# The Basics of Linear vs. Nonlinear Circuits

If you recall your first electronics class, it is likely that all the circuits you built were linear circuits, meaning they only contained resistors, capacitors, and non-ferritic inductors. While this is a great way to get an important introduction into the fundamentals of basic circuits, almost all modern circuits contain many more complicated elements and functionality.

Most modern circuits are run at a high enough signal level that they exhibit a nonlinear response. Consider a transistor, the fundamental building block of modern computing. We take advantage of the nonlinear response in its output current to define digital signals in a digital system. This nonlinear response leads to saturation in the current output, corresponding to an ON digital signal. There are plenty of other circuits you can construct in your next PCB that can provide functionality you might need.

## Nonlinear Circuits: The Basics

The key point that distinguishes a nonlinear circuit from a linear circuit is the relationship between the input and output signal. If you graph the output signal versus the input signal for a linear circuit, then the graph will be a straight line for all input signal level. With a nonlinear circuit, the output will not be a straight line. Instead, the output will be a curve.

Another possibility in a nonlinear circuit is that the output from the circuit is a piecewise function of the input. Note that piecewise functions are nonlinear, even if each region of the function may be linear within a certain range of inputs. Due to the discontinuity in the output signal in different regions, then the circuit is, by definition, nonlinear.

If this sounds confusing, consider a full wave rectifier circuit for power conversion. The output from the rectifier circuit (before being smoothed with a capacitor) is essentially the absolute value of the input, and the function abs(x) does not satisfy the mathematical definition of a linear function. Once the output is smoothed with a capacitor, the output is a DC signal plus some ripple wave that is not sinusoidal. You input an AC sinusoidal signal, but you get out a DC voltage plus a non-sinusoidal wave.

This type of nonlinear circuit takes advantage of the response in nonlinear circuit elements (in particular, 4 diodes) to produce the desired nonlinear response in the output. A typical circuit network can be divided into linear and nonlinear portions, which contain linear and nonlinear circuit elements, respectively. Once the output from a linear section of a network is input into a nonlinear circuit element, the overall output from the circuit will be nonlinear. There are very few exceptions to this (i.e., placing the output from a differentiator into an integrator).

*Half wave and full wave rectifiers are two common nonlinear circuits*

It is common for some folks to make the case that a linear circuit with a transient response (e.g., an RLC network) is actually a nonlinear circuit. In reality, this is not the case. The relationship between the output current and the driving voltage/current is still linear in this case. The transient response is a nonlinear function of time, not the input voltage or current, so this is still defined as a linear circuit. The response under AC driving is also a complex linear function of the input signal amplitude and frequency (i.e., simple multiplication), thus we still have a linear response.

## Signal Distortion and Feedback in Nonlinear Circuits

The ability for a nonlinear circuit to reshape (or distort) an input signal, whether it is sinusoidal or otherwise, is a central feature of a nonlinear circuit. The transition from the linear region at a low input voltage/current level the nonlinear region at high input can distort a signal, both in the frequency domain and the time-domain. This effect is quite important in circuits with saturation and feedback, both in the frequency domain and the time domain.

As an example, consider an operational amplifier, which is a common nonlinear circuit element. When driven with a low level input signal, the output will be a linear function of the input. At high input, the output will level off and saturate at a fixed value. This is useful for saturating an input sinusoidal signal once it rises above some saturation threshold, essentially converting the sinusoidal wave to a square wave (e.g., a Schmitt trigger circuit). This idea, where different frequencies can be modified as a function of the input signal strength, is a central feature of linear and nonlinear filter design.

Positive feedback in an amplifier or other circuit with nonlinear elements can induce instability in the nonlinear response. Note that instability is not confined to nonlinear circuits; certain linear time-invariant circuits can become unstable in the presence of feedback. No matter which type of circuit you are working with, you can identify regions of stability, both with DC and AC driving, by identifying the poles and zeros in the transfer function for your circuit. This can be done easily by hand for a linear circuit, or it can be done numerically using a circuit simulator for a nonlinear circuit.

*Many nonlinear circuits include amplifiers*

If you need to examine the behavior of nonlinear circuits in your PCB, you need the right PCB layout and design software that includes the circuit and signal analysis tools you need for your design. Allegro PCB Designer and Cadence’s full suite of design and analysis tools can help you design linear and nonlinear circuits. You’ll have access to a large component library with electrical models that easily interface with Cadence’s simulation features.

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