# Question about the negative resistance and the negative reactance

In physics, negative resistance (NR) is a property of some electrical circuits and devices in which an increase in voltage across the device's terminal results in a decrease in electric current through it.

What if a decrease in voltage cause an increase in current?

Will we also call it the negative resistance?

Plus can we define negative reactance also as "in which an increase in voltage across the device's terminal results in a decrease in electric current through it"?

Although reactance is the property of inductor or capacitor to oppose the flow of charge but i think same definition of negative resistance is applied to the negative reactance also. I guess capacitive reactance is called negative reactance because across a capacitor rise in voltage cause fall in current passing through it.

You have to be careful as to whether you are talking about resistance $R = \frac V I$ or small signal / incremental / dynamic resistance $R_{\rm incremental}= \frac {dV}{dI}$.

If the V_I characteristic graph of a circuit element or circuit is a straight line through the origin then the Resistance $\frac V I$ is a constant and is the same as the incremental resistance.

Negative resistance where the voltage drops as the current increases sometimes occurs in circuits which have voltage or current controlled voltage/current sources.

Negative incremental resistance where the the gradient of a V_TI characteristic graph is negative does occur sometimes and over a region of operation an increase/decrease in the voltage produces a decrease/increase in the current.

Now when it comes to reactance and impedance the situation is more complicated.

You wrote

I guess capacitive reactance is called negative reactance because across a capacitor rise in voltage cause fall in current passing through it.

but that is not so if you plot the instantaneous voltage across a capacitor $V_{\rm C}$ against the instantaneous current passing through it.

The graph is actually an ellipse and as time progresses a point on the graph moves in a clockwise direction.
This graph is the addition of two perpendicular simple harmonic motions which are at right angles to one another and $90^\circ$ out of phase.
For an inductor the time progression would be anti-clockwise.
If you were interested in the incremental reactance at a given time then it would be positive in the second and fourth quadrants and negative in the first and third quadrants.

So the statement you made about the capacitor is only true half of the time.

The introduction of a negative sign is really to do with the fact that the voltage across a capacitor lags the current through the capacitor by $90^\circ$.

When the frequencies are the same but there are phase differences it is convenient to represent voltages and currents as phasors or as complex numbers.

In this representation the reactance of a capacitor is $Z_{\rm C} = \frac {1}{j\omega C}$ and for an inductor $Z_{\rm L}= j \omega L$.
These are nothing to do with instantaneous values of the current and voltage rather the magnitude of the reactances is the ratio of peak voltage to peak current and the phase information is contained in the position of $j = \sqrt{-1}$.

When one is dealing with an inductor and a capacitor in series the impedance of the combination is $Z_{\rm LC}= j \omega L + \frac{1}{j \omega C} = j \left (\omega L - \frac {1}{\omega C} \right )=j(|Z_{\rm L}|-|Z_{\rm C}|)$ where $|Z_{\rm L}| = \omega L$ and $|Z_{\rm C}| = \frac{1}{\omega C}$

Pictorially

The minus sign in front of the capacitor is there because in a series circuit the instantaneous voltage across an inductor differs in phase from the voltage across a capacitor by $180^\circ$ and so to find the the total of these two voltages one only needs to take one from the other and so the same is true of the magnitudes of their reactances..

What I have tried to explain is that negative capacitor reactance is not the same as negative resistance.

• thanks alot @farcher. it was helpful. can you please further explain whats mean by negative impedence ? what does it mean if the net impedence in a circuit is negative ? – Alex Oct 10 '16 at 16:41
• in AC circuits the impedence is shown along with negative angle sometimes. what does this negative angle mean ? – Alex Oct 10 '16 at 17:08
• In the phaser diagram above the anti-clockwise direction is taken as positive. The voltage across the capacitor lags the current by $90_\circ}$. Relative to the current this is a negative angle. The reactance $X_{\rm C}$ is in the negative j direction because of the phase lag. – Farcher Oct 10 '16 at 20:24