# Nonlinear Devices and Network Analysis

### Key Takeaways

• Nonlinear devices are more complex to evaluate due to their lack of additivity and homogeneity (AKA superposition principle).

• Nonlinear operating ranges are present in all real-world components; accuracy of simulation models will depend on how well they capture this behavior.

• Nonlinear network analysis focuses on brute force computation to determine the state of the circuit over small time durations.

Diodes are nonlinear devices due to their exponential relationship between voltage and current

Focusing on linearity is an excellent approach when first studying network analysis; solving circuit equations is difficult enough without factoring in various real-world complicating factors. Of course, these nonideal behaviors are precisely what designers in the industry wish to simulate for better performance.

Most introductions focus on a strict distinction between linear and nonlinear devices – the truth is more complicated: within specific operating ranges, some “linear” devices can act nonlinearly (the reverse is also true, thankfully). Designers should understand the general cohort of nonlinear devices and methods that simplify nonlinear analysis for broader circuit applications.

### Nonlinear Op-Amp Considerations

 Saturation Output voltage must be below the positive rail and above the negative rail. Slew rate limited The maximum rate of change in voltage amplitude; additional power on the input will produce no further change in the output. Phase reversal Driving one of the inputs at a rail voltage can cause the output to flip polarity (and the feedback sign) in some op-amp styles.

## Defining and Evaluating Nonlinear Devices

Linear devices (and, by extension, their circuits) are “well-behaved” because their basic parameters do not fluctuate with current or voltage levels. Essentially, linear devices exhibit an eponymous linear relationship between current and voltage, depending on the exact physics of the device. Linearity is only partially accurate as an idealized form in certain operating conditions: every device has nonlinear behavior – some just show it more readily than others. This observation can also work backward, i.e., approximating nonlinear devices with linear behavior. The scope of nonlinear devices is much broader than that of linear devices, but it’s helpful to define linearity to establish the qualities that nonlinear devices don’t possess.

### Superposition Principle

Mathematically, linearity is the coexistence of additivity and homogeneity of two properties:

Linearity is highly desirable in circuit analysis because it allows the response of complex networks to be analyzed piecemeal – in other words, linearity does not require a numerical approach to approximate nonlinear systems. The simplest way to represent nonlinear devices is by determining the transfer function that maps a circuit block's input/output relationship.

Transfer function methodology.

Consider two of the basic nonlinear device families that designers are likely to encounter:

• Amplifiers - At varying signal input levels, the amplifier can boost the signal's amplitude up to its source according to the gain of the network. Critically, the amplifier output has limits according to its provided power rails: as the input signal increases in amplitude, the gain shrinks correspondingly. This behavior continues until the amplifier enters saturation, where no additional power on the input will provide any extra power on the output, i.e., the input is “banging against the rail.”

• Diodes - A diode fails regarding the most straightforward mathematical descriptor of linearity because its relationship between input voltage and output current is an exponential curve. Although most diode use focuses on its ability to regulate current flow unidirectionally, its performance is more complex than a simple one-way switch.

However, treating a nonlinear device linearly with certain restrictions is possible. For example, a BJT is exceedingly nonlinear, but designers (or modeling software) can linearize the relationship between the base-emitter voltage and collector-emitter current. To reiterate: there are no perfectly linear devices – even something as simple as a conductor wire introduces some nonlinearities to a design. It’s common to forego this level of rigor where components often function with near-uniform linearity (i.e., resistors); the level of accuracy comes down to the acceptable amount of error. This point isn’t to aggravate designers but to distinguish between idealized linearity and the nonlinear, nonidealized behavior of all circuit elements exhibited during operation.

## Techniques for Nonlinear Networks

These days, nonlinearity is less of an issue during analysis due to powerful modeling software: computer modeling can brute force even vast networks, given enough time. Analysis of nonlinear circuits (that is, any circuit that contains a device operating in a nonlinear region) focuses on two general methods:

• Transient analysis - Imagine a circuit’s parameters frozen in time. Then, in small, discrete time durations, allow the circuit to move forward, measure the state evolution, and solve the differential equations by working in the time domain. The accuracy of these results will depend on the sophistication of the model/completeness of the equivalent circuit elements and the length of the time duration (a shorter duration is more computationally intensive with better results, and vice versa).

• Perturbation analysis - A more heuristic approach to nonlinear circuits is to evaluate the response after an infinitely long steady state and determine the change in parameters with a slight adjustment to the signal. Notably, the response to the change in input signal produces a linear response, provided the signal is small relative to the original signal. With the isolated response of the perturbation input, designers can determine the circuit's transfer function.

Both methods have their uses when working by hand, but computer simulation/calculation will be the far more practical (and likely accurate) treatment beyond a few circuit elements. Perturbation is arguably more powerful as it ultimately produces a relationship between the input and output; however, these differences are negligible for the designer once the circuit complexity becomes significant enough to require software solutions.