# Nonlinear Equations: Analyzing Nonlinear Electronic Components and Circuits

LEDs and transistors are quintessential nonlinear electronic components

Nonlinear electronic components are commonplace in many systems and circuits. Readers should already be familiar with transistors, LEDs/diodes, and saturated core transformers. When taken together alongside linear passives, and nonlinear components run in a linear range, you can construct complicated nonlinear systems that enable a variety of unique applications.

Utilizing the proper nonlinear equations for your system can be difficult without the right tools. DC analysis of nonlinear electronic components and circuits is rather simple, but constructing and analyzing AC nonlinear circuits can be difficult without the right simulation tools. There are a number of SPICE-based analyses you can use to examine your circuit behavior and extract important insights about your system.

## Nonlinear Equations: Types of Electronic Analysis

Understanding the behavior of nonlinear electronic components and circuits requires identifying the right analysis to run. Misinterpretation of the results can be disastrous for any system as it leads to incorrect design choices. Examining individual nonlinear electronic components is usually quite easy and can be performed by hand. Similarly, very simple nonlinear circuits can be comparatively easy to examine by hand. Eventually, you may come upon a transcendental equation or system of transcendental equations without analytic solutions, and you’ll be forced to use numerical techniques.

The table below shows some of the most important fundamental behavior you can examine during circuit analysis with a SPICE-based simulator. These analyses are useful for a variety of nonlinear circuits, including nonlinear mixers, modulators, active filters, analog arithmetic circuits, and basically any circuit involving diodes and/or transistors.

 Behavior Analysis Notes Approach to steady state, time domain Transient analysis Involves DC, AC, or an arbitrary source. This will show distortion and harmonic/intermodulation generation; exact frequencies can be extracted with an FFT. Steady state behavior, frequency domain Harmonic balance This can account for a source with multiple harmonics, such as a modulated source or a source with spurious harmonics. DC transfer curve DC small signal analysis This is effectively an iterative method. Your results are most accurate when you use high resolution (small DC step size). AC transfer at multiple frequencies, single amplitude Small signal AC frequency sweep This only considers a single AC amplitude, but you can determine the behavior over a range of frequencies. This will not show clipping/distortion or harmonic/intermodulation generation. Nonlinear stability, time domain at individual frequencies and amplitudes Transient analysis at various frequencies Any nonlinear circuit with feedback and gain may exhibit instability (i.e., divergence), limit cycles, or relaxation behavior at various input amplitudes.

Note that not all SPICE-based simulators can perform these analyses with built-in functions, and you may need to program different steps in each analysis manually. The standard Gauss-Jordan technique is only useful for linear time-invariant circuits, or in nonlinear time-invariant circuits that have been approximated as linear circuits around a specific quiescent operating point. This should show the value of a SPICE-based simulator that can perform iterative sweeps with user-selected step sizes.

## Limitations of SPICE-based Nonlinear Analyses

By now, the power of the wide variety of SPICE-based circuit simulation tools should be obvious, especially when these tools are integrated into your schematic design tools. These tools are extremely useful for determining transient behavior, examining linear and nonlinear AC circuits in the frequency domain and time domain, respectively, smoke tests, and much more. However, there are some limitations of SPICE-based simulations that any designer should understand, particularly when trying to predict and explain nonlinear effects in complicated circuits.

As SPICE-based simulations are run directly from a schematic, most cannot account for parasitic effects in linear and nonlinear circuits. In PSpice, a user can assign parasitics to components in the schematic on the fly per the information present in datasheet. These effects can only be specified once you’ve arranged your layer stack and created a layout for your system. In order to account for these effects, you would need to include various capacitors and inductors in your board.

There are a number of parasitics extractors available on the market, but they still require creating a layout and layer stack. These parasitics become quite important in a variety of nonlinear circuits and are responsible for effects like crosstalk, undesired ringing/ripple in a PDN, transmission line behavior, and coupling. They can also create difficulties in impedance matching, especially in RF power amplifier circuits, which are typically run near saturation.

Problems like component lead and copper roughness, as well as manufacturing imperfections in passive components, can cause a linear component to behave like a nonlinear electronic component at high frequencies (e.g., at mmWave frequencies). With AC signals, modulated signals, analog pulses, and arbitrary analog signals, this nonlinear behavior generates intermodulation products and harmonic generation.

Manufacturing imperfections in RF mixers can produce nonlinear phenomena

Although the mathematics describing these phenomena is straightforward, they cannot be reliably predicted analytically, and these effects cannot be considered in the schematic or layout. Instead, they must be measured with real components. RF component manufacturers often provide passive intermodulation and harmonic generation data as functions of input voltage/current and frequency. This data can help you determine signal and frequency limits and which components are best for your system.

Nonlinear electronic components and circuits are much easier to design and analyze when you use the right PCB design and analysis software package. The toolset in OrCAD PSpice Simulator and the broader suite of analysis tools from Cadence are ideal for circuit design and analysis of linear and nonlinear circuits for a variety of applications. Rather than building circuits for every component by hand, you can access thousands of standard component models to build and analyze your circuits.