input topology, additional source resistance changes the effective input time constant and hence the upper differentiation boundary. Use the connected sensor model or a buffer where justified; a network calculated from Ri alone may behave differently near another part of the resistive track.

Differentiation emphasizes rapid voltage changes within its active band. Wiper bounce, interference and a connector event can produce a large rate-like output without corresponding movement. Suppressing these signals by moving the poles downward also changes genuine rate response. Preserve a separate physical plausibility check and distinguish a detection filter from a guarantee that every surviving transient is real motion.

Validate the active stage instead of assuming virtual ground

The transfer above uses an ideal amplifier. A real voltage-feedback amplifier has finite loop gain, input and output range, slew rate and loading constraints. The noise-gain behavior of the actual network determines the loop-stability question. Select and validate the amplifier using the complete connected circuit; the passive pole frequencies alone do not prove phase margin or freedom from oscillation.

For single-supply operation, define the reference bias and its impedance across the relevant frequencies. Check both polarities of expected rate around that bias and allow for offsets and transients. A passive resistor ratio cannot recover signal information lost through output clipping. Avoid carrying a voltage-feedback stability method directly to a current-feedback amplifier, whose absolute feedback resistance has a different role.

Specify a rate-valid operating region with the passive drawing

Provide the sensor transfer curve, connected source impedance, expected rate waveform and required frequency range. Allocate permitted magnitude error and phase departure before choosing the two RC time constants. Include component tolerance and operating-state variation in the corner analysis, then confirm the circuit with known input waveforms and a measurement chain fast enough to resolve the required phase.

ChipSimple can review a drawing-defined ceramic resistor network for this analog interface. The application and circuit owners validate the active stage and the relationship between voltage derivative and physical motion. Deliver the verified rate scale together with its frequency and amplitude boundaries, not a single derivative constant that silently assumes unlimited bandwidth.