Serial Connected RC Networks: Differentiator

When connecting the input to a capacitor for a passive RC differentiator circuit, the output voltage is taken from a resistance that is the opposite of the RC Integrator Circuit.

A passive RC differentiator is nothing more than a capacitance in a resistant series. In other words, it is a device connected to the frequency with serial recess (inverse of the integrator) with constant resistance. Like the integrator circuit, the output voltage depends on the RC time constant and input frequency of the circuits.

Therefore, at low input frequencies, the capacitor's recess, its XC blocks any dc high. Voltage or slow-changing input signals. when at high input frequencies, the recess of capacitors is low and allows rapidly changing pulses to pass directly from input to output.

This is due to the fact that the ratio of capacitive reactance (XC) to resistance (R) is different for different frequencies, and the lower the frequency, the less output there is. Therefore, for a certain time constant, as the frequency of input pulses increases, the output pulses are increasingly similar in shape to input pulses. We saw this effect in our tutorial on Passive High Pass Filters, and if the input signal is a sinus wave, it acts as an rc differentiator, a simple high pass filter (HPF) with a cut or corner frequency corresponding to the RC. time constant (tau, τ) of the serial network.

Therefore, when fed by pure sinus wave, an RC differentiator circuit acts as a simple passive high pass filter due to the XC = 1/(2πεC) standard capacitive reactate formula.

However, a simple RC network can also be configured to perform differentiation of the input signal. We know from previous tutorials that the current passing through a capacitor is a complex exponent given by iC = C(dVc/dt). The charging (or discharge) speed of the capacitor is directly proportional to the amount of resistance and capacitance that gives the circuit's time constant. Therefore, the time constant of an RC differentiator circuit is the time interval equal to the product of R and C. Consider the basic RC series circuit below.

RC Differentiator Circuit

Differentiator
RC Differentiator Circuit

The input signal for an RC differentiator circuit is applied to one side of the capacitor so that the output is taken through the resistance, after which VOUT is equal to VR. Since the capacitor is an element connected to the frequency, the amount of load formed between the plates is equal to the time zone integral of the current. This means that since the capacitor cannot charge instantly, it only takes a certain amount of time for the capacitor to fully charge, since it is only charged exponencially.

In our tutorial on RC Integrators, we found that when a one-step voltage pulse is applied to the input of an RC integrator, the output turns into a saw-tooth waveform if the RC time constant is long enough. The RC differentiator will also change the input waveform, but differently from the integrator.

Resistance Voltage

Previously, we said that the output for the RC differentiator is equal to the voltage on the resistance, that is: VOUT is equal to VR, and since there is a resistance, the output voltage can change instantly. However, the voltage on the capacitor cannot change instantly, it depends on the value of the capacitance, C, when trying to store an electrical charge along the Q plates. Then the current flowing into the capacitor depends on the speed at which the load changes along the plates. Therefore, the capacitor current is not proportional to the voltage, but to the time change: i = dQ/dt.

Since the amount of load on the capacitor plates is equal to Q = C x Vc, that is, capacitance multiplication voltage, we can deride the capacitor current equation as follows:

Capacitor Current

Differentiator

Therefore, the capacitor current can be written as follows:

Differentiator
Capacitor Current

Since VOUT is equal to VR, vr is equal here according to ohm law: iR x R. The current flowing from the capacitor must also flow through the resistance, since they are both connected in series. So:

Differentiator

RC Differentiator Formula

Differentiator
RC Differentiator Formula

Then we can see that the output voltage, the VOUT, the input voltage weighted by the RC constant, are derivatives of the VIN. Where the RC represents the time constant, the serial circuit's τ.

Single Pulse RC Differentiator

When a single-stage voltage pulse is first applied to the input of an RC differentiator, the capacitor initially "appears" as a short circuit to the fast-changing signal. This is because the dv/dt slope of the positive outbound edge of a square wave is too large (ideally infinite), so as soon as the signal appears, the entire input voltage passes to the output that appears along the resistance.

Differentiator

After the first positive edge of the input signal passes and the peak value of the input is constant, the capacitor begins to charge normally through the resistance in response to the input pulse at a rate determined by the RC time. constant, τ = RC.

When charging the capacitor, the voltage on the resistance and therefore the output are exponencially reduced until fully charged after a capacitor 5RC (5T) time constant, resulting in zero output throughout the resistance. Thus, the voltage on the fully charged capacitor is equal to the value of the input pulse: VC = VIN and this condition is valid as long as the size of the input pulse does not change.

Now, if the input pulse changes and returns to zero, the change rate of the negative edge of the pulse switches from capacitor to output because the capacitor cannot respond to this high dv/dt change. The result is an increase in output that goes negative.

After the first negative outgoing edge of the input signal, the capacitor recovers and begins to discharge normally, and the output voltage along the resistance, and therefore the output, begins to increase exponentially as the capacitor discharges.

Therefore, when the input signal changes rapidly, a voltage increase is produced at the output with the polarity of this voltage rise, depending on whether the input changes in a positive or negative direction. A positive rise is produced with a positive trend. the edge of the input signal and a negative increase produced as a result of the negative outgoing input signal.

Therefore, the RC differentiator output is a graph of the rate of change of the input signal, which has no resemblance to the square wave input wave but consists of narrow positive and negative spikes as the input pulse value changes.

The shape of the output pulses will change as shown by changing the T time period of the square wave input pulses according to the fixed RC time constant of the serial combination.

RC Differentiator Output WaveForms

Differentiator
RC Differentiator Output WaveForms

Then we can see that the shape of the output waveform depends on the ratio of pulse width to RC time constant. When the RC is much larger than the pulse width (greater than 10RC), the output waveform resembles the square wave of the input signal. When the RC is much smaller than the pulse width (less than 0.1RC), the output waveform takes the form of very sharp and narrow spikes, as shown above.

Thus, we can produce a series of different waveforms by changing the time constant of the circuit from 10RC to 0.1RC. In general, a smaller time constant is always used in RC differentiator circuits to provide good sharp pulses at output along R. Therefore, the differential (high dv/dt step input) of a square wave pulse is an infinitely short increase resulting in an RC differentiator circuit. .

Suppose there is a period of a square wavelength, T gives a pulse width of 20mS (divided by 20mS divided by 2) 10mS. The pulse width must be equal to the RC time constant, that is, RC = 10mS, for the spike to discharge up to 37% of the initial value. If we choose a C value of 1uF for the capacitor, R is equal to 10kΩ.

For the output to resemble the input, the pulse width of the RC must be ten times (10RC), so for a capacitor value of 1uF, this gives a resistance value of 100kΩ. Similarly, for the output to resemble a sharp pulse, the RC must have a tenth of the pulse width (0.1RC). Therefore, this resistance value is given for the same capacitor value 1uF: 1kΩ, etc.

RC Differentiator Example

Differentiator

Therefore, by having an RC value of one-tenth of the pulse width (and in our example above it is 0.1 x 10mS = 1mS), or with a lower value, we can produce the necessary spikes in output. The lower the RC time constant for a specific pulse width. Spikes are sharp. Thus, the exact shape of the output waveform depends on the value of the RC time constant.