The potential V shown at (a) is the potential at the common node of Z1 and Z2:
The following circuit shows how a simple voltage follower amplifier can have profound capabilities for simulating variable, floating monodirectional inductances, the very impedances that the elite societies had so long troubled themselves with. We'll assume the follower is of good quality, such as an op-amp wired in a follower configuration. At (a) is shown a general circuit we'll use to simulate a large variable monodirectional inductance that's continuously variable from 10 to 100 henries. The circuit at (b) shows the equivalent circuit representation as a monodirectional impedance, as we'll now show using elementary network analysis. Let at least one of Z1 and Z2 be very large in magnitude, so that its current will be negligible. This is an acceptable specification inasmuch as these are needed only to drive the voltage follower amplifier. As such, virtually any voltage source is capable of supplying the autonomous phasor potential EA, and the current I in Z0 is then the only significant current in the passive terminal of potential E. If this low-current restriction restricts options such as variability, then a follower can be used at either or both terminals f.
The potential V shown at (a) is the potential at the common node of Z1 and Z2:
The current I will be (V - E) / Z0, which is, on combining with the preceeding equation,
As such, the simulated impedance, as shown at (b), is
Let us now consider how to choose the values of Z to realize large and variable values of inductance. One option is to let Z2 be a large capacitance of, say, 10 microFarads (non-polarized), and let Z1 be a large potentiometer resistance R1, variable from 1 megOhm to 10 megOhms, and choose Z0 to be a small resistance r, such as the follower's output impedance. If r is 1 Ohm, then the simulated inductance will vary between 10 Henries and 100 Henries, as deternined by
The follower's output impedance r is the inductor's series resistance. 'Hardly a bad Q-factor for a floating inductor simulated with only one capacitor, one variable resistor, and only one op-Amp. A schematic of such a simulator is shown at (a) below, with its equivalent circuit at (b) and its bidirectional equivalent at (c).
The simplicity and versatility of this device emphasize the importance of Collier's methods and examples patented in U.S. Patent 4,963,845, and the circuits he presented in [1].
Another example of the foregoing device was also presented in [1]. It uses a small inductive reactance jwL0 to serve as Z0, and adjusts the value of Z2 as the tap resistance on a potentiometer of total resistance R1 + R2. This is Collier's CRIM simulator.
One other very versatile monodirectional impedance makes broader use of the op amp. It is ahown in (a), below, and its equivalent circuit is shown at (b). Here the impedances Z1 and Z2 are suitably large to insure that the voltage source supplying the autonomous phasor potential EA is not overloaded by the current demanded in Z1.
The opamp's output potential V is that value which, via feedback through Z2, will hold the inverting input potential equal to the noninverting input E. Specifically,
As such, the output current I is (V - E) / Z0, and the simulated impedance becomes
As such we can readily reverse the signs of and/or invert floating monodirectional impedances, AND: Because V is an autonomous voltage source, we can use another monodirectional impedance to simulate its Z0 value. Let
simulated by a second device, to couple the terminal of potential E to the potential V in the first simulator. Then the simulated monodirectional impedance becomes
Provided that Z1 is of very large magnitude, such that it does not overload the autonomous driving source of potential EA, then one can simulate a floating inductor by using a capacitor for either of the impedances z2 or Z2, and resistances (or variable resistances) for all other impedances. An example of this is shown in the drawing at (a) below, with its equivalent circuit at (b)
1. R. L. Collier, American Journal of Physics, 57, 4 (1989).
What IS an impedance?
What is a monodirectional impedance, and why is it so much easier to simulate?
How can one simulate a monodirectional impedance?
Some sample simulators