# Finding the input impedance of this gyrator

I'm having a little trouble with this. I'm supposed to show that the input impedance of this gyrator is $$\frac{R^2}{Z}$$ but the math just isn't working out.

The gyrator consists of negative impedance devices (NIC, shown below) whose input impedance I found to be -Z. Am I right to assume that an NIC in parallel with a resistor would have an impedance of R-Z?

I was trying to solve this as a series of voltage dividers, with the last R and Z element having an impedance of $$\frac{Z}{R+Z}$$ This would be in parallel with the NIC and R pointing downward to give $$\frac{Z/(R+Z)+(R-Z)}{\frac{Z}{(R+Z)}(R-Z)}$$

And then this could be solved using the voltage divider equation $$\frac{Z_2}{Z_1+Z_2}$$ where $$Z_1 = R-Z$$ and Z_2 was stated above.

Unfortunately the math isn't working out to the desired answer, so I'm not sure whats going wrong here. Any help is appreciated.  – uhoh
Jun 7 '17 at 0:10
• Before looking at anything else, your first equation is wrong. It can't be an impedance because it's unitless. Series impedance would be R+Z
– Chu
Jun 7 '17 at 0:24
• I get Zin = -Z + 1.5R as the impedance looking into the first network, assuming the impedance of the NIC's is -Z. Jun 7 '17 at 18:34

$$\frac{1}{\frac{1}{R}-\frac{1}{R+Z}}-R = \frac{R^2}{Z}$$