Digital Modeling of Analog Operations
Abstract Generated abstract
The paper describes an iterative method for simulating analog circuits on a digital computer by representing the output of each analog block as an expansion in a complete system of functions. It formulates circuit behavior through block operators and their composition, with convergence tied to the existence of a fixed point under contraction conditions. For power series bases, integration and multiplication reduce to simple index operations, while Chebyshev polynomial expansions are also considered for potentially faster convergence. The procedure specifies how input data, initial conditions, iteration termination, and stepwise shifting are handled, and illustrates the approach with an analog circuit generating sine and cosine approximations.
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DIGITAL MODELING OF ANALOG OPERATIONS
I. G. Vintizenko, V. A. Tartakovsky
(Presented by the scientific seminar of the Computer Engineering Department of the Research Institute of Automated Electric Drives at TPI)
A method is considered for modeling an analog circuit on a digital computer, based on iterations. In the iteration process, the values of the functions at the output of each of the analog blocks are computed. To each analog block there corresponds an operator \(p_m\); they form a composition \(P_k\). A feature of the method is that the successive approximations \(l_i \in L\) have the form
\[ l_i=\sum_n a_i^n \varphi_n , \tag{1} \]
where \(\varphi_n=\varphi_n(x)\) is a complete system defined on the set of real numbers from \([0,h]\).
From (1) follows the necessary property of the operators \(P_k\):
\[ P_k\left(l_{\in L}\right)=c_{\in L}. \tag{2} \]
As a result of the iterations, a fixed point \(y(x)_{\in L}\) of the mapping \(P_k(y)\) must be found. This point exists [1] (Banach theorem) if \(P_k\), for \(x\) from \([0,h]\), is a contraction operator.
Let \(\varphi_n=\dfrac{x^n}{n!}\), and let \(P_k\) include the operators of integration, addition, and multiplication. From their properties, and also from the fact that
\[ \int_0^x x^n\,dx=\frac{x^{n+1}}{n+1};\quad x^n\cdot x^m=x^{n+m}, \]
the validity of (1) and (2) follows.
Thus, integration is reduced to increasing the index by one, and multiplication—to adding indices.
The initial information for the digital computer is represented in the form of a list of operators \(p_m\), the connections between them, and also arrays of transfer coefficients and initial conditions. The iteration process is organized according to the following rules:
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The zero approximation at the first step is the initial conditions.
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Subsequent approximations are the result of the action of the block operator on the input quantities, taking into account the initial conditions.
As a result of the iterations, at the output of each of the blocks of the analog circuit a power series \(l_i\) is formed, which converges for \(x\) from \([0,h]\). [1, 2].
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The iteration process is terminated when the difference between the next and the preceding approximations is less than \(\varepsilon\).
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The zero approximation at the next step is obtained by shifting \(l_i\) to the left by \(h\), i.e.
\[ \sum_{k=0}^{n}\alpha_k x^k \]
passes into
\[ \sum_{i=0}^{n} a_i x^i, \]
where
\[ a_i=\sum_{k=i}^{n}\alpha_k \binom{k}{k-i} h^{k-i}; \]
after this we proceed to item 2.
In addition to the system \(x^n\), other systems may also be used, for example Chebyshev polynomials. For a broad class of functions, expansion in Chebyshev polynomials converges much faster than in any other system of orthogonal functions; [2] the following relations hold for them:
\[ \int_0^x T_0\,dx = T_1;\qquad \int_0^x T_1\,dx = \frac{1}{4}(T_2+T_0); \]
\[ n>1\quad \int_0^x T_n\,dx = \frac{1}{2} \left[ \frac{T_{n+1}}{n+1} - \frac{T_{n-1}}{n-1} + \frac{2nG}{n^2-1}T_0 \right]. \]
\(G\) depends on the last two binary digits of the number \(n\)
\[ G= \begin{pmatrix} 0&0\\ 0&1\\ 1&0\\ 1&0 \end{pmatrix} = \begin{cases} 0\\ +1\\ 0\\ -1 \end{cases}; \]
\[ T_m\cdot T_n=\frac{1}{2}\,[T_{m+n}+T_{m-n}]. \]
Fig. 1. Analog circuit for obtaining sine and cosine.
| No. of iteration | Block No. 1 | Block No. 2 | Block No. 3 |
|---|---|---|---|
| 0 | \(0\) | \(0\) | \(1\) |
| 1 | \(-x\) | \(+x\) | \(1-\dfrac{x^2}{2!}\) |
| 2 | \(-x+\dfrac{x^3}{3!}\) | \(+x-\dfrac{x^3}{3!}\) | \(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}\) |
| \(n\) | \(\simeq -\sin x\) | \(\simeq \sin x\) | \(\displaystyle \sum_{i=1}^{n}(-1)^i\frac{x^{2i}}{2!}\simeq \cos x\) |
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