What Are Time Varying Circuits?

Time varying circuits are electrical networks in which one or more component parameters, such as capacitance, inductance, resistance, or transconductance, change as a function of time. The governing equations for such circuits contain time-dependent coefficients, distinguishing them from linear time-invariant (LTI) circuits whose parameters are fixed. The change in parameters may be periodic, as in a switched-capacitor filter that samples and holds charge at a clock frequency, or aperiodic, as in a varactor diode whose capacitance tracks an applied control voltage. Time varying circuits arise throughout analog, mixed-signal, and radio-frequency design, and their analysis requires tools beyond the classical Laplace transform and phasor methods that apply only to time-invariant networks.

The field draws from linear system theory, signal processing, and analog circuit design. A significant practical impetus came from the desire to implement precise analog filters in integrated circuits, where the process variability of resistors and capacitors made RC time-constant accuracy difficult to guarantee without a switched, clocked architecture that maps filtering to ratios of capacitor values rather than their absolute magnitudes.

Circuit Elements with Time-Dependent Parameters

Several physical mechanisms produce time-dependent circuit parameters. Varactor diodes exploit the voltage-dependent junction capacitance of a reverse-biased p-n junction; the capacitance is controlled by an applied DC bias voltage that varies slowly relative to the signal of interest, making the varactor useful in voltage-controlled oscillators and tunable filters. Switches implemented with CMOS transistors or microelectromechanical systems (MEMS) create circuits whose topology changes periodically under clock control, as in switched-capacitor and switched-current circuits. Parametric amplifiers use a time-varying reactive element, historically a variable-capacitance diode pumped at a frequency equal to the sum of the signal and idler frequencies, to transfer energy from a pump source into a signal band with very low added noise. The IEEE Circuits and Systems Society discussion of linear time-varying circuits illustrates how these physical realizations share the same underlying mathematical framework.

Analysis Methods for Time-Varying Circuits

Standard frequency-domain techniques based on the Fourier or Laplace transform apply only to time-invariant systems; applying them directly to a time-varying circuit produces incorrect results because the system cannot be characterized by a single, input-independent transfer function. Instead, analysts use the modulation representation, which describes a periodically time-varying circuit by a doubly indexed transfer matrix relating input and output at multiples of the fundamental clock or pump frequency. The modulop Laplace transform and the harmonic transfer matrix are formal extensions of classical transforms that accommodate periodic time variation. The IEEE Xplore paper on demystifying time-varying circuits and systems provides a unified tutorial treatment of these methods and their application to noise, stability, and signal path analysis.

Switched-Capacitor and Switched-Current Circuits

Switched-capacitor circuits replace resistors with a clock-driven capacitor that periodically charges to an input voltage and discharges into a summing node. Because the effective resistance depends only on the clock frequency and the capacitor value, and because capacitor ratios in CMOS processes are accurate to better than 0.1 percent, switched-capacitor filters achieve precise frequency responses without trimming. The technique underpins analog-to-digital converters, sigma-delta modulators, and programmable gain amplifiers in modern integrated circuits. Noise analysis is a central design concern: thermal noise sampled onto a capacitor of value C has a minimum spectral density set by kT/C, a floor that cannot be lowered by circuit design alone. An IEEE tutorial on computer-aided analysis of switched-capacitor circuits covers state-variable formulation, nonideal operational amplifier effects, and practical simulation strategies.

Applications

Time varying circuits appear in a wide range of analog, RF, and mixed-signal applications, including:

• Switched-capacitor filters and sigma-delta analog-to-digital converters in consumer and professional audio equipment
• Voltage-controlled oscillators using varactor tuning in phase-locked loops and frequency synthesizers
• Parametric amplifiers in radio telescope receivers and satellite communication ground stations, where extremely low noise temperatures are required
• Chopper-stabilized amplifiers for precision DC measurement, which modulate the input, amplify in a frequency range free of 1/f noise, and demodulate to recover the signal
• MEMS-based tunable resonators for reconfigurable RF front ends in software-defined radios