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APPLICATIONS OF MATLAB IN SCIENCE AND ENGINEERING - PART 5

Trong hình 2 (b) có một phiên bản đơn giản của cùng một mạch với các nguồn điện áp điện áp kiểm soát (VCVS) có tăng điện áp. Đối với một opamp lý tưởng trong chế độ không đảo ngược được cho bởi | Low-Noise Low-Sensitivity Active-RC Allpole Filters Using MATLAB Optimization 201 Moschytz Horn 1981 . In Figure 2 b there is a simplified version of the same circuit with the voltage-controlled voltage source VCVS having voltage gain p. For an ideal opamp in the non-inverting mode it is given by p 1 Rf Rg . 10 Note that the voltage gain p of the class-4 circuit is positive and larger than or equal to unity. Voltage transfer function for the filters in Figure 2 expressed in terms of the coefficients aị i 0 1 2 is given by T s Y L Ns K. 2 a0 Vg D s s2 a1s a0 11a and in terms of the pole frequency Op the pole Q qp and the gain factor K by . V . T s Y K. Y O2 i2 ps O2 qp p 11b where a0 2 1 p R C Op a1 qp R1 C1 C2 R2C2 - pR1C1 R1R2C1C2 11c JRRCC qp R1 C1 C2 R2C2 - pR1C1 K p. a b Fig. 2. Second-order Sallen and Key LP active-RC filter. a With ideal opamp having feedback resistors Rf and Ra and nodes for transfer-function calculus. b Simplified circuit with the gain element replaced by VCVS p. To calculate the voltage transfer function T s Vout s Vg s of the Biquad in Figure 2 a consider the following system of nodal equations note that the last equation represents the opamp 202 Applications of MATLAB in Science and Engineering 1 V Vg . 1 1 1 _ ì 1 _ 2 -VA V l -E- - sC I-V3A-- V5sC1 0 1R1 2 R1 R2 1J 3 R2 51 -.1 . 1 _ ì 3 -V2-1- V31-1- SC2 I 0 12 R2 I R2 J 1 1 Ì 1 4 V3l J- J-I-V.J- 0 31 Rf Rg J 5 Rf 5 A-V - V5 V5 Vmt A 0 i- 0. The system of Equations 12 can be solved using Symbolic toolbox in Matlab. The following Matlab code solves the system of equations i. Matlab command syms defines symbolic variables in Matlab s workspace syms A R1 R2 C1 C2 RF RG s Vg V1 V2 V3 V4 V5 ii. Matlab command solve is used to solve analytically above system of five Equations 12 for the five voltages V1 to V5 as unknowns. The unknowns are defined in the last row of command solve. Note that all variables used in solve are defined as symbolic. CircuitEquations solve . V1 Vg . -V1 1 R1 V2 1 R1 1