Negative Impedance Converters

“Negative resistance” may seem like a purely academic concept, but can be easily realized in practice with a handful of common components. By adding a single resistor to a standard non-inverting op amp circuit, we can create a negative impedance converter, which has applications in load cancellation, oscillator circuits, and more.

Derivation of NIC Input Impedance

Here we will derive the input resistance of the following Negative Impedance Converter (NIC) and show that it is equal to:

\(
\begin{equation}
\mathbf{R}_{IN} = \mathbf{-} \frac{\mathbf{R}_{1}\mathbf{R}_{3}}{\mathbf{R}_{2}}\\
\end{equation}
\)

Negative Impedance Converter (NIC)

Negative Impedance Converter (NIC)

The applied input voltage, Vin, causes some resulting current, I1. If we can predict I1 for a given Vin, then we can use Ohm’s law to calculate the effective resistance, Rin, seen by the voltage source at Vin.

No current flows into an (ideal) op amp’s inputs, so all of current I1 must pass through resistor R1.

Applying Ohm’s law, the I1 current is equal to the voltage drop across R1 divided by its resistance:

\(
\begin{equation}
\mathbf{I}_{1} = \frac{\mathbf{V}_{IN} – \mathbf{V}_{OUT}}{\mathbf{R}_{1}} \tag{1}\\
\end{equation}
\)

Knowing that the op amp is in a non-inverting configuration, and assuming an ideal voltage source for Vin, we know that the output voltage Vout is:

\(
\begin{equation}
\mathbf{V}_{OUT} = \mathbf{V}_{IN} (1 + \frac{\mathbf{R}_{2}}{\mathbf{R}_{3}}) \tag{2}\\
\end{equation}
\)

Substituting equation #2 into equation #1, we can factor Vin out of the numerator and simplify:

\(
\begin{equation}
\mathbf{I}_{1} = \frac{\mathbf{V}_{IN} – \mathbf{V}_{IN} (1 + \frac{\mathbf{R}_{2}}{\mathbf{R}_{3}})}{\mathbf{R}_{1}} \tag{3}\\
\end{equation}
\)

\(
\begin{equation}
\mathbf{I}_{1} = \frac{\mathbf{V}_{IN} [1 – 1 (1 + \frac{\mathbf{R}_{2}}{\mathbf{R}_{3}})]}{\mathbf{R}_{1}} \tag{4}\\
\end{equation}
\)

\(
\begin{equation}
\mathbf{I}_{1} = \frac{\mathbf{V}_{IN} [1 – (1 + \frac{\mathbf{R}_{2}}{\mathbf{R}_{3}})]}{\mathbf{R}_{1}} \tag{5}\\
\end{equation}
\)

\(
\begin{equation}
\mathbf{I}_{1} = \frac{\mathbf{V}_{IN} [1 – 1 – \frac{\mathbf{R}_{2}}{\mathbf{R}_{3}}]}{\mathbf{R}_{1}} \tag{6}\\
\end{equation}
\)

\(
\begin{equation}
\mathbf{I}_{1} = \frac{-\mathbf{V}_{IN} \frac{\mathbf{R}_{2}}{\mathbf{R}_{3}}}{\mathbf{R}_{1}} \tag{7}\\
\end{equation}
\)

We can now divide both sides by Vin and simplify the complex fraction on the right-hand side of the equation:

\(
\begin{equation}
\frac{\mathbf{I}_{1}}{\mathbf{V}_{IN}} = \frac{- \frac{\mathbf{R}_{2}}{\mathbf{R}_{3}}}{\mathbf{R}_{1}} \tag{8}\\
\end{equation}
\)

\(
\begin{equation}
\frac{\mathbf{I}_{1}}{\mathbf{V}_{IN}} = \mathbf{-} \frac{\mathbf{R}_{2}}{\mathbf{R}_{1}\mathbf{R}_{3}} \tag{9}\\
\end{equation}
\)

Since Ohm’s law defines resistance as voltage divided by current, we just have to flip both sides of the equation to finally arrive at:

\(
\begin{equation}
\mathbf{R}_{IN} = \frac{\mathbf{V}_{IN}}{\mathbf{I}_{1}} = \mathbf{-} \frac{\mathbf{R}_{1}\mathbf{R}_{3}}{\mathbf{R}_{2}} \tag{10}\\
\end{equation}
\)

This is the input resistance seen looking into the input of the NIC circuit.
If R2 and R3 are made equal to each other, the input resistance is simply equal to -R1.
If R1 and R2 are made equal to each other, the input resistance is simply equal to -R3.

References and Additional Reading

Description Reference
Negative Impedance Converters Negative Impedance Converter, Wikipedia
Use of NIC as an active load Negative Resistor Cancels Op Amp Load, Maxim Application Note 1868
Chua chaotic oscillator Improved Implementation of Chua’s Chaotic Oscillator Using Current Feedback Op Amp, A.S. Elwakil & M.P. Kennedy
The use of impedance converters in active filters The Filter Wizard
issue 18: Gee, I see! The Ins and Outs of Generalized Impedance Converters
, Kendall Castor-Perry

5 thoughts on “Negative Impedance Converters

  1. FYI – to see the math you may need to disable your adblock software. Chrome with uBlock showed [Math processing error] until I disabled it on this site.

    Great article, reminds me of the Frequency Dependent Negative Resistor used in filter designs – http://www.analog.com/media/en/training-seminars/design-handbooks/Op-Amp-Applications/Sections5-5-to-5-8.pdf, section 5.6

    The whole op-amp handbook is here for free:
    http://www.analog.com/en/education/education-library/op-amp-applications-handbook.html

    I highly recommend Analog Devices’ material for learning practical, realizable, analog and mixed signal circuits.

    • Looks fine to me with uBlock Origin and Firefox ESR 52.6. I suppose it depends on whether one applies one too many blocklists or not, but the default seems ok…

  2. Pingback: Negative impedance converters – gStore

  3. 1. Why do I have to allow Scripting AND Cloudflare on this site to see the equations? This is really BAD Web design. Grow up! We’re tired of Web sites tracking and spying on us just to see the content.

    2. Why is there a YouTube video about the SAME topic? Click-Bait/Join push maybe? Like many I DO NOT HAVE TIME to watch YouTube videos on simple subjects that can be rendered on simple Web pages.

    FAIL…

    • 1) I use MathJax, which I’ve found quite useful for rendering LaTeX-style equations. This does require that your browser load the MathJax javascript file, which is hosted on Cloudflare.

      2) Many of my posts on this site, this one included, are simply here to provide supporting content for my YouTube videos.

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