VNA FAQ: What are Linear Devices?

Engineering is hard. RF engineering is even harder. Complicated concepts, tons of terminology—it’s information overload. Even the simplest of questions have convoluted answers, making it difficult to know how to even get started learning.

Welcome to the second installment of the “VNA FAQ” blog series! In each blog of this series, I’ll provide bite-sized answers to a handful of related, common VNA questions. Today, we’ll take a look at device linearity.

Passive devices we call linear

So, what on earth is a “linear” device? Because one thing we know is linear is…a line, right? You got “m” as your slope, your “x” value, and “b” as your y-intercept (Figure 1). Super simple. Another thing we know is linear is a resistor. And we know that because the math checks out and so do the graphs.

*Figure 1: The line equation tells us what a linear relationship is supposed to look like. *

The characteristic equation for a resistor is old, reliable Ohm’s Law, V = IR. We do a little arithmetic to see that Ohm’s Law fits into our line equation (Figure 2).

*Figure 2: With a little arithmetic manipulation, we can see that Ohm’s Law fits the line equation format. *

When we graph it, the IV curve—or the DC linear response—of the resistor behavior traces a straight line (Figure 3).

Figure 3: Linear relationship between voltage and current in a DC resistor circuit.

And resistor with a variable voltage source, or in an AC circuit, maintains this linear relationship between voltage and current regardless of input frequency (Figure 4). Everything looks clean.

Figure 4: AC circuit response for a resistor showing that voltage and current traces are in phase.

But do you know what we also call “linear”? Capacitors and inductors. How does that make sense? They are passive devices, sure. That means that they have predictable performance based on their physical properties—dimensions, material. I got that. But when we look at their characteristic equation and graph, how do we consider them “linear”?

*Figure 5: Capacitor and inductor characteristic equations. *

All of a sudden, we are in calculus now, because these derivatives popped out of nowhere (Figure 5). Looking at the DC analysis just continues the confusion. At time = 0, the capacitor acts like a short, but then quickly charges up until no current is flowing and it is just acting like an open circuit (Figure 6).

Figure 6: Capacitor DC transient response showing the current at voltage relationship after a switch has just been closed and then opened.

Likewise, the inductor at t = 0 acts like an open circuit with no current flowing, but then transitions to acting like a short as time goes on (Figure 7). As you can see, this does not graph a straight line.

Figure 7: Inductor DC transient response showing what happens just after a switch has been closed.

Things do not get any better when we look at an AC circuit either. The current and voltage are all out of phase (Figure 8). So, where is the linearity? What is going on?

*Figure 8: Inductor AC circuit response showing the relationship between voltage and current given a variable source. *

What makes a device linear:

Well, to save you from all the confusion, heartache, and turmoil I went through when first learning about these component concepts, I will tell you. The first thing I wish I knew is that just because a relationship is linear does not mean it’s straightforward. Let us take another look at the characteristic equations.

For the resistor, we changed the equation format to show that the relationship between the current and voltage, given resistance, is just a scaling factor by the reciprocal of that resistance. This function graphs a line of slope 1 / R for the DC response (Figure 9). The frequency response of the resistor is constant since the relationship between voltage and current has no time dependence (Figure 4). This is all pretty straightforward. Capacitors and inductors are not as straightforward in their voltage-current relationships, but they are still linear.

Figure 9: The relationship between the current and voltage is linear given resistance.

What makes inductors and capacitors more intimidating and seemingly complicated is their time dependence. So, instead of the characteristic equations defining a relationship between two constants, capacitor and inductor equations define a relationship between a constant and a rate of change. The nature of that relationship, however, is still linear. In fact, for a capacitor, similar to resistors, the relationship between current and the rate of change of voltage graph as a line of slope C (Figure 10).

Figure 10: The relationship between the rate of change of voltage and current given resistance is linear.

Similarly, for an inductor, graphing the voltage against the derivative of current with respect to time results in a line of slope L (Figure 11). \

Figure 11: The relationship between the voltage and the rate of change of current is linear.

When we take a look at the AC circuit response for these devices, we more clearly see that the relationship between voltage and current for the resistor, capacitor, and inductor are all constant. Then what is going on with the phase? While the resistor’s AC response traces are in-phase because there is no time dependence involved (Figure 4), the shift in phase between the two traces for the capacitor and inductor responses indicates a time-dependent relationship. In the capacitor response, the voltage lags the current. This makes sense because the capacitor resists a change in the voltage (Figure 12). The inductor resists a change in current, and unsurprisingly we see that the current lags the voltage (Figure 8). The relationship between the voltage and current is still linear, regardless of the phase.

*Figure 12: The AC circuit response of a capacitor shows the time dependence of the voltage and current relationship. *

The second thing I wish I knew is why we even care to classify a component’s behavior as “linear”? Why does it matter? If you search this online, the explanation you might find is that identifying component behavior as “linear” is significant because it tells us that the behavior is mathematically deterministic. It is “closed form.” Basically, if you tell me about the voltage across a resistor or the rate of change of voltage across a capacitor, I can calculate the current flowing through each. Same thing for inductors—if you tell me the rate of change of the current, I can tell you the voltage.

This is important because it allows us to create component specifications, or expectations, against which we evaluate. We can solve circuits made from linear elements exactly. So, classifying a component as linear means that we can mathematically predict the behavior of that component in every scenario, which impacts the stability of our system designs. But, while this answer is true, it is not really the full story. I mean, we can just assess a bunch of nonlinear devices and get an accurate behavioral profile that way—including the nonlinear behavior.

Practically, the real reason we care to classify device behavior as “linear” and “nonlinear” is because we need to know if the relationship between the input and the output is directly proportional, or if there are additional factors at play. If there are other factors impacting the output, then we need to test our device under those conditions to accurately understand its behavior.

This is the same logic for identifying devices as “passive” or “active.” The output produced by a passive device is governed by the input signal and the intrinsic characteristics of the passive device—again, the dimensions, materials, and whatnot. Active device behavior, on the other hand, is based on the internal transistor’s operating conditions as well as the input signal and materials. We can “bias” active devices, which basically means we can finetune them to have a more refined operation. Again, the test engineers must account for more variables. The more variables to account for, the more robust your testing needs to be.

Are all passive devices linear?

If all active devices behave nonlinearly, then do all passive devices behave linearly? Are “passive” and “linear” redundant terms? Does calling a device “passive” and calling it “linear” mean the same thing? Well, while it is true that all active devices behave nonlinearly past a certain power level, the same principle applies to passive devices. The difference is that active devices require less signal power to go nonlinear. They are more sensitive.

While uncommon, nonlinear behavior from passive devices happens when they receive high enough power. A passive filter, for example, behaves nonlinearly when driven with power high enough to change its dielectric or saturate its inductor.

For most common passive elements, note that pushing them with so much power that they act nonlinearly will likely damage or destroy the component. Defective or damaged passive devices, or those that need maintenance due to corrosion, might also behave nonlinearly. However, corner-case passive networks display nonlinear behavior more readily than standard passive components, so particular care should be taken when testing them.

Filters implemented with inductors using ferrite cores, for example, behave nonlinearly due to inherent nonlinearities of the core material long before the wire melts from excessive power. Another example is passive intermodulation distortion (PIM). Base-station operators test for passive IMD on base station antenna feeds. The device under test in this case is the junction between two dissimilar metals. These two different metals cause a diode effect, which causes PIM. It takes lots of power to observe the PIM, but not enough to damage the feed lines.

To clarify, the terms “linear” and “nonlinear” describe the behavior of the device. Engineers determine device behavior by comparing the input and output signals. Calling a component “passive” or “active” describes the internal operation of that component. The internal operation of a device includes everything happening between the input and output ports.

How does the behavioral linearity of your device impacts network analysis?

Characterizing passive components has fewer variables at play than characterizing active components. This means that the network analyzers and software applications needed to characterize the devices are more straightforward. When working with passive devices, engineers do not need to test nonlinear behavior, use sophisticated compensation techniques for distortion, or optimize the operational region through biasing.

This list provides an overview of VNAs that specialize in passive device characterization:

With more variables to consider, active devices require more robust testing such as pulsed-RF, noise figure, distortion, and EVM measurements. Because actives device more readily exhibits nonlinear behavior, the best network analyzers for active device characterization need distortion and distortion compensation analysis capabilities.

This list provides an overview of VNAs that specialize in active device characterization:

Conclusion:

At first glance, capacitors and inductors do not seem to share linear characteristics with resistors. However, looking a little harder at the characteristic equations reveals that all three components simply scale their independent variable, regardless of whether that independent variable is a constant or a derivative. So, all three components are, in fact, linear, passive devices.

If you like what you learned here today, be sure to check out more of our RF Explained videos to learn about other engineering topics. If you want more in-depth, free educational resources I would also highly recommend checking out Keysight University, which is our online education platform for all things engineering. \