A short tutorial of nonlinear circuit theory - part 2

In a previous post, we look at Kirchoff's laws and device constituency relations to derive state equations of an electrical circuit.  Today we will look at some simple circuits and their corresponding state equations.  The simplest dynamic circuit must have at least one capacitor (or inductor).  Let us consider a circuit with one linear capacitor and one nonlinear resistor.

  Case 1: Assume that the resistor is voltage-controlled ($i = g(v)$)

The state equations can then be written as

\begin{equation} \frac{dq}{dt} = -g(v) = -g(q/C) \end{equation}

The trajectories can be visualized on the dynamic route, where we plot $\frac{dq}{dt}$ against $q$ and the arrows indicate how the trajectory will evolve:

Case 2: assume that the resistor is current controlled ($v = f(i)$).  

We have $v = f(i) = f(-dq/dt)$.  Since $v = q/C$ we get $q = Cf(-dq/dt)$.  Using the dynamic route interpretation and depending on the shape of $f$, it is possible to find impasse points, where the trajectory does not know what to do.  

How do we resolve such nonphysical situations?  By adding parasitic elements. Suppose we add a parasitic inductor L in series to the nonlinear resistor.

The resulting state equations will be given by: \[ \begin{array}{lcl}\frac{dq}{dt} & = & i_L = \frac{\phi}{L} \\ \frac{d\phi}{dt} & = & v_L = v_R - v_C = f(i_R) - q/c = f(\phi/L) - q/c \end{array} \] 

By introducing the variable $i_C = -i_L = -\phi/L$ we get the state equations: \[ \begin{array}{lcl}\frac{dq}{dt} & = & i_C \\ \frac{di_C}{dt} & = & -\frac{1}{L}(q/C - f(-i_C)) \end{array} \] By drawing the phase portrait, we see that at the curve $q = Cf(-i_C)$, $di_C/dt = 0$. In fact, we have an oscillator.  

For the nonlinear resistor with a constituency relation as shown above, the existence of the limit cycle can be proved more rigorously by using the Poincaré -Bendixson theorem and the fact that the system has a bounded trapping region and a single equilibrium point that is unstable.

For $f$ an odd function, a second order system in this form is a Liénard equation [5] and the Levinson-Smith theorem [6] allows us to prove the existence of a stable limit cycle when the nonlinear resistor is of a certain form:

Theorem [Levinson-Smith 1942]: Consider the second order equations
\[ \begin{array}{lcl} dx/dt & = & y - F(x) \\ dy/dt & = & -g(x) \end{array} \] Suppose that

• $F$ is an odd $C^1$ function (i.e. $F(-x) = -F(x)$),
• $g$ is an odd $C^0$ function with $xg(x)>0$ for all $x\neq 0$, 
• There is a constant $a>0$ such that $F(x) < 0$ on $0 < x < a$, $F(x) > 0$ on $x > a$, and $F'(x) > 0$ on $x>a$,
• $\int_0^x g(s)ds \rightarrow\infty$ as $|x|\rightarrow \infty$ and $F(x) \rightarrow\infty$ as $x\rightarrow \infty$. 

Then there is a unique stable limit cycle around the origin in the system.

Note that our circuit above satisfies the hypothesis of the Levinson-Smith theorem if our nonlinear resistor has the form $v = f(i)$ where $f$ is a cubic polynomial of the form $x^3-ax$ for $a>0$.  In particular, we get exactly the Van der Pol equation $x'' + \alpha (x^2-1)x' + x = 0$ [8], an equation used to model oscillations occurring in a circuit involving vacuum tubes.

Another condition for guaranteeing the existence of a unique periodic solution is the following result [7]:

Theorem: Consider the second order equations
\[ \begin{array}{lcl} dx/dt & = & y - F(x) \\ dy/dt & = & -g(x) \end{array} \] Suppose that

• $F$ is an odd $C^1$ function and is zero only at $x=0$,$x=a$ and $x=-a$ for some $a>0$,
• $F(x) \rightarrow \infty$ monotonically as $x\rightarrow \infty$ for $x >a$.
• $g$ is an odd $C^0$ function with $xg(x)>0$ for all $x\neq 0$, 

Then the system has a unique periodic solution.

References:

[1] L. O. Chua, Introduction to nonlinear network theory, McGraw-Hill, 1969.
[2] L. O. Chua, C. A. Desoer, E. S. Kuh, Linear and nonlinear circuits, McGraw-Hill, 1987. 
[3] L. O. Chua, Memristor-The missing circuit element, IEEE Transactions on Circuit Theory, Sept. 1971, vol. 18, no. 5, pp. 507-519.
[4] D. B. Strukov, G. S. Snider, D. R. Stewart1 & R. S. Williams, The missing memristor found, Nature 453, 1 May 2008, pp. 80-83. 
[5] A. Liénard,  Étude des oscillations entretenues, Revue génér. de l'électr. 23, (1928), 901 - 902; 906 - 954.
[6] N. Levinson, O. Smith, A general equation for relaxation oscillations Duke Math. J. 9, (1942), 382 - 403.
[7] D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, Oxford University Press, 1977.
[8] Van der Pol, B. and Van der Mark, J., Frequency demultiplication, Nature, vol. 120, pp. 363-364, 1927.