Electronics Guide

Analog Test Equipment and Techniques

Introduction

Analog test equipment forms the foundation of electronic measurement and characterization, providing the means to generate precise stimuli and capture accurate responses from circuits and systems under test. Unlike digital testing where signals occupy discrete states, analog testing demands instrumentation capable of handling continuous waveforms with exacting requirements for amplitude accuracy, frequency response, noise performance, and dynamic range.

Modern analog test environments employ sophisticated instruments ranging from source-measure units that combine stimulus and measurement in single precision instruments, to complex automatic test equipment (ATE) systems that integrate dozens of specialized subsystems. Understanding these instruments and their proper application is essential for design verification, production testing, and failure analysis of analog and mixed-signal electronics.

Mixed-Signal ATE Systems

Automatic Test Equipment (ATE) for mixed-signal devices combines digital pattern generation with precision analog measurement capabilities, enabling comprehensive testing of integrated circuits that contain both digital logic and analog circuitry.

ATE Architecture Overview

Mixed-signal ATE systems integrate multiple subsystems coordinated by a central controller:

  • Digital subsystem: High-speed pattern generators and comparators for digital interface testing, typically supporting multiple voltage levels and timing formats
  • Analog subsystem: Precision DC sources, arbitrary waveform generators, digitizers, and measurement instruments for analog parameter testing
  • Device power supplies: Programmable supplies with fast transient response and accurate current measurement for power consumption testing
  • Timing system: Precision clock generation and distribution ensuring synchronized operation across all subsystems
  • Interface hardware: Device interface boards (DIBs) or load boards that connect the tester to the device under test

The system controller executes test programs that orchestrate stimulus generation and measurement across all subsystems while managing data collection and bin sorting decisions.

Digital Pin Electronics

Digital pin electronics provide the high-speed interface for testing device digital ports:

  • Driver circuits: Programmable output drivers with adjustable voltage levels (VIH, VIL) and edge rates for generating test patterns
  • Comparators: High-speed comparators with programmable threshold levels (VOH, VOL) for capturing device responses
  • Per-pin timing: Independent programmable delays on each pin enabling skew compensation and timing margin testing
  • PMU capability: Parametric measurement units integrated with pin electronics for DC parametric testing
  • Active load: Programmable termination and load current capability for interface testing

Modern digital pin electronics achieve timing resolution below 10 picoseconds and support data rates exceeding several gigabits per second for high-speed serial interface testing.

Mixed-Signal Test Strategies

Testing devices with both analog and digital content requires coordinated strategies:

  • Digital stimulus, analog measurement: Use digital patterns to configure DACs or other analog outputs, measure with precision instruments
  • Analog stimulus, digital measurement: Apply precision analog signals to ADC inputs, capture digital outputs with pattern comparators
  • Loopback testing: Connect analog outputs to analog inputs through the tester, verifying end-to-end signal chain performance
  • Built-in self-test (BIST): Leverage on-chip test circuitry to reduce external test resources and improve coverage

Coherent sampling techniques are essential for mixed-signal testing, ensuring that stimulus frequencies and sample rates maintain integer relationships to eliminate spectral leakage in FFT-based measurements.

Test Floor Deployment

Production deployment of ATE systems involves significant infrastructure considerations:

  • Environmental control: Temperature and humidity control to maintain measurement accuracy and repeatability
  • Vibration isolation: Isolation from floor vibration for sensitive analog measurements
  • EMI shielding: Shielded enclosures to prevent interference from nearby equipment
  • Calibration infrastructure: Regular calibration traceable to national standards
  • Handler integration: Mechanical handling systems for high-volume production testing

Source-Measure Units

Source-Measure Units (SMUs) combine precision voltage and current sourcing with accurate measurement in a single instrument, making them indispensable for device characterization and parametric testing.

SMU Operating Principles

SMUs employ feedback-controlled output stages with integrated measurement:

  • Voltage source mode: Force a precise voltage while measuring the resulting current through the load
  • Current source mode: Force a precise current while measuring the resulting voltage across the load
  • Four-quadrant operation: Source or sink current at positive or negative voltages, enabling both power delivery and electronic load functions
  • Compliance limiting: Automatic protection limiting prevents damage when forcing into fault conditions

The feedback architecture maintains accuracy across wide dynamic ranges while protecting sensitive devices from overcurrent or overvoltage conditions.

Key SMU Specifications

Critical specifications determine SMU suitability for various applications:

  • Source accuracy: Typically specified as percentage of output plus fixed offset; high-end units achieve 0.01% or better
  • Measurement accuracy: Often better than source accuracy; sub-femtoampere current measurement available in specialized units
  • Resolution: Minimum step size for sourcing and measurement, often 6 to 8 digits
  • Settling time: Time to reach specified accuracy after step changes; critical for throughput
  • Output impedance: Ideally zero in voltage mode, infinite in current mode; actual values affect measurement at extremes
  • Voltage and current ranges: Maximum ratings; power is typically limited to a few hundred watts

SMU Applications

SMUs serve diverse applications in electronics test and measurement:

  • Semiconductor characterization: IV curves, threshold voltage, leakage current, and breakdown testing of transistors and diodes
  • Component testing: Resistance measurement, capacitor leakage, inductor DC resistance
  • Power device testing: High-current characterization with pulsed techniques to manage heating
  • Solar cell characterization: IV curve tracing under illumination for efficiency measurement
  • Battery testing: Charge and discharge cycling with precise current control and voltage monitoring
  • Low-current measurement: Leakage testing, photodetector characterization, and sensor evaluation

Advanced SMU Techniques

Sophisticated measurement techniques extend SMU capabilities:

  • Pulsed measurements: Short duty-cycle pulses minimize self-heating during high-power characterization
  • Quasi-static CV: Combining DC bias with AC analysis for capacitance-voltage profiling
  • Multi-channel synchronization: Coordinated operation of multiple SMUs for multi-terminal device testing
  • Kelvin connections: Four-wire sensing eliminates lead resistance errors in low-resistance measurements
  • Guarding: Active guard shields minimize leakage currents in high-impedance measurements

Arbitrary Waveform Generators

Arbitrary Waveform Generators (AWGs) synthesize complex analog waveforms with precise control over amplitude, frequency, and waveshape, enabling stimulus generation for sophisticated analog and mixed-signal testing.

AWG Architecture

Modern AWGs combine digital waveform memory with high-performance DACs and analog output conditioning:

  • Waveform memory: Stores sample points defining the output waveform; capacity ranges from kilosamples to gigasamples
  • Sample clock: High-frequency, low-jitter clock determines the playback rate and output bandwidth
  • Digital-to-analog converter: Converts stored samples to analog voltages; resolution typically 12 to 16 bits
  • Reconstruction filter: Low-pass filter removes sampling artifacts from the output
  • Output amplifier: Provides gain, offset adjustment, and output drive capability

Sample rates exceeding 10 gigasamples per second enable generation of waveforms with frequency content to several gigahertz.

AWG Specifications

Critical specifications define AWG performance:

  • Sample rate: Maximum clock frequency; determines highest possible output frequency
  • Vertical resolution: DAC bit depth; affects spurious-free dynamic range and distortion
  • Analog bandwidth: Usable frequency range of the output path; typically one-third to one-half of sample rate
  • Memory depth: Number of samples stored; determines waveform duration at given sample rate
  • Spurious-free dynamic range: Ratio of signal to largest spurious component
  • Phase noise and jitter: Timing purity of the sample clock affects output spectral purity

Waveform Generation Techniques

AWGs generate waveforms through various methods:

  • Direct digital synthesis: Mathematical computation of waveform points; provides frequency accuracy limited only by clock reference
  • Waveform import: Load captured or simulated waveforms for playback; enables replay of real-world signals
  • Equation-based generation: Define waveforms using mathematical expressions
  • Sequence playback: Link multiple waveform segments with conditional branching for complex stimulus patterns
  • Real-time streaming: Continuously stream waveform data from external source for unlimited duration

AWG Applications

AWGs support diverse test and measurement applications:

  • Data converter testing: Generate precise multi-tone signals for ADC dynamic testing
  • Communication system testing: Create modulated waveforms including QAM, OFDM, and spread spectrum
  • Radar and electronic warfare: Synthesize complex pulsed and chirped waveforms
  • Sensor simulation: Emulate sensor outputs for system-level testing
  • Jitter and timing testing: Add controlled jitter to clock signals for margin testing
  • Power supply testing: Generate transient waveforms for load step and stability testing

Digitizers and Waveform Capture

Digitizers capture analog waveforms with high resolution and accuracy, converting continuous signals to digital data for analysis. While oscilloscopes are familiar digitizing instruments, specialized digitizers offer superior performance for precision measurement applications.

Digitizer Architecture

High-performance digitizers optimize the signal acquisition chain:

  • Input conditioning: Attenuators, coupling capacitors, and protection circuits prepare the input signal
  • Track-and-hold amplifier: Captures instantaneous signal value for conversion; bandwidth must exceed input frequency
  • Analog-to-digital converter: Converts held voltage to digital code; architecture choices include SAR, delta-sigma, and pipeline
  • Sample clock: Low-jitter clock determines sampling moments; jitter degrades effective bits at high frequencies
  • Acquisition memory: Deep memory enables long capture durations at high sample rates

Digitizer Specifications

Key specifications determine digitizer suitability for various applications:

  • Resolution: ADC bit depth; 12 to 24 bits typical, though effective bits are lower at high frequencies
  • Sample rate: Samples per second; ranges from kilosamples to gigasamples depending on application
  • Analog bandwidth: Input frequency range; determined by front-end amplifier performance
  • Effective number of bits (ENOB): Actual resolution accounting for noise and distortion; decreases with input frequency
  • Spurious-free dynamic range: Ratio of fundamental to largest spurious component
  • Input voltage range: Full-scale input; often programmable with multiple ranges

Sampling Techniques

Different sampling approaches address various measurement needs:

  • Real-time sampling: Captures single-shot events at the full sample rate; memory depth limits capture duration
  • Equivalent-time sampling: Reconstructs repetitive waveforms using multiple acquisitions; achieves effective sample rates far exceeding the actual clock
  • Random interleaved sampling: Variant of equivalent-time using random trigger delay for improved frequency response flatness
  • Segmented acquisition: Divides memory into segments triggered independently; captures multiple events efficiently
  • Continuous streaming: Transfers data to host memory during acquisition; enables unlimited record length

Waveform Analysis

Post-capture analysis extracts meaningful measurements from acquired waveforms:

  • Time-domain measurements: Amplitude, rise time, fall time, period, duty cycle, and pulse parameters
  • Frequency-domain analysis: FFT-based spectral analysis for harmonic distortion, SNR, and spurious content
  • Statistical analysis: Histogram and trend analysis for characterizing variation
  • Protocol decode: Extract data content from serial communication waveforms
  • Eye diagram analysis: Overlay multiple bit periods to visualize signal quality and jitter

Spectrum and Network Analyzers

Spectrum analyzers and network analyzers characterize electronic systems in the frequency domain, providing essential insight into signal spectral content and circuit frequency response that time-domain instruments cannot easily reveal.

Spectrum Analyzer Fundamentals

Spectrum analyzers display signal amplitude versus frequency:

  • Swept-tuned architecture: Superheterodyne receiver sweeps a local oscillator to down-convert different frequency bands to an IF filter
  • FFT-based architecture: Digitizes the input and computes the spectrum mathematically; offers speed advantages for wide spans
  • Real-time spectrum analysis: Continuous FFT processing captures transient spectral events without gaps
  • Resolution bandwidth: Width of the measurement filter; narrower bandwidths reveal closer spectral components but slow measurement
  • Video bandwidth: Post-detection smoothing filter; reduces noise at the expense of response time

Spectrum Analyzer Applications

Spectrum analyzers serve diverse measurement needs:

  • Harmonic distortion measurement: Quantify harmonic content of oscillators and amplifiers
  • Spurious signal detection: Identify unwanted signals from mixing products, clock harmonics, or interference
  • Modulation analysis: Measure occupied bandwidth, adjacent channel power, and modulation quality
  • EMI pre-compliance: Identify emissions that may cause regulatory compliance failures
  • Noise measurement: Characterize noise spectral density and noise figure

Vector Network Analyzer Principles

Vector Network Analyzers (VNAs) measure the complex ratio of incident and reflected signals to characterize circuit response:

  • S-parameter measurement: Scattering parameters describe reflection and transmission in terms of incident and scattered waves
  • Magnitude and phase: VNAs measure both amplitude ratio and phase difference, unlike scalar analyzers
  • Swept frequency: Measurements across frequency range reveal filter response, resonances, and bandwidth
  • Calibration: Systematic error correction using known standards; essential for accurate measurements
  • Time-domain transformation: Convert frequency-domain data to time-domain for fault location and step response

VNA Applications

VNAs are essential for RF and microwave characterization:

  • Filter characterization: Measure passband response, insertion loss, return loss, and group delay
  • Amplifier testing: Gain, gain flatness, stability analysis, and compression measurement
  • Antenna measurement: Input impedance, VSWR, and efficiency characterization
  • Cable and connector testing: Loss, return loss, and fault location
  • Component modeling: Extract accurate models for simulation from measured S-parameters
  • Material characterization: Determine permittivity and permeability of materials

Advanced Network Analysis

Specialized network analysis techniques address complex measurement challenges:

  • Multi-port measurements: Characterize devices with more than two ports using switch matrices or multi-port VNAs
  • Balanced device measurement: Analyze differential circuits using mixed-mode S-parameters
  • Pulsed measurements: Characterize devices that cannot sustain continuous operation
  • Hot S-parameters: Measure S-parameters of active devices under operating bias
  • Noise parameter measurement: Determine optimal source impedance for minimum noise figure

Time-Domain Reflectometry

Time-Domain Reflectometry (TDR) characterizes transmission line impedance and discontinuities by analyzing reflections of a fast edge or pulse, providing essential insight into signal integrity and interconnect performance.

TDR Operating Principles

TDR systems inject a fast signal transition and observe the reflected waveform:

  • Step generator: Produces fast-edge step function with rise time typically 30-50 picoseconds for high-resolution instruments
  • Sampling receiver: High-bandwidth sampler captures incident and reflected waves
  • Impedance calculation: Reflection coefficient at each point determines local impedance: Z = Z0 * (1 + rho) / (1 - rho)
  • Distance resolution: Determined by system rise time; faster edges resolve smaller discontinuities
  • Distance accuracy: Depends on knowledge of propagation velocity in the transmission medium

The reflected waveform provides a spatial map of impedance along the transmission line, with time delay corresponding to physical position.

TDR Applications

TDR serves critical applications in electronics test:

  • PCB impedance verification: Verify controlled-impedance traces meet design specifications
  • Connector characterization: Evaluate connector impedance match and discontinuities
  • Cable testing: Locate faults, measure characteristic impedance, and assess cable quality
  • Package characterization: Measure bond wire inductance and package parasitics
  • Via characterization: Assess via impedance and optimize via stub lengths
  • Fault location: Locate opens, shorts, and impedance anomalies in cables and interconnects

TDR Measurement Techniques

Advanced techniques enhance TDR measurement capabilities:

  • Differential TDR: Characterize differential transmission lines using odd-mode and even-mode excitation
  • TDT (Time-Domain Transmission): Measure transmitted signal to characterize loss and delay
  • De-embedding: Remove effects of fixtures and adapters from measurements
  • Impedance profiling: Mathematical processing to separate multiple reflections and reveal true impedance profile
  • S-parameter extraction: Convert TDR data to frequency-domain S-parameters for simulation

TDR Limitations and Considerations

Understanding TDR limitations ensures accurate interpretation:

  • Rise time effects: Finite rise time limits spatial resolution; features smaller than effective pulse width appear smoothed
  • Multiple reflections: Reflections from distant discontinuities can interfere with closer measurements
  • Loss effects: Transmission line loss attenuates reflections from distant features
  • Dispersion: Frequency-dependent propagation velocity causes pulse spreading in lossy lines
  • Calibration requirements: Accurate reference plane definition essential for meaningful results

Jitter and Phase Noise Measurement

Jitter and phase noise characterize timing imperfections in clock and data signals, representing complementary time-domain and frequency-domain views of the same underlying phenomena. These measurements are critical for high-speed digital systems and communication equipment.

Understanding Jitter

Jitter is the time-domain deviation of signal edges from their ideal positions:

  • Random jitter (RJ): Unbounded Gaussian distribution from thermal noise, shot noise, and other random processes
  • Deterministic jitter (DJ): Bounded, repeatable timing errors from specific causes
  • Data-dependent jitter (DDJ): Varies with data pattern; includes inter-symbol interference effects
  • Periodic jitter (PJ): Sinusoidal components at specific frequencies; often from power supply or crosstalk
  • Duty-cycle distortion (DCD): Asymmetry between positive and negative half-periods

Total jitter (TJ) at a specified bit error rate combines random and deterministic components, with RJ contribution extrapolated to account for rare events.

Jitter Measurement Techniques

Different measurement approaches provide various insights into jitter behavior:

  • Time interval analysis: Measure period or edge-to-edge timing with high-resolution time interval analyzers
  • Real-time oscilloscope: Capture long sequences for statistical analysis; enables jitter decomposition
  • Equivalent-time oscilloscope: Achieve sub-picosecond timing resolution for repetitive signals
  • Phase detector method: Mix signal with clean reference to convert phase variations to amplitude
  • BERT-based measurement: Bit error rate testing scans threshold timing to construct bathtub curve

Understanding Phase Noise

Phase noise is the frequency-domain representation of timing instability:

  • Definition: Spectral density of phase fluctuations, expressed as dBc/Hz at specified offset from carrier
  • Relationship to jitter: Integration of phase noise over relevant bandwidth yields RMS jitter
  • Noise floor: Far-from-carrier phase noise typically limited by measurement system noise floor
  • Close-in phase noise: Near-carrier noise from 1/f noise processes in oscillator components
  • Spurious signals: Discrete frequency components appearing as spurs on phase noise plot

Phase noise directly impacts system performance in communication systems, where it causes reciprocal mixing and constellation rotation.

Phase Noise Measurement Methods

Several techniques measure phase noise with varying sensitivity and capability:

  • Direct spectrum analysis: Measure carrier and sideband power with spectrum analyzer; limited by analyzer phase noise
  • Phase detector method: Compare DUT to clean reference in quadrature; mixer output proportional to phase difference
  • Frequency discriminator: Delay line converts frequency fluctuations to amplitude; enables single-source measurement
  • Cross-correlation: Use two measurement channels to average out uncorrelated instrument noise; improves sensitivity
  • Dedicated phase noise analyzers: Integrated instruments combining multiple techniques for optimal performance

Jitter and Phase Noise Applications

These measurements are critical across diverse applications:

  • High-speed serial links: Verify transmitter jitter meets specifications; assess receiver jitter tolerance
  • Data converter clocking: Clock jitter directly degrades ADC and DAC SNR at high frequencies
  • PLL characterization: Measure loop bandwidth, phase margin, and output phase noise
  • Oscillator qualification: Specify and verify crystal oscillator and synthesizer performance
  • Radar systems: Phase noise limits range resolution and clutter rejection capability

Load-Pull Systems

Load-pull measurement systems characterize RF and microwave power amplifiers and oscillators by varying the load impedance presented to the device and measuring performance at each impedance condition.

Load-Pull Fundamentals

Load-pull systems map device performance across the Smith chart:

  • Impedance tuning: Present controllable impedance to device output through mechanical or electronic tuners
  • Contour mapping: Measure output power, efficiency, gain, or linearity at each impedance point
  • Optimal impedance identification: Locate the load impedance that maximizes desired parameter
  • Trade-off analysis: Understand performance trade-offs between power, efficiency, and linearity
  • Matching network design: Use load-pull data to design output matching networks

Load-pull results are essential for power amplifier design, where optimal load impedance cannot be predicted from small-signal S-parameters alone.

Tuner Technologies

Different tuner types offer various capabilities and trade-offs:

  • Mechanical slide-screw tuners: Probe positioned along transmission line creates adjustable reflection; excellent for high-power fundamental load-pull
  • Electronic tuners: PIN diode or FET-based switched networks provide fast impedance changes for automated measurement
  • Hybrid tuners: Combine mechanical pre-matching with electronic fine tuning for coverage and speed
  • Active load-pull: Inject synthesized signals to create virtual loads; achieves impedances beyond passive tuner capabilities including negative resistance
  • Harmonic tuners: Independent control of fundamental and harmonic impedances using multiplexed tuner paths

Load-Pull System Configuration

Complete load-pull systems integrate multiple subsystems:

  • Input tuner: Matches source impedance to device input for maximum power transfer or noise optimization
  • Output tuner: Primary measurement tuner presenting variable load impedance
  • Bias tees: Inject DC bias while maintaining RF path integrity
  • Power meters: Measure incident and reflected power at device ports
  • Vector network analyzer: Characterize tuner impedance and system calibration
  • Signal sources and receivers: Generate stimulus and measure device output for gain and linearity

Advanced Load-Pull Techniques

Sophisticated techniques address complex characterization needs:

  • Source-pull: Vary source impedance while measuring noise figure or input match sensitivity
  • Harmonic load-pull: Control impedance at second and third harmonics for waveform engineering
  • Mixed-mode load-pull: Characterize differential amplifiers with common-mode and differential-mode impedance control
  • Modulated load-pull: Measure performance with realistic modulated signals rather than CW
  • Time-domain load-pull: Capture voltage and current waveforms at device terminals for deep design insight

Load-Pull Applications

Load-pull measurements enable critical design decisions:

  • PA matching design: Determine optimal load impedance for power, efficiency, or linearity targets
  • Device model validation: Verify that transistor models accurately predict large-signal behavior
  • Process characterization: Establish device capability for power and efficiency across process variations
  • Oscillator design: Find load conditions that sustain oscillation with desired frequency and power
  • Doherty amplifier design: Characterize transistors at multiple load conditions required for Doherty operation

Measurement Accuracy and Uncertainty

Understanding and managing measurement uncertainty is fundamental to valid test results. Every measurement has associated uncertainty from multiple sources that must be quantified and controlled.

Sources of Measurement Uncertainty

Uncertainty arises from various factors:

  • Instrument accuracy: Specified accuracy of the measurement instrument including linearity, offset, and gain errors
  • Environmental effects: Temperature, humidity, and electromagnetic interference affecting measurements
  • Interconnect losses: Cable attenuation, connector mismatch, and contact resistance
  • Loading effects: Instrument input impedance affecting the circuit under test
  • Noise: Random fluctuations limiting measurement resolution
  • Repeatability: Variation in repeated measurements under identical conditions

Calibration and Traceability

Calibration ensures measurement accuracy through reference to known standards:

  • Traceability chain: Unbroken series of calibrations linking to national metrology institutes
  • Calibration intervals: Periodic recalibration maintains confidence in measurement accuracy
  • In-situ calibration: System-level calibration in the measurement configuration
  • Transfer standards: Stable, characterized devices used to transfer calibration between instruments
  • Calibration verification: Regular checks using known references to confirm calibration validity

Guardbanding and Test Limits

Test limits must account for measurement uncertainty to ensure specifications are truly met:

  • Simple guardbanding: Tighten test limits by the measurement uncertainty to ensure specification compliance
  • Risk-based guardbanding: Adjust guardbands based on acceptable probability of incorrect decisions
  • Shared risk: Supplier and customer agree to share risk of marginal devices being accepted or rejected
  • Statistical guardbanding: Use population statistics to optimize guardbands for yield while maintaining quality

Best Practices for Analog Measurement

Following established best practices ensures reliable, repeatable measurements across analog test applications.

Signal Integrity Considerations

Maintaining signal integrity throughout the measurement path:

  • Impedance matching: Maintain matched impedances to minimize reflections and standing waves
  • Shielding: Use shielded cables and enclosures to prevent interference pickup
  • Ground loops: Avoid ground loops that introduce 50/60 Hz interference and low-frequency noise
  • Cable routing: Separate sensitive measurement paths from high-power or noisy signals
  • Connector maintenance: Clean and inspect connectors regularly; torque to specification

Measurement Technique Selection

Choosing appropriate techniques for specific measurements:

  • Match instrument to measurement: Select instruments with specifications exceeding measurement requirements
  • Consider frequency content: Ensure bandwidth exceeds signal frequency content including harmonics
  • Averaging and filtering: Use appropriate averaging to reduce noise while maintaining measurement bandwidth
  • Dynamic range considerations: Position measurement within instrument's optimal range, avoiding both noise floor and saturation
  • Coherent sampling: For FFT-based measurements, choose sample rates that achieve integer periods

Documentation and Reproducibility

Ensuring measurements can be reproduced and validated:

  • Measurement procedures: Document detailed step-by-step procedures for critical measurements
  • Equipment configuration: Record all instrument settings including range, coupling, filtering, and averaging
  • Environmental conditions: Note temperature, humidity, and other relevant conditions
  • Uncertainty statements: Include uncertainty estimates with measurement results
  • Version control: Track changes to procedures and configurations over time

Related Topics