Integrator

OBJECTIVE:

To study the effect of integrator on different waveforms and measuring the frequency and peak to peak voltages of the output waveforms.

THEORY :

As its name implies, the Integrator Amplifier is an operational amplifier circuit that performs the mathematical operation of Integration that is we can cause the output to respond to changes in the input voltage over time and the integrator amplifier produces a voltage output which is proportional to that of its input voltage with respect to time. In other words the magnitude of the output signal is determined by the length of time a voltage is present at its input as the current through the feedback loop charges or discharges the capacitor. The circuit of the inverting-input terminal represents a virtual ground,

and
For an ideal op amp, current at inverting input i_- = 0, so i_f must equal i₁. Equating the above two expressions and integrating both sides of the result yields,
In which the initial value of the output voltage is determined by the voltage on the capacitor at t=0, V_o(0)=V_f(0). Thus the voltage at the output of this circuit at any time t represents the initial capacitor voltage plus the integral of the input signal from the start of the integration interval, chosen in this case to be t=0.


Figure 1: The Integrator Circuit

When a voltage, Vin is firstly applied to the input of an integrating amplifier, the uncharged capacitor C has very little resistance and acts a bit like a short circuit (voltage follower circuit) giving an overall gain of less than 1, thus resulting in zero output. As the feedback capacitor C begins to charge up, its reactance Xc decreases and the ratio of Z_f/R₁ increases producing an output voltage that continues to increase until the capacitor is fully charged. At this point the ratio of feedback capacitor to input resistor Z_f/R₁ is infinite resulting in infinite gain and the output of the amplifier goes into saturation as shown below. (Saturation is when the output voltage of the amplifier swings heavily to one voltage supply rail or the other with no control in between).


Figure 2: Output voltage for a step-function input with V_f (0)=0.

The rate at which the output voltage increases (the rate of change) is determined by the value of the resistor and the capacitor, "RC time constant". By changing this RC time constant value, either by changing the value of the Capacitor, C or the Resistor, R, the time in which it takes the output voltage to reach saturation can also be changed for example. If we apply a constantly changing input signal such as a square wave to the input of an Integrator Amplifier then the capacitor will charge and discharge in response to changes in the input signal. This results in the output signal being that of a saw tooth waveform whose frequency is dependent upon the RC time constant of the resistor/capacitor combination. If we apply a constantly changing input signal such as a square wave to the input of an Integrator Amplifier then the capacitor will charge and discharge in response to changes in the input signal. This results in the output signal being that of a saw tooth waveform whose frequency is dependent upon the RC time constant of the resistor/capacitor combination. This type of circuit is also known as a Ramp Generator and the transfer function is given below.

Figure 3: Transfer function of the ramp input integrator