NIC (Negative Impedance Converter)

Introduction

Negative impedance ? What on earth is that ? How can an impedance be negative, and why would we need it ?
These surely aren't surprising questions when encountering this mysterious concept for the first time. Don't worry, it has nothing to do with black magic.
Firstly, let's take a look at what a negative impedance means. To keep things simple, we apply the concept to a resistor and check what it would mean to have a negative resistance.

We are familiar with positive (normal) resistance.
In the figure below, positive (normal) resistance is shown. When looking into the circuit from the input-port, we see a certain impedance Zin. In this case it is a common resistor, so impedance Zin of the circuit is simply resistive. When increasing the voltage Vin over the positive resistance, the current I flowing into the circuit will increase linear with a slope depending on the value of the resistor. A low resistor value will have a steeper voltage/current slope than a higher resistor value.

General note:
In some circuit schematics, I left out the power supply connections for the OPAMPs.
All the gyrator circuits with OPAMPs need a symmetrical (positive and negative) power supply voltage.

Now, let's assume that we have a negative resistance.
In the figure below, a negative resistance is shown. When we increase the voltage Vin, the current will increase just like with a positive resistance, but now the current is not flowing into the circuit but out of the circuit. So the direction of the current is opposite to a common resistor. The circuit sources current instead of sinking current. 

But this is theory. Great, but how could we achieve something like that in practice ? How can we turn a normal resistor into a negative resistor that does not consume current, but instead generates current ?
See the picture below of how this can be achieved using an amplifier. What we need to do to turn a resistor into a negative resistor is 'lift' the voltage at the bottom terminal of the resistor in proportion with the input voltage. When we force a voltage, being twice the input voltage, on the bottom terminal of the resistor, the current will be the same as with a normal resistor, but it will flow in the opposite direction.
In the circuit below, you see the amplifier with a gain of 2 that amplifies the input voltage and puts it on the lower terminal of the resistor. So on the lower terminal we have a voltage of 2 x Vin and on the upper terminal a voltage equal to Vin. The current flowing in upward direction through R will thus be ((2 x Vin )- Vin) / R = Vin / R. Because we are talking about an ideal amplifier, the current flowing into the amplifier is zero. This means that the current through the resistor will entirely flow out of the circuit instead of into the circuit.
The circuit now sources the same current as a single resistor would do, but in the opposite direction.
This is called a negative resistance : the current increases proportional to the input voltage, but instead of flowing into the resistor, it flows out of the resistor. The impedance that is seen when looking into the circuit from the input terminals Zin is a negative resistance because it does not consume current but delivers current proportional to the voltage that is put over it.

When we implement this amplifier above in a practical circuit, we use an OPAMP in a non-inverting configuration with a gain of 2 as shown in the picture below. The non-inverting configuration around the OPAMP gives us a gain of 2 because R1 = R2.
The circuit will convert any resistance R into a negative resistance with value R. 

Mathematical derivation

This principle of converting an impedance to its negative version can not only be applied to resistors but also to any other type of impedance.
It is also possible to generate negative capacitors or negative inductors.
In the figure below, a generalized NIC (Negative Impedance Converter) is shown that is capable of generating any kind of negative impedance.
For the mathematical derivation, we do some assumptions about the direction of i1. 

Right of the circuit, the mathematical derivation for the input impedance Zin of the circuit is shown.
Vin / I1 = Zin = -Z1 × (Z3 / Z2)
The negative sign indicates that either the current i1 can be negative (flowing out of the circuit instead of into) or that
Vin is negative.

When Z3 = Z2, then Zin = -Z1.
When Z1 = Z2, then Zin = -Z3.

See table below for some useful combinations of impedance's and the resulting negative impedance Zin:

INIC and VNIC

As we explained above, the clue of the NIC is that the OPAMP output is twice the input voltage Vin when Z2 = Z3.
Because Vout >> Vin, the current i1 will flow out of the circuit.
So the negative impedance is created by sensing the input voltage and generating an inverted current equal to the sensed voltage divided by the impedance that is converted. This also called an INIC or Current Inverting NIC.

When we would exchange the non-inverting and inverting inputs of the OPAMP in the circuit above, it would still be a NIC and the whole mathematical derivation would be exactly the same. The only difference would be that Zin is now connected to the inverting input of the OPAMP, causing a negative voltage on the output of the OPAMP. Current i1 would actually flow into the circuit instead of out of the circuit as with the INIC.
Because Vout is negative, the voltage at the node between Z2 and Z3, that forms a voltage divider, will also be negative. Thus, Vin must also be a negative voltage.
So this kind of NIC with the OPAMP inputs exchanged creates a negative impedance by sensing the input current and generating an inverted voltage equal to the sensed current times the impedance that is converted.
This is also called a VNIC (Voltage Inverting NIC).

In the picture below, a INIC and a VNIC are shown next to each other with all the currents drawn in their actual flowing direction (by convention, the flow of the holes and not the electrons :-)).

Note:
The base impedance Z, that is converted into negative impedance, can also be R3 in the INIC and R1 in the VNIC shown below.

Practical example of a INIC and VNIC generating negative resistance

In the 2 pictures below, you see an example of an INIC and a VNIC of -1k (negative 1k) that both cancel out a normal (positive) resistor of 1k.
The INIC does this by putting the INIC in parallel to the resistor that we want to cancel out, and the VNIC does this by putting the VNIC in series with the resistor.
In the first picture, the current that the voltage source of 1V is delivering to the circuit is 0mA. So the voltage source 'sees' an infinite impedance as if R3 was not there at all ! The INIC pushes the same current into the circuit as R3 would be drawing from the voltage source
In the second picture, the current that the voltage source of 1V is delivering to the circuit is 1mA. So the voltage source only 'sees' a resistance of 1k to ground. This is because the VNIC has cancelled out R5 completely. At the node between R4 and R5, the voltage is 0V ! The VNIC makes its input negative, so the current through R5 is the same as the current that would flow when R5 was not there.

In practice the OPAMP is of course not an ideal amplifier, so there will be a small current flowing into the inputs and the output will have an offset voltage,
but this can be compensated for.

Note: The resistors in the examples below are chosen so the OPAMP with a symmetrical power supply of + and - 10V will not saturate.

Note: Care should be taken that the OPAMP is able to generate the required current. Most common OPAMPs can only source or sink low currents.

Note: Because a real OPAMP is not an ideal amplifier, there will be some current flowing into its inputs and there will some offset voltage at the output. 

Negative capacitance and negative inductance

What does it mean to be a negative capacitance or negative inductance ? Let's compare the behavior of a negative capacitance when it is implemented in an RC network to make f.e. an integrating (low pass) filter to the same RC network but with a positive (normal) capacitance.

In the graph below, we check the response of the RC low pass network when the C is a positive (normal) capacitance (green curve) and when C is a negative capacitance (red curve) when using a square wave input (blue curve). We use an input waveform with a frequency that is higher than the cut-off frequency of the RC low pass filter, so we see the influence of the capacitor. What we see is that the voltage over the positive (normal) capacitance increases when the input waveform is positive and decreases when the input voltage is 0. This is the capacitor charging and discharging while trying to follow the input voltage. This is what we are familiar with. But when we check the voltage over the negative capacitance, we see that the voltage over the negative capacitance gets more negative when the input voltage is positive and becomes less negative when the input voltage is 0. 

So a negative capacitance does 'consume' current when trying to change the voltage over it, but instead it sources a current that is proportional to the rate of change of the voltage over it. The same goes for a negative inductor : the voltage over a negative inductor will be negative and is proportional to the rate of change of the current flowing through it.

Below is a Bode graph with the amplitude and phase plot of a positive (normal) capacitance and a negative capacitance when implemented in a low pass RC network. It clearly shows that the amplitude behavior is the same for a positive or negative capacitance, because the amplitude shows the absolute value/magnitude and not the polarity. But in the phase plots, we see that the negative capacitance behaves opposite to the positive (normal) capacitance. The phase between output and input of an RC network with a positive (normal) capacitance goes from 0 to -90 degrees, while the phase of the negative capacitance goes from 0 to +90 degrees. 

Practical example of a negative inductance

In the circuit below, a VNIC is used to generate a negative inductance. The VNIC converts C2 to a negative inductance of - 1H.
The formula to calculate the inductance value is shown at the top of the picture.
The negative inductance forms an RL circuit with R4. This RL circuit is a 1st order high pass filter with a cut-off frequency determined by R4 and the negative inductance.
The cut-off frequency of the filter is given by : F = 1 / (2 × pi ×  (L/R))

LTpice simulation of a negative inductance

Here is an LTSpice simulation to check out the bode plots of an RL circuit using a positive and a negative inductance: 
Synthetic inductor of 1H with positive and negative inductance

The following graph shows the bode plot generated using the LTSpice simulation. The RL circuit is using a 1k resistor and a positive inductance of 1H or a negative inductance of -1H.
The amplitude plots are the same for both the RL circuit with the negative or positive inductance.
The phase plot of the RL circuit with the negative inductance shows that the phase is inverted when compared with the normal inductance.
The RL circuit with a normal inductance has a phase that changes from +90 degrees to 0 when increasing the frequency.
The RL circuit with a negative inductance has a phase that changes from -90 degrees to 0 when increasing the frequency.

Practical example of a positive inductance by combining two NICs

In the circuit below, two VNICs are used to implement a positive inductance of 1H.
The top VNIC would be a negative inductance of -1H when the bottom VNIC would be replaced with a grounded resistor of 10k, as we have seen in the previous example (the grounded resistor in the previous example is R3 = 10k).  By replacing what would normally be a positive resistance of 10k by a VNIC (the bottom one) that generates a negative resistance of -10k, we transform what would normally be a negative inductance of -1H into a positive inductance of 1H.
Right of the circuit, the mathematical derivation is shown.
Together with R4, the positive inductance forms a high pass RL circuit. When we do an AC analysis on this RL circuit, the bode plot will show exact the same amplitude and phase plot as we would have with a normal inductor of 1H.

Note: The interesting part is that we created an inductor of 1H that we can tune simply by changing f.e. the value R1 or R1' using a potmeter or any other resistive element (NTC, PTC, LDR ...). 

Applications where a NIC is used

So, all in all, these mysterious circuits have a fascinating behavior, that puts everything you know about passive capacitors and inductors upside down.
But why would we need circuits that behave this way ?

There are applications where they are very useful :

• Negative resistance can be used to cancel out or decrease undesired resistance

• Negative capacitance can be used to cancel out or decrease undesired parasitic capacitance.

• Negative inductance can be used to cancel out or decrease undesired inductance.

• Negative resistance is used for LC oscillators to compensate the resistance in the LC circuit that dissipates the energy that is needed to keep the oscillations going.

• A combination of multiple NICs can be used to make an inductance or capacitance multiplier.

• The Howland current source (differential or single ended) is a negative resistance in parallel with the load resistance in order to get "infinite" impedance. The negative resistance is used to cancel out the load resistance. An ideal current source has infinite impedance.

• The Deboo integrator is a positive resistor followed by a grounded capacitor and a negative resistance in parallel over the capacitor. In fact, it is a Howland current source that drives a capacitive load. The negative resistance compensates for the positive resistance in a way that the current through the capacitor depends only on the input voltage and the positive resistor and not on the voltage over the capacitor. This way you get a very good integrator