What is Linear and Nonlinear Resistance?

November 22, 2019 Cadence PCB Solutions

Linear and nonlinear resistance components on a PCB

Do you know which of these components exhibits linear and nonlinear resistance?

 

I rarely see the term nonlinear resistance used in textbooks, but it often gets used by the electronics enthusiast practicing the scientific method. The difference between linear and nonlinear resistance forms the basis for understanding different circuit simulations, as well as knowing when to implement them. 

If you’re unsure about the difference between these two terms, you just need to think about the relationship between the current and voltage in any circuit element.

I-V Curves: Linear and Nonlinear Resistance

Every electronic component can be classified as a linear or nonlinear component, depending on whether it has linear or nonlinear resistance, respectively. In the simplest sense, a component with linear resistance will always provide the same resistance, regardless of the input voltage level. In contrast, the current in a component with nonlinear resistance will be some function of the voltage drop across the component.

If it is not clear what this means, it helps to look at the behavior of each type of component on a graph. A graph of current vs. voltage, also called an I-V curve, will immediately tell you the difference between linear and nonlinear resistance. A component with linear resistance will have an I-V curve that is actually a straight line. In contrast, a component with nonlinear resistance will have a nonlinear I-V curve. The image below shows I-V curves for a resistor and a diode driven with a DC voltage source.

 

Linear and nonlinear resistance in a resistor and diode

I-V curves for a resistor and diode, illustrating their linear and nonlinear resistance, respectively.

 

From this curve, we can easily extract the resistance of the resistor from the slope of the line. In this case, we have a 5 Ohm resistor. If we draw a tangent line against the red curve, we can extract the resistance of the component at the particular operating point. In this case, the inverse of the slope of the tangent line is equal to the resistance at the particular voltage. From the data above, and similarly for any component for which you have measurements, you can simply calculate the derivative of the voltage with respect to current; the resulting curve is the component’s resistance at every input signal level. This is shown below for the resistor and diode in the graphs above.

 

Linear and nonlinear resistance in a resistor and diode

Linear and nonlinear resistance of the resistor and diode shown above, respectively.

 

From a conceptual standpoint, you can quickly use these curves to examine the DC behavior in linear and nonlinear circuits. Consider an example where our hypothetical resistor and diode are placed in series with a voltage source. If we look at the resistance curves in the second graph, we see that we can satisfy the maximum power transfer theorem when the input voltage is ~1.05 V, because this is the exact input voltage at which the resistances of both components are equal. This is one interesting case where the maximum power transfer depends on the input voltage because one of the components is nonlinear, i.e., it does not obey Ohm’s law.

Note that we’ve discussed resistors in terms of linear resistance, but the same idea applies to impedance. The impedance of ideal inductors and capacitors is linear, meaning that it is always the same for any applied voltage and at a specific frequency. Capacitors, however, have impedance that is inversely proportional to frequency.

Despite this inverse proportionality, the three fundamental components discussed here are can be combined in any way you like to produce a linear circuit. In other words, the equivalent resistance of the circuit, which must exist according to Thevenin’s theorem, is a constant value that does not depend on the input voltage. We call this type of circuit a linear circuit (i.e., composed of only linear components). Any circuit that contains at least one nonlinear component will have a nonlinear equivalent resistance, thus we call it a nonlinear circuit.

The Effects of Nonlinear Resistance

The inverse of nonlinear resistance (a.k.a., nonlinear conductance) of a component is actually called the transconductance. The second graph above shows the inverse of the transconductance (in DC circuits) or transadmittance (in AC circuits). This value is important in small-signal analysis, where the behavior of a circuit is simulated around a specific operating point.

The nonlinear resistance of a circuit is responsible for a number of effects that are particular to AC circuits and switching digital circuits. Because the current in the circuit and voltage in a nonlinear component are related by a nonlinear function, the component will produce additional harmonic content when an oscillating or switching signal propagates in the component. This leads to distortion of the signal, i.e., the output from the component no longer matches the input. As an example, this is a particular aspect of nonlinear circuits limits the useful input signal level in power amplifiers. Similar effects are seen in other components.

 

Nonlinear resistance in an amplifier

The amplifier IC has nonlinear resistance

 

In reality, no component has truly linear resistance up to an infinite voltage and current. The three fundamental components (resistors, capacitors, and inductors) will exhibit nonlinear resistance once the input voltage or input frequency are large enough. This occurs for a number of reasons, such as the skin effect and roughness in electrical leads and manufacturing imperfections in components. In particular, roughness in electrical leads and in copper traces on the surface of a PCB is known to be responsible for passive intermodulation.

Linear and Nonlinear Components in Circuit Simulations

The above concepts are important when defining simulations during the circuit and schematic design phases. Small-signal analysis was mentioned above, but other important analyses are only defined in terms of linear circuits. Two common examples include transfer functions and pole-zero analysis and AC sweeps in the frequency domain. In general, if a simulation technique can be used for a nonlinear circuit, then it can also be used for a linear circuit.

The analysis listed above is inherently a SPICE-based analysis, thus it cannot account for signal integrity problems that arise due to parasitics in a real PCB layout. Such layout-dependent signal integrity problems can include skew and high frequency resonances due to the fiber weave effect, noise coupling in traces and vias due to crosstalk, and EMI throughout your board. It also cannot account for power integrity problems, which are related to the PDN topology and are responsible for many signal integrity problems.

Analyzing the linear and nonlinear resistance of your components and laying them out in your PCB doesn’t have to be difficult. You can examine how these components affect circuit behavior in pre and post-layout simulation tools when you use the right PCB design and analysis software. Allegro PCB Designer and Cadence’s full suite of analysis tools allow you to examine how these components affect electrical behavior throughout your board.

If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.

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