What is dynamic resistance? Is it related to differential resistance or to incremental resistance?

Introduction

Yes, all those terms denote the same concept, that of differential (or incremental, or dynamic) parameter of a device.

In the following I will explain such a concept initially in the context of dynamic resistance, but it will be apparent that the same concept has a wider application.

Note: the explanation uses some calculus concepts, some of which may have been taught at high-school level (derivatives) and some others are definitely college/university-level (Taylor expansion, partial derivatives).

Although for a deep understanding of the subject those math concepts are necessary, the explanation is written so that you can simply take the formulas for granted and skip to the essential points. I.e. the only math that appear on the text below that you must understand is some basic algebra and geometry, and the concept of function and its graphical representation.

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2D-curve case (single independent variable)

When a non-linear device is described by a I-V characteristic curve (like the one of semiconductor diodes), or a family of curves (like the output characteristics of BJTs and MOSFETs), it is often useful to relate the changes of voltage to the corresponding changes of current (these changes are often referred to as increments of the variables).

So if the curve can be expressed in functional form^[1] as $V = V(I)$, we know from calculus that we can approximate the curve $V(I)$ around a point $Q_0=(I_0,V_0)$ as a linear function (1st order Taylor expansion):

\[ V \approx V_0 + \biggl(\frac{dV}{dI}\biggr)_{Q_0} \cdot\; (I - I_0) \]

where $V_0=V(I_0)$.

If we introduce the so-called increments of the variables $\Delta V = V - V_0$ and $\Delta I = I - I_0$, we can rewrite that relation as:

\[ \Delta V \approx \biggl(\frac{dV}{dI}\biggr)_{Q_0} \cdot\; \Delta I \]

In electronics $Q_0$ is called operating point or, in some situations, quiescent point.

In other words, the operating point is the point around which we choose to perform the approximation (how we choose that point is a whole different problem). When dynamical parameters are given in a datasheet they are always accompanied by additional info that describe the operating point where those parameters were calculated or, more often, measured.

The derivative in that formula has the dimensions of a resistance and that is what we refer to as dynamic resistance at the chosen operating point, i.e. calculated when $I=I_0$^[2].

\[ r_0 = \biggl(\frac{dV}{dI}\biggr)_{Q_0} = \biggl(\frac{dV}{dI}\biggr)(I_0) \]

Therefore we can write:

\[ \Delta V \approx r_0 \cdot \Delta I \]

It's very important to note that this is an approximation whose accuracy increases as the increment $\Delta I$ gets smaller^[3].

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Interpretations

You can express the same concept in words in many equivalent ways:

• The dynamic resistance represents the slope of the curve near $Q_0$.

• The dynamic resistance represents the slope (angular coefficient) of the straight line that is tangent to the curve at $Q_0$.

• The dynamic resistance tells us how steep the curve is near $Q_0$.

• The dynamic resistance tells us how quickly with the current the voltage varies near $Q_0$.

• The dynamic resistance tells us how much the voltage changes for a given current increment near $Q_0$.

And I'm sure there are some other interpretations that I'm missing right now.

Note that in all formulations is important to understand that the dynamic resistance is a local property of the curve, i.e. changing the point $Q_0$ may change the value of $r_0$.

However, this doesn't mean that it must change. There are important cases where it does not change appreciably or at all, even for large variations of $Q_0$. Knowing when and how a dynamic parameter changes with the quiescent point is often a key part of the knowledge about an electronic device.

From its definition we can also understand the rationale behind its names:

• Differential resistance because it's a derivative.

• Incremental resistance, because it expresses the relationship between the increments of voltage and current.

• Dynamic resistance, because it expresses a relationship between variations (if nothing varies, i.e. everything is static, that parameter tells us nothing).

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Generalization: inverse relationships

The same reasoning can be applied to a curve which expresses the IV relationship as $I = I(V)$, i.e. inverting the dependency between I and V, like in the curve of a semiconductor diode.

In that case we could define a dynamic conductance $g_0 = \dfrac{1}{r_0}$ and we would find that:

\[ \Delta I \approx g_0 \cdot \Delta V = \frac{1}{r_0} \cdot \Delta V \]

where:

\[ g_0 = \biggl(\frac{dI}{dV}\biggr)_{V_0} = \biggl[\, {\biggl(\frac{dV}{dI}\biggr)_{I_0}} \;\, \biggr]^{-1} \]

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Generalization: multiple independent variables; families of curves

This also works for a family of curves, like those of a BJT, where the current depends on more than one variable. The only problem is that in this case we have to use a multivariable Taylor expansion.

For example, assume $I_C = I_C(V_{CE},I_B)$ is the analytic form of the output curves of a BJT. In the $(I_C,V_{CE},I_B)$ space we could choose an operating point $Q_0=(I_{CQ},V_{CEQ},I_{BQ})$ and write:

\[ \Delta I \quad \approx \quad \underbrace { \biggl( \frac{\partial I_C}{\partial V_{CE}} \biggr)_{Q_0} } _{ \textcolor{red}{ \Large h_{oe} }} \cdot \; \Delta V_{CE} \quad + \quad \underbrace { \biggl( \frac{\partial I_C}{\partial I_B} \biggr)_{Q_0} }_{\textcolor{red}{ \Large h_{fe} }} \cdot \; \Delta I_{B} \]

The parameters $h_{oe}$ and $h_{fe}$ are the traditional BJT parameters for the hybrid-h model, and they are all dynamic/differential/incremental parameters.

In particular, $h_{oe}$ represents the dynamic output conductance of the BJT in the point Q. It is also the slope of the specific $V_{CE}-I_{C}$ curve relative to the chosen $I_{BQ}$ base current.

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1. It is not necessary to actually have an explicit analytical form, i.e. an equation, for the curves. It's enough to know that the curves could be expressed in some form that is amenable to analytical reasoning. So these concepts are applicable also to devices for which we only know the curves experimentally, i.e. as a graph of measured values. ↩︎

2. Note that the derivative of V(I) is itself a function of I, so in general it has different values for different values of I. ↩︎

3. From a mathematical point of view, that formula is exact only if the increments are infinitesimal. ↩︎