A negative impedance converter (NIC) is a clever circuit, the analysis of which is a favorite exercise set by engineering lecturers everywhere. It’s not often used in practice, but it’s well worth having in your design kit-bag. I’ll show you two real-life applications once we get through the analysis—which is not as hard as it might seem.
Figure 1 shows the circuit. We are interested in analyzing the impedance as seen from the input terminal. Let’s start with the input current iIN. This is simply iIN = (vIN – v’)/Zf. R1 and R2 form a voltage divider where v’ =vIN(1 + R1/R2) since the inverting input of the op amp will be at the same voltage as the non-inverting input. If we substitute this equation into the earlier one, we can eliminate v’. Now we have iIN = -vIN(R1/(Zf ✕ R2)).
Now we can calculate the input impedance ZIN = vIN/iIN . We get the result ZIN = -Zf ✕ (R2/R1). If R1 = R2 the input impedance is simply -Zf. If Zf is a resistor we see a negative resistance at the input terminal. In the same way we can also get a “backwards” capacitor or inductor. They will be backwards in that their voltage and current will be 180 degrees out from their usual position, however their impedance will decrease or increase with frequency respectively.
The first time I ever used this circuit in a real design was long ago in an audio (anyone remember wired telephones?) circuit that switched a signal through a series of analogue cross-connect switches. The source expected a 600Ω termination, so the usual practice was to terminate the line with a resistor of sufficient value that the sum of the switch on-resistances and the terminating resistor summed to 600Ω.
One brilliant piece of design I came up with resulted in the sum of on-resistances of the analog switches summing to a bit more than 700Ω. What to do? I consulted our resident engineering guru who suggested terminating the string with a negative 100Ω resistance. I thought he was joking until he drew up the circuit of Figure 1. Problem solved.
Another example of the use of this circuit is to effectively boost the drive capacity of a precision op-amp. Sometimes the op amp with the input characteristics we really want to use does not have sufficient drive capacity if the load impedance is low. Take for example the circuit of Figure 2. Imagine the load resistor RL is 150Ω but our precision op amp does not have sufficient drive capability. If we put a negative impedance of -150Ω in parallel with the load the combined impedance will be infinite (you can check the math). Our drive problems are solved.
Note that the op amp used in the negative impedance converter must be equivalent in bandwidth as the precision op amp and be able to drive a 150Ω load, but otherwise it does not have to be particularly special.
“Negative Impedance Converter.” In Wikipedia, April 13, 2020. https://en.wikipedia.org/w/index.php?title=Negative_impedance_converter&oldid=950615079.
“Negative Resistor Cancels Op Amp Load.” Accessed January 15, 2021. https://www.maximintegrated.com/en/design/technical-documents/app-notes/1/1868.html.