Amplitude-Frequency Characteristic Analysis of Servo Systems: Principles, Testing Methods and Applic

Amplitude-frequency characteristic analysis is a fundamental frequency-domain method used to evaluate the dynamic response, bandwidth, and stability of servo systems. By measuring the gain ratio between output and input signals across a range of excitation frequencies, engineers can identify resonances, verify control loop performance, and optimize servo tuning. This article explains the technical principles, testing samples, test items, measurement methods, and typical applications of servo system amplitude-frequency characteristic analysis.

Technical Overview

The amplitude-frequency characteristic describes how the magnitude of a servo system's output response varies with the frequency of a sinusoidal input command. Mathematically, it is the modulus of the closed-loop (or open-loop) frequency response function, expressed as the gain ratio between output amplitude and input amplitude at each excitation frequency. The result is typically presented as a Bode magnitude plot, with gain in decibels plotted against frequency on a logarithmic scale.

For a servo system, the amplitude-frequency characteristic is one of the most important indicators of dynamic performance. It directly reveals the usable bandwidth, the low-frequency gain that determines tracking accuracy and stiffness, the resonant peaks caused by mechanical compliance or structural modes, and the high-frequency roll-off that governs noise rejection. Because servo systems operate as closed-loop control systems, an excessive resonant peak in the amplitude response often indicates insufficient stability margin, which may lead to oscillation, audible noise, and accelerated mechanical wear.

Key parameters derived from the amplitude-frequency characteristic include the -3 dB bandwidth, resonant peak magnitude and frequency, anti-resonance notch frequencies, passband gain flatness, low-frequency gain, and high-frequency attenuation slope. Together with the phase-frequency characteristic, these parameters form a complete frequency-domain portrait of the servo system and serve as the basis for controller design, gain scheduling, and notch filter placement.

Testing Samples

Amplitude-frequency characteristic analysis is applicable to a wide range of servo products and motion subsystems, including:

• AC permanent magnet synchronous servo motors with matched servo drives
• DC brushed and brushless servo motors and servo amplifiers
• Linear servo motors and direct-drive rotary torque motors
• Integrated servo actuators combining motor, gearbox, brake, and encoder
• Servo hydraulic and electro-hydrostatic actuators
• Precision positioning stages, rotary tables, and gimbal assemblies
• Complete closed-loop motion axes in CNC machines and robots

Sample Preparation Requirements

Prior to testing, samples should be run in and thermally stabilized, since bearing lubrication, magnet temperature, and drive amplifier settings all influence the measured response. The sample must be rigidly mounted on a calibrated test bench, connected to the standard servo drive and feedback devices used in the intended application, and configured with the control loop parameters to be evaluated.

Testing Items

A typical amplitude-frequency characteristic test program for a servo system covers the following items:

• Current loop bandwidth: gain response of the innermost torque-producing loop, usually in the range of several hundred hertz to several kilohertz
• Velocity loop bandwidth: gain response of the speed control loop, typically from tens to hundreds of hertz
• Position loop bandwidth: gain response of the outer positioning loop, usually from a few hertz to several tens of hertz
• -3 dB amplitude bandwidth: the frequency at which the output amplitude falls to 70.7 percent of the low-frequency reference value
• Resonant peak magnitude and resonant frequency: maximum gain value and its corresponding frequency within the passband
• Anti-resonance frequencies: notch points introduced by two-inertia coupling between motor and load
• Low-frequency gain: gain plateau in the working band, reflecting steady-state tracking stiffness
• Passband gain flatness: ripple or gain variation within the specified working frequency range
• High-frequency roll-off slope: attenuation rate beyond the cutoff frequency, indicating noise and vibration suppression capability

Testing Methods

The most widely used approach is sinusoidal excitation testing. A pure sine command of known amplitude is injected into the control loop reference input, and the resulting output signal is measured with high-resolution feedback sensors. The ratio of output amplitude to input amplitude is calculated at each frequency point, and the test is repeated across the frequency range of interest to construct the amplitude-frequency curve.

Stepped Sine Sweep Method

The system is excited at discrete frequency points with logarithmic spacing. At each point, excitation continues until steady-state oscillation is reached, and the amplitude ratio is extracted using synchronous detection or curve fitting. This method offers excellent signal-to-noise ratio and high accuracy at resonance, at the cost of longer test duration.

Chirp and Broadband Excitation Method

A fast frequency sweep (chirp) signal or a broadband random signal such as pseudo-random binary sequence excitation is applied, and the frequency response function is computed through FFT-based spectral analysis of the input-output pair. This method significantly reduces test time and is well suited for automated production testing, although frequency resolution and near-resonance accuracy may be lower than stepped sine testing.

Instrumentation and Procedure

A complete test setup typically consists of a dynamic signal analyzer or data acquisition system, a function generator or drive-based signal injection port, high-resolution incremental or absolute encoders, laser displacement sensors or accelerometers for mechanical response, torque transducers, and a dynamometer or representative mechanical load. The standard procedure is summarized as follows:

• Mount the sample on the test bench and connect the servo drive, feedback devices, and measurement sensors
• Warm up the system and verify that position, velocity, and current loops are functioning correctly
• Define the frequency span, excitation amplitude, and frequency resolution according to the loop under test
• Inject the excitation signal and simultaneously acquire input command and output response data
• Compute the amplitude ratio at each frequency and plot the magnitude curve in decibels
• Extract characteristic parameters such as -3 dB bandwidth, resonant peak, and gain flatness
• Compare results with design specifications or acceptance criteria and issue the test report

Applications

Amplitude-frequency characteristic analysis plays a decisive role in the development, commissioning, and acceptance of servo-driven equipment across many industries:

• CNC machine tools: verifying feed-axis bandwidth to ensure contouring accuracy at high cutting feed rates
• Industrial robotics: evaluating joint servo stiffness and resonance behavior to prevent vibration during rapid motion
• Aerospace systems: qualifying flight control surface actuators and thrust vector control servos against stringent dynamic specifications
• Gimbal and optical tracking platforms: confirming sufficient bandwidth for target tracking and image stabilization
• Semiconductor manufacturing: characterizing wafer stage and lithography motion axes where nanometer-level tracking is required
• Servo drive commissioning: identifying mechanical resonances and tuning notch filters, gain scheduling, and feedforward parameters
• Quality control and incoming inspection: verifying that delivered servo motors and drives meet declared dynamic performance ratings

FAQ

What is the difference between amplitude-frequency and phase-frequency characteristics? The amplitude-frequency characteristic describes how the gain magnitude varies with frequency, while the phase-frequency characteristic describes the phase shift between input and output. Both are obtained from the same frequency response measurement and are usually plotted together on a Bode diagram.

Why is bandwidth defined at the -3 dB point? At the -3 dB point, the output amplitude has dropped to 70.7 percent of the low-frequency value, which corresponds to half the power transfer. It is a widely accepted convention that marks the practical upper limit of useful tracking frequency for the loop under test.

What causes resonant peaks in the amplitude-frequency curve? Resonant peaks usually originate from mechanical compliance between the motor and load, structural modes of the machine frame, coupling backlash, or insufficient damping in the control loop. Excessive peaks indicate poor stability margin and require retuning or notch filtering.

Which loop should be tested first? Testing proceeds from the innermost loop outward: current loop first, followed by velocity loop and position loop. Each outer loop can only achieve stable bandwidth when the inner loop response is already verified.

How often should amplitude-frequency characteristic testing be performed? It should be performed during product development, after any change to control parameters or mechanical configuration, during type testing and acceptance, and periodically in service when performance degradation or vibration issues are suspected.