# Parametric Amplifier Circuit Simulation and Applications

June 4, 2020

### Key Takeaways

• A parametric amplifier circuit is an example of a nonlinear circuit where the circuit parameters vary with an input signal.

• An input signal is used to vary the reactance of a nonlinear component, usually a varactor.

• Because the circuit parameters vary with the input signal, you can easily simulate a parametric amplifier circuit using SPICE as long as you can properly model the reactance of the nonlinear elements. This motor control board could benefit from a switching regulator vs linear regulator.

Hidden within the parabolic dishes at the Very Large Array telescope in New Mexico is a circuit for capturing and amplifying very low level VHF/UHF signals. An electronic parametric amplifier circuit (and their optical cousins) is used with very high frequencies, where the input frequency to be amplified may vary. These circuits are also useful where a conventional amplifier does not provide sufficient gain to detect a signal above the noise floor.

Now that typical telecom frequencies have been pushed into the 10s of GHz by inexpensive GaAs FETs and HEMTs, there are practical solutions for low-noise amplification in integrated circuits. Despite the usefulness of transistor-based amplifiers, they’ll be limited at micron wavelengths unless more sophisticated amplifiers are used.

Today, the relevant frequencies lie in the ~100 GHz range, and we may see a resurgence in these circuits when we get deeper into 5G and 6G.  Here’s how you can simulate the design of a parametric amplifier circuit in the time domain and how your circuit will function. ## How a Parametric Amplifier Circuit Works

A parametric amplifier circuit provides adjustable amplification and up/down conversion of an input analog signal. This is done by applying a second sinusoidal signal, called a pump signal, to a nonlinear reactive circuit element. The pump signal causes the reactance of the nonlinear circuit element to vary sinusoidally at the same frequency as the pump signal frequency.

If the idea of a “nonlinear reactive circuit element” sounds vague, remember that you have two options:

• Ferrite inductor or transformer run at magnetization saturation

• A varactor diode, which has capacitance that is a nonlinear function of the voltage drop across the diode

Ferrites need to run at high magnetic fields (i.e., input current) in order to reach saturation. As a result, varactor diodes are normally used in a parametric amplifier circuit to provide nonlinear reactance. In principle, any component that has nonlinear reactance with input voltage/current can be used in a parametric amplifier circuit. A basic parametric amplifier circuit is shown below. Simple parametric amplifier circuit

In the circuit above, the varactor diode is element C3. The RLC circuit legs on the pump and input signals can be designed to be underdamped, so the pump and input signals will have some resonance. This allows the capacitance modulation on the varactor diode to be changed by simply adjusting the input and pump frequencies.

By Kirchhoff’s voltage law, because the input source, its RLC leg, R2, and C3 form a closed loop, the loop forms a damped oscillator with periodic modulation in the natural frequency and damping. This type of system, known as a parametric oscillator, is described with the damped Mathieu equation.

## The Damped Mathieu Equation

Depending on the topology of your parametric amplifier circuit, the damping, natural frequency, or both could vary sinusoidally as the pump signal oscillates. In simulating a parametric amplifier circuit, the role of a SPICE simulator is to solve the damped Mathieu equation in the time-domain.

The damped Mathieu equation is a general equation that describes oscillators with sinusoidally-varying parameters, and it finds applications in mechanics, population dynamics, quantum mechanics, economics, and astrophysics. The damped Mathieu equation is shown below: Damped Mathieu equation for a parametric amplifier circuit.

Note that the time dependence in the above equation arises because the circuit parameters can both be functions of the two voltage sources, both of which oscillate in time. The goal in this circuit is to amplify (or decrease) the input signal by adjusting the amplitude and/or frequency of the pump signal.

The parametric amplifier circuit above will produce higher and lower order harmonics through frequency mixing. These harmonics are generated at C3 and are output into the idler circuit. This frequency mixing only occurs through the use of a nonlinear reactive component (the varactor diode above). Each harmonic generated by the circuit and the gain for each component are defined in the equations below. Output frequencies and gain from the parametric amplifier circuit shown above.

The idler circuit receives all frequencies generated by C3. The point of the idler circuit is to provide bandpass filtration. Finally, the idler circuit provides bandpass filtration which then outputs the desired frequency.

## Time Domain or Frequency Domain Simulations

Simulations with a time-variant circuit need to be performed in the time domain. For nonlinear components, you need to derive a circuit model in the frequency domain. A table of Fourier transforms only helps so much as the Fourier transform of the voltage-dependence for the nonlinear reactive component may not exist, or it is defined as a convolution of two functions. Simulation of nonlinear time-variant systems remains an active research topic.

Whichever type of simulation you opt to perform, you should use verified component models for your components. If you’re skilled at defining circuit models for components in SPICE, you can manually create a circuit model for the varactor diode. Both approaches give you a flexible way to create a time-domain SPICE simulation for a time-variant circuit. Note that this is a simple approach in a SPICE simulation for a parametric amplifier because the voltage-dependence for the nonlinear reactive component is easily defined in terms of the input and pump voltages.

No matter which type of amplifier circuit or oscillator you want to design, you can create your schematics and simulate your system with the best PCB design and analysis software. The front-end design features from Cadence integrate with the powerful PSpice Simulator to create an ideal system for evaluating your parametric amplifier circuit.

Once you’ve created a layout and are ready to examine noise and thermal behavior, you can use Cadence’s suite of SI/PI Analysis Point Tools for post-layout verification and simulation. You’ll have all the features you need to design a variety of RF circuits and systems.

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