Differentiator

In electronics, a differentiator is a circuit that is designed such that the output of the circuit is approximately directly proportional to the rate of change (the time derivative) of the input. An active differentiator includes some form of amplifier. A passive differentiator circuit is made of only resistors and capacitors.

Passive differentiator

Figure 1: Capacitive Differentiator

Figure 2: Inductive Differentiator

A true differentiator cannot be physically realized, because it has infinite gain at infinite frequency. A similar effect can be achieved, however, by limiting the gain above some frequency. Therefore, a passive differentiator circuit can be made using a simple first-order high-pass filter, with the cut-off frequency set to be far above the highest frequency in the signal. This is a four-terminal network consisting of two passive elements as shown in Figures 1 and 2.

Transfer function

The analysis here is for the capacitive circuit in Figure 1. The inductive case in Figure 2 can be handled in a similar way.

The transfer function shows the dependence of the network gain on the signal frequency for sinusoidal signals.

According to Ohm's law,

Y=X{\frac {Z_{R}}{Z_{R}+Z_{C}}}=X{\frac {R}{R+{\frac {1}{j\omega C}}}}=X{\frac {1}{1+{\frac {1}{j\omega RC}}}},

where $X$ and $Y$ are input and output signals' amplitudes respectively, and $Z_{R}$ and $Z_{C}$ are the resistor's and capacitor's impedances. Therefore, the complex transfer function is

K(j\omega )={\frac {1}{1+{\frac {1}{j\omega RC}}}}={\frac {1}{1+{\frac {\omega _{0}}{j\omega }}}},

where

\omega _{0}={\frac {1}{RC}}.

The amplitude transfer function

H(\omega )\triangleq |K(j\omega )|={\frac {1}{{\sqrt {1+\left({\frac {\omega _{0}}{\omega }}\right)^{2}}}}},

and the phase transfer function

\phi (\omega )\triangleq \arg K(j\omega )=\arctan {\frac {\omega _{0}}{\omega }},

which are both shown in Figure 3.

Figure 3: Amplitude and phase transfer functions for a passive differentiator circuit

Transfer functions for the second circuit are the same (with $\omega _{0}={\frac {R}{L}}$ ).

Impulse response

The circuit's impulse response, which is shown in Figure 4, can be derived as an inverse Laplace transform of the complex transfer function:

h(t)={\mathcal {L}}^{{-1}}\left\{K(p)\right\}=\delta (t)-\omega _{0}e^{{-\omega _{0}t}}=\delta (t)-{\frac {1}{\tau }}e^{{-{\frac {t}{\tau }}}}

where $\tau ={\frac {1}{\omega _{0}}}$ is a time constant, and $\delta (t)$ is a delta function.

Figure 4: An impulse response of a passive differentiator circuit

Active differentiator

A differentiator circuit consists of an operational amplifier, resistors are used at feedback side and capacitors are used at the input side. The circuit is based on the capacitor's current to voltage relationship:

V=V(\infty )+[(V(0+)-V(\infty )]e^{-{\frac {t}{\tau }}}

I=C{\frac {dV}{dt}}

where I is the current through the capacitor, C is the capacitance of the capacitor, and V is the voltage across the capacitor. The current flowing through the capacitor is then proportional to the derivative of the voltage across the capacitor. This current can then be connected to a resistor, which has the current to voltage relationship:

I={\frac {V}{R}}

where R is the resistance of the resistor.

Note that the op amp input has a very high input impedance (it also forms a virtual ground) so the entire input current has to flow through R.

If V_out is the voltage across the resistor and V_in is the voltage across the capacitor, we can rearrange these two equations to obtain the following equation:

V_{{{\text{out}}}}=-{R}{C}{\frac {dV_{{{\text{in}}}}}{dt}}

From the above equation following conclusions can be made:

Output is proportional to the time derivative of the input – Hence, the op amp acts as a differentiator;
Above equation is true for any frequency signal.

Thus, it can be shown that in an ideal situation the voltage across the resistor will be proportional to the derivative of the voltage across the capacitor with a gain of RC.

Operation

Input signals are applied to the capacitor C. Capacitive reactance is the important factor in the analysis of the operation of a differentiator. Capacitive reactance is X_c = 1/2πfC. Capacitive reactance is inversely proportional to the rate of change of input voltage applied to the capacitor. At low frequency, the reactance of a capacitor is high and at high frequency reactance is low. Therefore, at low frequencies and for slow changes in input voltage, the gain, R_f/X_c, is low, while at higher frequencies and for fast changes the gain is high, producing larger output voltages.

If a constant DC voltage is applied as input then the output voltage is zero. If the input voltage changes from zero to negative, the voltage output voltage is positive. If the applied input voltage changes from zero to positive, the output voltage is negative. If a square wave input is applied to a differentiator, then a spike waveform is obtained at the output.

The active differentiator isolates the load of the succeeding stages, so it has the same response independent of the load.

At high frequencies this simple differentiator circuit becomes unstable and starts to oscillate. This high frequency gain of the circuit is reduced by adding a small value capacitor across feedback resistor R_f or a resistor in series with the capacitor. In exchange for stability, the circuit has a reduced high-frequency capability.^[1]

Uses

The differentiator circuit is essentially a high pass filter. It can generate a square wave from a triangle wave input, and will produce alternating-direction voltage spikes when a square wave is applied. In ideal cases, a differentiator will reverse the effects of an integrator on a waveform, and vice versa. Differentiators are an important part of electronic analogue computers and analogue PID controllers.

A passive differentiator circuit is one of the basic electronic circuits, being widely used in circuit analysis based on the equivalent circuit method.

References

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- http://www.wisc-online.com/objects/index_tj.asp?objID=SSE5203 wisc-online.com.

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