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| 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.
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| ==Passive differentiator==
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| [[Image:Passive_differentiator_circuit_1.png|thumb|150px|Figure 1: Capacitive Differentiator]]
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| [[Image:Passive_differentiator_circuit_2.png|thumb|150px|Figure 2: Inductive Differentiator]]
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| A '''passive differentiator circuit''' is a four-terminal network consisting of two passive elements as shown in Figures 1 and 2. It is a simple first-order [[high-pass filter]].
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| === Transfer function ===
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| The analysis here is for the capacitive circuit in Figure 1. The inductive case in Figure 2 can be handled in a similar way.
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| The [[transfer function]] shows the dependence of the network gain on the signal frequency for sinusoidal signals.
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| According to [[Ohm's law]],
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| :<math>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}},</math> | |
| where <math>X</math> and <math>Y</math> are input and output signals' amplitudes respectively, and <math>Z_R</math> and <math>Z_C</math> are the [[resistor|resistor's]] and [[capacitor|capacitor's]] [[Electrical impedance|impedance]]s.
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| Therefore, the complex transfer function is
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| :<math>K(j \omega)=\frac{1}{1+\frac{1}{j \omega RC}}=\frac{1}{1+\frac{\omega_0}{j \omega}},</math>
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| where
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| :<math>\omega_0=\frac{1}{RC}.</math>
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| The amplitude transfer function
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| :<math>H(\omega)\triangleq|K(j \omega)|=\frac{1}{\sqrt{1+\left(\frac{\omega_0}{\omega}\right)^2}},</math>
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| and the phase transfer function
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| :<math>\phi (\omega)\triangleq\arg K(j \omega)=\arctan \frac{\omega_0}{\omega},</math>
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| which are both shown in Figure 3.
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| [[Image:Passive_differentiator_circuit_transfer_function.png|center|thumb|350px|Figure 3: Amplitude and phase transfer functions for a passive differentiator circuit]]
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| Transfer functions for the second circuit are the same (with <math>\omega_0=\frac{R}{L}</math>).
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| === Impulse response ===
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| The circuit's [[impulse response]], which is shown in Figure 4, can be derived as an inverse [[Laplace transform]] of the complex transfer function:
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| :<math>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}}</math>
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| where <math>\tau=\frac{1}{\omega_0}</math> is a time constant, and <math>\delta (t)</math> is a [[Dirac delta function|delta function]].
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| [[Image:Passive_differentiator_circuit_impulse_response_1.png|center|thumb|350px|Figure 4: An impulse response of a passive differentiator circuit]]
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| ==Active differentiator==
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| [[Image:opampdifferentiating.svg|300px]] | |
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| A differentiator circuit consists of an [[operational amplifier]], [[resistor]]s are used at feedback side and [[capacitor]]s are used at the input side. The circuit is based on the capacitor's [[electric current|current]] to [[voltage]] relationship:
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| :<math>I = C \frac{dV}{dt}</math>
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| where ''I'' is the [[Electric Current|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:
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| :<math>I = \frac{V}{R}</math>
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| where ''R'' is the [[Electrical resistance|resistance]] of the resistor.
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| 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''.
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|
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| If ''V<sub>out</sub>'' is the voltage across the resistor and ''V<sub>in</sub>'' is the voltage across the capacitor, we can rearrange these two equations to obtain the following equation:
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| :<math>V_{\text{out}} = -{R}{C}\frac{dV_{\text{in}}}{dt}</math>
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| From the above equation following conclusions can be made:
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| *Output is proportional to the time derivative of the input – Hence, the op amp acts as a differentiator;
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| *Above equation is true for any frequency signal.
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| 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''.
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| ===Operation===
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| Input signals are applied to the capacitor C. Capacitive [[Electrical reactance|reactance]] is the important factor in the analysis of the operation of a differentiator. Capacitive reactance is ''X<sub>c</sub>'' = {{sfrac|2''πfC''}}. 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, {{sfrac|''R<sub>f</sub>''|''X<sub>c</sub>''}}, is low, while at higher frequencies and for fast changes the gain is high, producing larger output voltages.
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| 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.
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| The active differentiator isolates the load of the succeeding stages, so it has the same response independent of the load.
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| 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<sub>f</sub>'' or a resistor in series with the capacitor. In exchange for stability, the circuit has a reduced high-frequency capability.<ref>*http://www.wisc-online.com/objects/index_tj.asp?objID=SSE5203 wisc-online.com.</ref>
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| ==Uses==
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| 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 computer]]s and analogue [[PID controller]]s.
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| A passive differentiator circuit is one of the basic [[electronic circuit]]s, being widely used in circuit analysis based on the [[equivalent circuit]] method.
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| ==See also==
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| *[[Integrator]]
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| *[[Operational amplifier applications#Inverting differentiator|Inverting differentiator]] at op amp applications
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| ==References==
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| {{reflist}}
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| [[Category:Analog circuits]]
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