Variation range based simplification for meaningful symbolic analysis of analog integrated circuits

Abstract

This paper presents a variation range based symbolic simplification technique, where each circuit parameter is represented via a range of variation. We generate a database to handle the variation ranges of the circuit parameters and perform simulated annealing algorithm to achieve the most compact symbolic expressions. In contrast to the existing techniques, the main advantage of the proposed method is that the simplified symbolic expressions have an acceptable generalizability in the whole ranges of variations of the circuit parameters. The maximum allowable simplification error can be controlled by the user based on the required accuracy level that he needs. Simulation results demonstrate that the proposed method outperforms the conventional techniques in terms of expression complexity, accuracy and generalizability of the simplified expressions. The resultant expressions are only as complicated as necessary to give the required accuracy in the results, which help in giving a better insight into the circuit behavior.

Appendices

Appendices

1.1 Appendix 1: Operators among range of variations

Given two symbolic parameters y _i  ∊ [y _iL , y _iH ] and y _j  ∊ [y _jL , y _jH ], the operators among ranges including addition, product, modulus, reciprocal, lower and upper, can be defined as

$$y_{i} \pm y_{j} \in \left[ {y_{iL} \pm y_{jL} ,y_{iH} \pm y_{jH} } \right].$$

(19)

$$y_{i} \times y_{j} \in \left[ {min\left( {y_{iL} y_{jL} ,y_{iL} y_{jH} ,y_{iH} y_{jL} ,y_{iH} y_{jH} } \right),max\left( {y_{iL} y_{jL} ,y_{iL} y_{jH} ,y_{iH} y_{jL} ,y_{iH} y_{jH} } \right)} \right] .$$

(20)

$$\left| {y_{i} } \right| \in \left[ {min\left( {\left| {y_{iL} } \right|,\left| {y_{iH} } \right|} \right),max\left( {\left| {y_{iL} } \right|,\left| {y_{iH} } \right|} \right)} \right] .$$

(21)

$$\frac{1}{{y_{i} }} \in \left[ {\frac{1}{{y_{iH} }},\frac{1}{{y_{iL} }}} \right].$$

(22)

$$L\left( {y_{i} } \right) = y_{iL} ,$$

(23)

$$U\left( {y_{i} } \right) = y_{iH} ,$$

(24)

1.2 Appendix 2: Inefficiency of classical simplification criteria

Let us consider the transfer function of Eq. (1). If the same error ɛ _M is exactly occurred for all polynomials in the numerator and denominator of the transfer function, the simplified expression would become:

$$H^{new} (s,{\mathbf{x}}) = \frac{{\left( {1 - \varepsilon_{M} } \right)f_{0} ({\mathbf{x}}) + \left( {1 - \varepsilon_{M} } \right)sf_{1} ({\mathbf{x}}) + \left( {1 - \varepsilon_{M} } \right)s^{2} f_{2} ({\mathbf{x}}) + \cdots + \left( {1 - \varepsilon_{M} } \right)s^{A} f_{A} ({\mathbf{x}})}}{{\left( {1 - \varepsilon_{M} } \right)g_{0} ({\mathbf{x}}) + \left( {1 - \varepsilon_{M} } \right)sg_{1} ({\mathbf{x}}) + \left( {1 - \varepsilon_{M} } \right)s^{2} g_{2} ({\mathbf{x}}) + \cdots + \left( {1 - \varepsilon_{M} } \right)s^{B} g_{B} ({\mathbf{x}})}}{\kern 1pt} .$$

(25)

It is obviously that there is no change in magnitude or phase and neither zero nor pole displacement. However, since simplification is a discrete process, the actual errors are different for the different coefficients:

$$H^{new} (s,{\mathbf{x}}) = \frac{{\left( {1 - \varepsilon_{0N} } \right)f_{0} ({\mathbf{x}}) + \left( {1 - \varepsilon_{1N} } \right)sf_{1} ({\mathbf{x}}) + \left( {1 - \varepsilon_{2N} } \right)s^{2} f_{2} ({\mathbf{x}}) + \cdots + \left( {1 - \varepsilon_{AN} } \right)s^{A} f_{A} ({\mathbf{x}})}}{{\left( {1 - \varepsilon_{0M} } \right)g_{0} ({\mathbf{x}}) + \left( {1 - \varepsilon_{1M} } \right)sg_{1} ({\mathbf{x}}) + \left( {1 - \varepsilon_{2M} } \right)s^{2} g_{2} ({\mathbf{x}}) + \cdots + \left( {1 - \varepsilon_{BM} } \right)s^{B} g_{B} ({\mathbf{x}})}}.$$

(26)

It may lead to significant pole/zero displacements where the roots are very sensitive to coefficient variations. To have a better understanding, we mathematically demonstrate this drawback in the following. The magnitude and phase of the transfer function in the frequency s = jω ₀ for a given nominal point ${\mathbf{x}}_{0}$ can be calculated as

$$\left| {H\left( {s = j\omega_{0} } \right)} \right| = \frac{{\sqrt {A^{2} + B^{2} } }}{{\sqrt {C^{2} + D^{2} } }} ,$$

(27)

$$\angle H\left( {s = j\omega_{0} } \right) = tan^{ - 1} \left( {\frac{B}{A}} \right) - tan^{ - 1} \left( {\frac{D}{C}} \right) ,$$

(28)

where

$$A = f_{0} - f_{2} \omega_{0}^{2} + \cdots \pm f_{{\left( {2\left[ {\frac{M}{2}} \right]} \right)}} \omega_{0}^{{\left( {2\left[ {\frac{M}{2}} \right]} \right)}} ,$$

(29)

$$B = f_{1} \omega_{0} - f_{3} \omega_{0}^{3} + \cdots \pm f_{{\left( {2\left[ {\frac{M - 1}{2}} \right] + 1} \right)}} \omega_{0}^{{\left( {2\left[ {\frac{M - 1}{2}} \right] + 1} \right)}} ,$$

(30)

$$C = g_{0} - g_{2} \omega_{0}^{2} + \cdots \pm g_{{\left( {2\left[ {\frac{N}{2}} \right]} \right)}} \omega_{0}^{{\left( {2\left[ {\frac{N}{2}} \right]} \right)}} ,$$

(31)

$$D = g_{1} \omega_{0} - g_{3} \omega_{0}^{3} + \cdots \pm g_{{\left( {2\left[ {\frac{N - 1}{2}} \right] + 1} \right)}} \omega_{0}^{{\left( {2\left[ {\frac{N - 1}{2}} \right] + 1} \right)}} .$$

(32)

As mentioned above, the maximum-allowable error generated in the each simplified polynomial (via one of the four classical criteria) can be approximated by ɛ. Therefore, it is assumed that: $\left| {\Delta f_{i} } \right| \le \varepsilon \times \left| {f_{i} } \right|$ and $\left| {\Delta g_{j} } \right| \le \varepsilon \times \left| {g_{j} } \right|$, in which, $\Delta f_{i}$ or $\Delta g_{j}$ represents the change in the value of f _i or g _j , due to the simplification of f _i or g _j , respectively. Therefore, the extreme values for the simplified polynomials can be formulated as: f ^new _i  = f _i (1 ± ɛ) and g ^new _j  = g _j (1 ± ɛ). Similar to the exact transfer function (see Eq. 1), the simplified transfer function in expanded format can be shown as follows:

$$H^{new} (s,{\mathbf{x}}) = \frac{{f_{0}^{new} ({\mathbf{x}}) + sf_{1}^{new} ({\mathbf{x}}) + s^{2} f_{2}^{new} ({\mathbf{x}}) + \cdots + s^{M} f_{M}^{new} ({\mathbf{x}})}}{{g_{0}^{new} ({\mathbf{x}}) + sg_{1}^{new} ({\mathbf{x}}) + s^{2} g_{2}^{new} ({\mathbf{x}}) + \cdots + s^{N} g_{N}^{new} ({\mathbf{x}})}}.$$

(33)

Therefore, magnitude and phase of the simplified transfer function for the specific frequency s = jω ₀ in the nominal point ${\mathbf{x}}_{0}$ can be calculated as follows:

$$\left| {H^{new} (s = j\omega_{0} )} \right| = \frac{{\sqrt {A^{new2} + B^{new2} } }}{{\sqrt {C^{new2} + D^{new2} } }} ,$$

(34)

$$\angle H^{new} (s = j\omega_{0} ) = tan^{ - 1} \left( {\frac{{B^{new} }}{{A^{new} }}} \right) - tan^{ - 1} \left( {\frac{{D^{new} }}{{C^{new} }}} \right) .$$

(35)

A _new can be formulated and simplified according to Eq. (36), where $A_{new} = A \pm\Delta A$.

$$A_{new} = f_{0}^{new} - f_{2}^{new} \omega_{0}^{2} + \cdots \pm f_{{\left( {2\left[ {\frac{M}{2}} \right]} \right)}}^{new} \omega_{0}^{{\left( {2\left[ {\frac{M}{2}} \right]} \right)}} = A + \varepsilon \times \left( { \pm f_{0} \pm f_{2} \omega_{0}^{2} \pm \cdots \pm f_{{\left( {2\left[ {\frac{M}{2}} \right]} \right)}} \omega_{0}^{{\left( {2\left[ {\frac{M}{2}} \right]} \right)}} } \right).$$

(36)

Therefore, the extreme values for $\Delta A$ can be formulated as follows:

$$\Delta A_{max} = \varepsilon \times \left( {\left| {f_{0} } \right| + \left| {f_{2} \omega_{0}^{2} } \right| + \cdots + \left| {f_{{\left( {2\left[ {\frac{M}{2}} \right]} \right)}} \omega_{0}^{{\left( {2\left[ {\frac{M}{2}} \right]} \right)}} } \right|} \right),$$

(37)

where $\Delta A$ is the maximum allowable difference between A and A _new , in the worst case. By similar formulations for B _new , C _new and D _new , the extreme values for $\Delta B$, $\Delta C$ and $\Delta D$ can be written as follows:

$$\Delta B_{max} = \varepsilon \times \left( {\left| {f_{1} \omega_{0} } \right| + \left| {f_{3} \omega_{0}^{3} } \right| + \cdots + \left| {f_{{\left( {2\left[ {\frac{M - 1}{2}} \right] + 1} \right)}} \omega_{0}^{{\left( {2\left[ {\frac{M - 1}{2}} \right] + 1} \right)}} } \right|} \right),$$

(38)

$$\Delta C_{max} = \varepsilon \times \left( {\left| {g_{0} } \right| + \left| {g_{2} \omega_{0}^{2} } \right| + \cdots + \left| {g_{{\left( {2\left[ {\frac{M}{2}} \right]} \right)}} \omega_{0}^{{\left( {2\left[ {\frac{M}{2}} \right]} \right)}} } \right|} \right),$$

(39)

$$\Delta D_{max} = \varepsilon \times \left( {\left| {g_{1} \omega_{0} } \right| + \left| {g_{3} \omega_{0}^{3} } \right| + \cdots + \left| {g_{{\left( {2\left[ {\frac{M - 1}{2}} \right] + 1} \right)}} \omega_{0}^{{\left( {2\left[ {\frac{M - 1}{2}} \right] + 1} \right)}} } \right|} \right) .$$

(40)

Finally, the extreme values for the magnitude of $H^{new} \left( {s = j\omega_{0} } \right)$ can be calculated as follows:

$$\left| {H^{new} (s = j\omega_{0} )} \right|_{min} = \frac{{\sqrt {min(A_{new}^{2} ) + min(B_{new}^{2} )} }}{{\sqrt {max(C_{new}^{2} ) + max(D_{new}^{2} )} }},$$

(41)

$$\left| {H^{new} (s = j\omega_{0} )} \right|_{max} = \frac{{\sqrt {max(A_{new}^{2} ) + max(B_{new}^{2} )} }}{{\sqrt {min(C_{new}^{2} ) + min(D_{new}^{2} )} }} .$$

(42)

And the extreme values for the phase of $H^{new} \left( {s = j\omega_{0} } \right)$ can be written as follows:

$$\angle \left. {H^{new} (\omega_{0} )} \right|_{min} = min\left( {tan^{ - 1} ({{B_{new} } \mathord{\left/ {\vphantom {{B_{new} } {A_{new} }}} \right. \kern-0pt} {A_{new} }})} \right) - max\left( {tan^{ - 1} ({{D_{new} } \mathord{\left/ {\vphantom {{D_{new} } {C_{new} }}} \right. \kern-0pt} {C_{new} }})} \right),$$

(43)

$$\angle \left. {H^{new} (\omega_{0} )} \right|_{max} = max\left( {tan^{ - 1} ({{B_{new} } \mathord{\left/ {\vphantom {{B_{new} } {A_{new} }}} \right. \kern-0pt} {A_{new} }})} \right) - min\left( {tan^{ - 1} ({{D_{new} } \mathord{\left/ {\vphantom {{D_{new} } {C_{new} }}} \right. \kern-0pt} {C_{new} }})} \right) .$$

(44)

Here, we demonstrate the drawback of the classical criteria through a simple numerical example. An optional exact transfer function for a given nominal point ${\mathbf{x}}_{0}$ in an optional frequency ω ₀ is assumed as follows:

$$H(s = j\omega_{0} ,{\mathbf{x}} = {\mathbf{x}}_{0} ) = \frac{{(5 \times 10^{7} ) + j(1000) + j^{2} (5 \times 10^{7} )}}{{(50) + j(10^{5} ) + j^{2} ( - 50) + j^{3} (10^{5} )}} .$$

(45)

As well as the assumption of ɛ = 0.01, it can be calculated that A ^new = (5 × 10⁷–5 × 10⁷) ± 10⁶ = 0 ± 10⁶, B ^new = 1000 ± 10, C ^new = (50 + 50) ± 1=100 ± 1 and D ^new = (10⁵–10⁵) ± 2000 = 0±2000. Therefore, the magnitude and phase of the exact transfer function for the nominal point ${\mathbf{x}}_{0}$ in the frequency s = jω ₀ are derived as 20 decibel and 90 degree, respectively. However, the magnitude of the simplified transfer function can be varied from −6 decibel to 80 decibel. Furthermore, the phase of the simplified transfer function can vary from −87 degree to 267 degree. Hence, the maximum magnitude error and the maximum phase error in $s = j\omega_{0}$ can be 60 decibel and 177 degree, respectively, for the worst case.