Symbolic SPICE TM Circuit Analyzer and Approximator

Transcription

1 ymbolic PICE ymbolic PICE TM Circuit Analyzer and Approximator Application Note AN-001: eries Resonant Circuit by Gregory M. Wierzba Rev 07010

2 A) Introduction The schematic shown below in Fig. 1 is a series RLC circuit. This simple circuit is capable of acting as a low-pass, high-pass, band-pass and band-stop filter [1]. A Ppice input file, rlc.cir, is given in Table 1. The same file is also accepted by ymbolic PICE TM. Table 1. Ppice/ymbolic PICE Input File Figure 1. eries RLC circuit ERIE RLC CIRCUIT V 1 0 AC 1 R L 3 1M C U.AC DEC 500 5K.5MEG.PROBE.END B) Numerical Program Results Running Ppice, we have the following numerical results with the element nodes indicated that create low-pass, high-pass, band-pass and band-stop filters: Figure. Low-pass response

3 Figure 3. High-pass response Figure 4. Band-pass response 3

4 Figure 5. Band-stop response C) Running ymbolic PICE Although a numerical program can show you what a circuit is capable of doing, ymbolic PICE can show you why. ymbolic PICE can also provide design equations. Running [] the same input file shown in Table 1, the following are the prompts. The user responses are shown in bold: ymbolic PICE - Circuit Analyzer and Approximator Demo Version 3.1 (C) Copyright 010 by Willow Electronics, Inc. INPUT FILE NAME [.cir] : rlc OUTPUT FILENAME [rlc.det] : (hit enter) Determinant string sorted according to orders of some variable? (y/n) : n Numerical evaluation of the results? (y/n) : y Discard terms if their magnitude falls below a threshold? (y/n) : n Check and solve for second order filter functions? (y/n) : y FILTER FUNCTION FILE NAME [rlc.fun] : (hit enter) olve for a variable or expression? (y/n) : y Available Unknowns: 4

5 V1 V V3 V5 *Ignore nodes 4 and higher if present. They are used for internal numbering. Valid Operators: +, -, *, /, ( ), { }, [ ] Equation: v3 olve for another variable or expression? (y/n) : y Equation: v(,3) olve for another variable or expression? (y/n) : y Equation: v(1,) olve for another variable or expression? (y/n) : y Equation: v() olve for another variable or expression? (y/n) : n D) ymbolic PICE Determinant Output The output file rlc.det listed in Table is the matrix written by ymbolic PICE and the transfer functions requested. The program formulates this matrix by using admittances [3]. For example, the resistance R is modeled as a conductance G with a value of 1/ É, the capacitance C is modeled as an admittance sc with a value of (0.01:)s and the inductance L is modeled as an admittance 1/(sL ) with a value of 1/[(1m)s] where s is the Laplace operator. All symbols are echoed as uppercase letters except for the Laplace operator. ERIE RLC CIRCUIT Table. ymbolic PICE Output File rlc.det [0 ] [-G G ][V1 ] [0 ][0 -sl sc+1 sl-1 ][V ] [1 ] [ ][V3 ] [0 ] [0 sl+1-1 -sl ][V5 ] *Ignore nodes 4 and higher if present. They are used for internal numbering. Numerator of: v3 s**0 terms: + G * s**0 5

6 Denominator of: v3 s** terms: + sl*sc*g s**1 terms: + sc s**0 terms: + G e-014 * s** + 1e-008 * s** * s**0 Numerator of: v(,3) s** terms: + sl*sc*g e-014 * s** Denominator of: v(,3) s** terms: + sl*sc*g s**1 terms: + sc s**0 terms: + G e-014 * s** + 1e-008 * s** * s**0 6

7 Numerator of: v(1,) s**1 terms: + sc + 1e-008 * s**1 Denominator of: v(1,) s** terms: + sl*sc*g s**1 terms: + sc s**0 terms: + G e-014 * s** + 1e-008 * s** * s**0 Numerator of: v() s** terms: + sl*sc*g s**0 terms: + G e-014 * s** * s**0 7

8 Denominator of: v() s** terms: + sl*sc*g s**1 terms: + sc s**0 terms: + G e-014 * s** + 1e-008 * s** * s**0 ymbolic PICE s format is a collection of strings of symbols sorted by the (default) Laplace operator s. This is usually not how most people view equations, so you may need to do some minor editing. For example, the last transfer function v() is actually V / V V / V 1 V since V 1 1. Thus we have 1 V V sl sc G sc G s L C G sc G s s LG LC s + slscg + G s LCG + G LC s + LC R 1 s + s + L L C Likewise for the numerical results, 1 14 V.1766e s s + 100G V e s + 1e s s + 470k s+ 100G E) ymbolic PICE Filter Function Output ymbolic PICE will examine any transfer function to determine if its denominator is second order in the Laplace operator s. If this is the case then it finds the parameters associated with the different filter functions. ymbolic PICE finds formulas for Q o, T o and H i where some of the various cases of i are listed on the following page: 8

9 Low-pass filter T lp s Hlp ωo ω o + s + ω Q o o High-pass filter T hp s Hhp s ω s Q + o + ωo o Band-pass filter T bp s H ωο s Qo ω bp o + s + ω o Q o Band-stop filter T bs z Hbs ( s + ω ) ω s s Q + o + ωo o The output file rlc.fun listed in Table 3 contains the filter parameters solved for by ymbolic PICE. Table 3. ymbolic PICE Output File rlc.fun ERIE RLC CIRCUIT ECOND ORDER FILTER PARAMETER: Qo is: ( + G)*QRT{ + L} QRT{ + C} Wo** is: ( + 1) ( + L*C) 9

10 fo 5039.Hz **************************************** There exists a LOW PA filter for : v3 LOW PA GAIN (Hlp) is: ( + 1) **************************************** There exists a HIGH PA filter for : v(,3) HIGH PA GAIN (Hhp) is: ( + 1) **************************************** There exists a BAND PA filter for : v(1,) BAND PA GAIN (Hbp) is: ( + 1) **************************************** There exists a BAND TOP filter for : v() BAND TOP GAIN (Hbs) is: ( + 1) BAND TOP FREQUENCY (Wz**) is: ( + 1) ( + L*C) fz 5039.Hz **************************************** 10

11 F) Comparison of Results From Fig. 5, we can read the numerical value of f o f z 50.31k Hz. The exact value of f o is 50.39k Hz. This is a difference of 0.195% which is approximately the default accuracy of Ppice. Likewise we can calculate the numerical value of Q o from the data [1] in Fig. 5 where Q o f o /(-3dB bandwidth) 50.31k / (100.75k k) The exact value is This is a difference of 0.40%. Besides the numerical results, we now have the formulas for f o f z ( π ) 1 LC and L 1 L Qo G C R C To turn these formulas into a design procedure[1]: Pick L tandard inductor value then C 1 ( o) π f L and R 1 L Q C o G) Conclusion ymbolic PICE can be used to analyze passive filters. ymbolic PICE can recognize second order transfer functions in the Laplace operator s. It can also determine the filter type such as low-pass, high-pass, band-pass and band-stop and then solve for the filter parameters T o, Q o and H i. 11

12 H) References [1] G. M. Wierzba, ECE 0: Electric Circuits and ystems II - Class Notes, Ch.1, pp. 6-4.This e-book is available at [] G. M. Wierzba, ymbolic PICE User Manual, p. 6. This e-book is available at [3] G. M. Wierzba, ymbolic PICE User Manual, p