Abstract
The time dependent master equation from the seminal article by Ragazzini, Randall and Russell (Ragazzini et al 1947 Proc. of the I.R.E. 35, 444–452) is recovered as a necessary tool for the analysis of contemporary circuits with operational amplifiers. This equation gives the relation between time dependent the output voltage U0(t) and the difference between the input voltages ( and ). The crossover frequency f0 is represented with the time constant in this equation. The work of the master equation is illustrated by two typical examples: a) the stability criterion of the devices with negative impedance converters, which we consider as a new result b) the frequency dependence of the amplifiers with operational amplifiers given in the technical specifications without citations of the time dependent equation. A simple circuit for determination of f0 is suggested and the method is illustrated by determination of crossover frequency for the low-noise and high speed ADA4898 operational amplifier. It is concluded that for an exact calculation of the pass bandwidth of amplifiers with active filters the 70 years old master equation is a useful technique implicitly included in the contemporary software. The frequency dependent formulae for the amplification coefficient of inverting and non-inverting amplifiers are given for the case of non-zero conductivity between the inputs of the operational amplifiers.
Highlights
Inspired by Heaviside operational calculus 70 years ago, Ragazzini, Randall and Russell [1] introduced the idea and coined the generic term operational amplifier [2], and introduced U+ − U− = G−1U0, G −1 = 1 G0 + τ0 s, (1)which describes the relation between input voltages U+(t) and U−(t) and output voltage U0(t) Here sis the time differentiating operator d s =, (2)dt which for exponential time dependence of the voltages U ∝ est is reduced to its eigenvalue sest = sest, and for the reciprocal open loop-gain we have
Here we will present only two typical examples illustrating the applicability of the master equation Eq (1) to the negative impedance converter (NIC) and 2) the determination of crossover frequency f0 by frequency dependence of the amplification of a non-inverting amplifier analyzed in the two sections
We give a prescription for determination of the crossover frequency Eq (27) and we consider this not redundant since operational amplifiers vendors provide this parameter in their specifications, often with 50% difference
Summary
Inspired by Heaviside operational calculus 70 years ago, Ragazzini, Randall and Russell [1] introduced the idea and coined the generic term operational amplifier [2] (enabling summation, integration and differentiation), and introduced. The equation from the work of Ragazzini, Randall and Russell [1] exists in Fourier representation in the modeling of operational amplifiers since 1970 [3,4,5,6,7,8] but never as a differential equation describing time dependent voltages. The name of the equation itself is not even finalized cf [10] and in many applications the frequency dependence is neglected leaving no trace of any time or frequency relation Another important example is the stability problem of circuits with operational amplifiers. Here we will present only two typical examples illustrating the applicability of the master equation Eq (1) to the negative impedance converter (NIC) and 2) the determination of crossover frequency f0 by frequency dependence of the amplification of a non-inverting amplifier analyzed in the two sections
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