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CIRCUITS and SYSTEMS – part II Prof. dr hab. Stanisław Osowski Electrical Engineering (B.Sc.) Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie Lecture 14 Operational amplifier circuits Ideal operational amplifier U 2 AU U at A •Infinite gain A •Zero output impedance •Infinite input impedance •These features valid for frequency from 0 to infinity. 3 2-port model of ideal op-amp Hybrid description I 1 0 0 U 1 U A 0 I 2 2 Voltage adding circuit Voltage adding circuit - equations Kirchhoff’s equations G2 U 2 U G0U G1 U 1 U G0U G f U U 0 U U After simplification we get G2 U U2 G0 G 2 G2 G0 G1 G f G1 U0 U2 U1 Gf G0 G 2 Gf U 0 k1U 1 k 2U 2 Voltage adding circuit - gains G1 k1 Gf G G2 0 G1 G f k2 Gf G0 G2 G2 k2 Gf if G0 G1 G f G0 G2 Integrator Transfer function U 1 RI 1 U2 I sC U2 1 H (s) U1 sRC Differentiator Transfer function 1 U1 I sC U 2 RI U2 H ( s) sRC U1 Phase shifter Kirchhoff’s equations 1 U1 R I 2 sC U 1 2R f I1 U 2 U 2 R f I1 1 I2 sC Phase shifter (cont.) The currents U1 R 1 / sC U U2 I1 1 2R f I2 Output voltage U 2 R f U1 U 2 U1 1 s 1 / RC U1 2R f sC R 1 / sC s 1 / RC Transfer function U 2 s 1 / RC H ( s) U1 s 1 / RC Negative impedance converter (NIC) Kirchhoff’s equations U1 U 2 R1 I 1 R2 I 2 Chain matrix description 1 0 U U 1 R2 2 I 0 I 1 R1 2 Gyrator Kirchhoff’s equations I 1 G z U 1 U 1 U 2 G zU 2 I 2 G z U 2 U 2 U 1 G zU 1 Admittance matrix description I1 0 I G 2 z Gz U1 0 U 2 Mason signal flow graph (SFG) Basic notions: • Node – the point of graph associated with variable x • Branch – the directed arch joining 2 nodes • Gain – the transfer function describing branch • Loop – the sequence of identically directed branches forming closed loop • Gain of the loop – the product of gains of branches of the loop • Source node – the node from which the branches can only start • Cascade – the sequence of identically directed branches from the source node to the output node. Example of SFG Set of linear equations Transformation to Mason form Mason SFG a11x1 a12 x2 F1 a21x1 a22 x2 F2 x1 (a11 1) x1 a12 x2 F1 x2 a21x1 (a22 1) x2 F2 Mason gain formula Transfer function T Tk k k Δ - main determinant of SFG 1 Gi GiG j GiG j Gk ... i Gains of all loops i, j i, j ,k Gains of non-touching loops Tk – gain of kth cascade from source to output node Δk - determinant of graph after eliminating kth cascade from SFG Example Graph Transfer function T aej (1 l ) bgk (1 h) adgk (1 h) adfj(1 l ) bcej (1 l ) bfj(1 l ) 1 cd h l cdh cdl hl cdhl Direct construction of SFG for passive elements connection n Ysk Y0k Yi i 1 Circuit Its SFG Direct construction of SFG for op-amp Op-amp Its SFG Vo AV1 AV2 Example SFG of the circuit T U wy U we Y3 Y1 Ys 2 Ys1 Y YY A 5 A 4 3 Ys 2 Ys1Ys 2 A Y32 1 Ys1Ys 2 After simplification at A we finally get Y1Y3 T Y5 Y1 Y2 Y3 Y4 Y3Y4