We will now learn about techniques for analyzing dynamic circuits, that are governed by differential equations. We will emphasize fundamental concepts behind dynamic nonlinear networks, time domain analysis of nth-order nonlinear networks, frequency response concepts, circuit analysis techniques for memristive networks and energy approaches (Lagrangian, Hamiltonian). We cannot hope to cover all the analysis techniques for dynamic nonlinear networks in detail in one chapter. Nevertheless, this chapter should prepare the reader for picking up advanced techniques for analyzing dynamic nonlinear networks from any specialized references.
Simulated (blue) and experimental (red) limit cycle of a Van der Pol oscillator derived from Chua’s circuit [1]
We mean a network containing only two-terminal fundamental circuit elements and independent sources. No dependent sources, ideal transformers, gyrators, etc. are allowed.
2.
With respect to \(\mathscr {D}\) with memristors, the concept of using (ϕM, qM) to determine the degree of complexity and write network equations is further explored in Sect. 4.4.1.
3.
We will primarily focus on capacitor circuits in this section since the corresponding dual inductor circuit(s) can be easily derived using the ideas of duality discussed in Sect. 4.1.2. The reader is encouraged to derive the results for the dual inductor case as they read this section, to enhance their conceptual understanding.
4.
It would be helpful to review Sect. 1.9.3, specifically the memory and continuity properties.
5.
Stability is a system property, not a signal property. We say the signals associated with a stable system are bounded. In system terminology, we are using the concept of bounded-input bounded-output (BIBO) stability.
6.
The delta function is used to model point charges in physics. Using the theory of distributions from advanced mathematics, the unit impulse can be rigorously defined as a “generalized” function imbued with most of the standard properties of a function. In particular, most of the time, δ(t) can be manipulated like an ordinary function.
7.
We say “differentiating in the distribution sense” to emphasize that whenever we differentiate a function which has a jump discontinuity at t = t0, i.e., f(t) jumps from \(f(t_0^-)\) to \(f(t_0^+)\), we must include the corresponding impulse in the derivative:\(f'(t_0)=[f(t_0^+)]-f(t_0^-)]\delta (t-t_0)\).
8.
Historically, relaxation oscillators were designed using only two vacuum tubes, or two transistors, such that one device is operating in a “cut-off” or relaxing mode, while the other device is operating in an “active” or “saturation” mode.
9.
In fact, Fig. 4.26a could model the classic 555 timer, since the nonlinear DP characteristic can also be obtained by simply using two BJTs.
10.
We will utilize the ideas from this section and Sect. 4.3 to derive important small-signal AC characteristics of memristors in Sect. 4.4.2.
11.
If Eq. (4.69) have several solutions, we choose one and stick to it.
12.
We will implicitly assume that the system is stable in the neighborhood of Q. For details, please refer to [12].
13.
In fact, this is also true for linear capacitors and inductors.
14.
Dr. Muthuswamy thanks Dr. Jevtic for valuable discussions over the years, including suggesting Needham’s excellent text on “Visual Complex Analysis.”
15.
This is a consequence of the Fundamental Theorem of Algebra: a cubic will have at least one real root.
16.
Bombelli is generally regarded as the father of complex numbers.
17.
A phasor is essentially a complex number written in exponential or Euler form.
18.
The convention is to say “current leads/lags voltage,”’ not “voltage lags/leads current.”
19.
It is important to note that we do not say frequency response since that is a term reserved for linear systems.
20.
Many thanks to Dr. Jevtic and Dr. Thomas for reviewing and correcting errors in this section.
21.
We will only focus on linear elements, for brevity. Specifically with respect to action and coaction definitions for the memristor, please see [4, 22].
22.
The following interpretations are meaningful only for small-signal sinusoidal excitations at a fixed frequency. Such interpretations however often provide valuable information for circuit designers in their analysis of physical nonlinearities. The main point is: depending on the operating point and the operating frequency, the small-signal model of a device may be either resistive, inductive, or capacitive.
23.
This change in the steady-state dynamic behavior of a circuit as one (or more) parameters are varied is called a bifurcation. The parameter that is being varied is called the bifurcation parameter. A detailed study of bifurcations is beyond the scope of this book.
24.
Although the circuit could theoretically oscillate when \(\mathscr {N}_R\) is a short circuit (passive but not strictly passive), no oscillation is possible in practice because the connecting wire always has some small but nonzero resistance.
25.
If a trajectory were to intersect itself at \((\hat {x}_1,\hat {x}_2)\), then its slope \(\frac {dx_2}{dx_1}\) would have two different values at \((\hat {x}_1,\hat {x}_2)\). This is impossible since our system of equations is deterministic, not stochastic.
26.
Our reasoning does not prove that all trajectories must tend towards a unique limit cycle, although this is actually the case for the particular v − i characteristic. The particular question of the number of limit cycles for a second order autonomous ODE is unsolved and is famously referred to as “Hilbert’s sixteenth problem.”
References
Ambelang, S., Muthuswamy, B.: From Van der Pol to Chua: an introduction to nonlinear dynamics and chaos for second year undergraduates. In: Proceedings of the 2012 IEEE International Symposium on Circuits and Systems, pp. 2937–2940 (2012)
Elwakil, A.S., Kennedy, M.P.: Chaotic oscillator configuration using a frequency dependent negative resistor. J. Circuits Syst. Comput. 9(3,4), 229–242 (1999)
