Engineering Dictionary of Electronic Terms
"A" "B" "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M",
"N", "O", "P", "Q", "R", "S", "T", "U", "V", "W", "X", "Y", "Z"

### RC Differentiator

 An RC Differentiator is a series RC circuit in which the input is applied to a capacitor and the output is taken from the resistor. RC Differentiating Circuit

The circuit shows the standard schematic for the RC differentiator and the classic input and output waveforms used to explain the operation.

#### RC Differentiator Introduction

The output signal from the differentiator is proportional to the rate of change of the input signal. When the input signal rises or goes positive at the input of the differentiator, the circuit generates a positive going spike. When the input signal falls or goes negative at the input of the differentiator, the circuit generates a negative going spike. When the input is constant or steady the output is zero.

The RC circuit generates a first order derivative of the signal applied to the input. A second order, another RC node, would need to be appended to the first circuit to generate a second derivative of the input. Additional circuits could be concatenated to produce more complex outputs, but anything after the second order becomes unpractical.

In fact because it's a passive RC network the circuit is sensitive to frequency and input pulse width. The input capacitor will block any DC bias, so the output will always be centered at zero volts. Because the capacitor is non-ideal, the output will always be less than the input; however the peak-to-peak output voltage may be greater because the capacitor produces a negative going spike, at least for a square wave input.

In addition; non-symmetrical waveforms will also output voltages less than the input voltages. A sine wave will be effected by a shift in phase and a reduction in amplitude, but otherwise appear as a sine wave.

#### RC Differentiator Operation

The circuit begins with no voltage applied to the input and an output of zero voltages. When the first pulse is applied to the circuit the capacitor instantaneously charges to the input voltage. Or at least charges as fast as the physical make-up of the capacitor will allow. However, there is a finite amount of time required to charge the capacitor. During that time a large current flows through the RC circuit, its that current surge that develops across the resistor and produces the first spike at the output. So the leading edge of the square wave produces the leading edge of the output. As the input voltage reaches its peak and flattens out current flow into the capacitor slows by an exponential rate. The slow exponential rise in voltage across the capacitor occurs with the exponential fall in current through the resistor. It's that reducing current and voltage drop across the resistor that produces the exponential decay in output voltage.

Once the capacitor voltage reaches it's peak and stays there, no further current flows in the circuit and the output voltage settles at zero volts. As long as the input voltage stays at its peak value, current flow is zero and the output remains at zero volts. When the input waveform reaches the end of the first pulse width, a negative going pulse is feed to the differentiator. The identical, but opposite thing occurs within the circuit.

The capacitor discharges through the resistor producing a large negative going spike. The same voltage as before, but in the opposite polarity. The capacitor voltage exponential decays to zero volts, which causes an exponential reduction in the current flowing through the resistor. As the capacitor reaches zero volts the current through the resistor again levels of to zero, producing a zero voltage drop across the resistor.

#### RC Component Selection

Component selection is straight forward, but frequency dependent. Comparing the input square wave to the output details the basic operation of a capacitor. The capacitor instantly charges to the applied input voltage step, or at least appears to when compared to the fall time of the spike. The value of the capacitor should offer no resistance [Xc] to the frequency of the input signal [Capacitor Selection]. So the capacitor value should be selected to be 10x higher than the incoming frequency. That is the capacitor should not effect the input signal frequency until at least a decade above the frequency of interest. Unfortunately that statement applies to both the period of the waveform and the rise time of the pulse.

Reactance [X] is the opposition offered to the flow of an alternating current by the capacitance [XC] in the circuit. The reactance is the imaginary part of the total complex impedance formula; Z = R + jX [shown in the right side-bar]. The reactance of a capacitor decreases with frequency.

The resistor value effects the RC time constant of the circuit, or determines the charge and discharge times of the capacitor [Resistor Selection]. The wattage would be determined by the current through the resistor as the capacitor charges or discharges.

Note if the period of the waveform is faster then the charge time of the capacitor [TC], than the capacitor will not be allowed to completely charge or discharge. The effect will be a bias voltage on the output. The output voltage will not return to zero, because current is still flowing in the circuit.

#### RC Networks

Note that this two component RC network is found as a input to many circuits, and outputs if the resistor is considered as a load. In a transistor circuit the capacitor might be used as a bias blocking capacitor followed by a Base bias resistor [shown as C1 and R2 in the transistor circuit to the right]. A first order passive RC high pass filter could also appear with the same configuration, as found on the Passive Filters page. However its the frequency of operation and the component values that determine the function of the network, not that it has the same configuration.