# Current Source Built with a Negative Resistance

## Introduction

I have seen quite a few schematics lately that are using an operational amplifier (opamp) configured as a negative resistor. I thought it would be interesting to analyze this circuit and show how it can provide an economical solution to a common sensor interface scenario.

## Background

Figure 1 illustrates a common sensor interface scenario. This scenario consists of:

• Sensor
The sensor has a resistance characteristic that varies according to some external parameter, e.g. temperature, ambient light level, gas concentration.
• Current Source
A current source drive can be desirable for a number of reasons: (1) The sensor may be calibrated for a specific current level, (2) The sensor may have a linear response for a given current drive, (3) The sensor output voltage range needs to be restricted in some way.
• Amplifier
The amplifier is often needed to provide isolation from the monitoring circuitry and to scale the output for further signal processing.

Figure 1: Common Sensor Interface Scenario.

## Negative Resistance Opamp Circuit

Figure 2 shows an operational amplifier connected as a negative resistance.

Figure 2: Negative Resistance Circuit.

Equation 1 shows the derivation of the negative resistance equation.

 Eq. 1 ${{v}_{I}}={{v}_{O}}+i\cdot {{R}_{F}}$ ${{v}_{I}}={{v}_{O}}\cdot \frac{{{R}_{1}}}{{{R}_{1}}+{{R}_{2}}}$ ${{v}_{I}}={{v}_{I}}\cdot \frac{{{R}_{1}}+{{R}_{2}}}{{{R}_{1}}}+i\cdot {{R}_{F}}$ $-{{v}_{I}}\cdot \frac{{{R}_{2}}}{{{R}_{1}}}=i\cdot {{R}_{F}}\Rightarrow {{Z}_{IN}}\triangleq \frac{{{v}_{I}}}{i}=-{{R}_{F}}\cdot \frac{{{R}_{1}}}{{{R}_{2}}}$

Equation 2 shows the derivation of the voltage gain equation.

 Eq. 2 ${{v}_{I}}={{v}_{O}}\cdot \frac{{{R}_{1}}}{{{R}_{1}}+{{R}_{2}}}\Rightarrow {{A}_{v}}\triangleq \frac{{{v}_{O}}}{{{v}_{I}}}=\frac{{{R}_{1}}+{{R}_{2}}}{{{R}_{1}}}$

## Introduction

My application example is shown in Figure 4.

Figure 3: Negative Resistance Example Application.

The basic idea here is to use negative input resistance of the opamp circuit to cancel out the resistance of RS. Figure 4 illustrates this source transformation.

Figure 4: Source Transformation.

The parameters of this case are the following:

• $V_{CC} = 10 V$
• $I = 1\text{mA}$
• $A_v = 10$

Figure 4 shows my solution in Mathcad.

Figure 5: Example Solution in Mathcad.

## Conclusion

I worked through some basic design equations for the application of a negative resistance circuit to a sensor interface example. I have been seeing this circuit quite a bit lately and it is an interesting application of basic linear circuit analysis.

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