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

3 Ideal operational amplifier Infinite gain A Zero output impedance Infinite input impedance These features valid for frequency from 0 to infinity.

2-port model of ideal op-amp Hybrid description

Voltage adding circuit

Voltage adding circuit - equations Kirchhoff’s equations After simplification we get

Voltage adding circuit - gains

Integrator Transfer function

Differentiator Transfer function

Phase shifter Kirchhoff’s equations

Phase shifter (cont.) The currents Output voltage Transfer function

Negative impedance converter (NIC) Kirchhoff’s equations Chain matrix description

Gyrator Kirchhoff’s equations Admittance matrix description

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

Mason gain formula Transfer function Δ - main determinant of SFG T k – gain of kth cascade from source to output node Δ k - determinant of graph after eliminating kth cascade from SFG Gains of non-touching loopsGains of all loops

Example Graph Transfer function

Direct construction of SFG for passive elements connection CircuitIts SFG

Direct construction of SFG for op-amp Op-amp Its SFG

Example

SFG of the circuit After simplification at we finally get