Unified Robotics I: Actuation

Operational Amplifier

Mathematical Model

Now for a circuit using an op-amp.

Figure 2-1 - Operational Amplifier Diagram

Note that all of the inputs and outputs are referenced to ground as shown in the diagram.

The output equation for this diagram is given by the formula vo = AOL(v1 - v2). As you can see, the output  is given by the difference between the non-inverting input and inverting input multiplied by the constant AOL. AOL is the 'open-loop' gain. The open-loop gain and for most op-amps is a very large value, often as large as a million or more.

Let’s think about the implications of that statement for a second. Suppose the difference between V0 and V1 is just one volt and the open-loop gain AOL is 1 million..  That means the output voltage is going to be one million volts! We’ll, actually no.  Let’s look at the next figure to see what really happens.

 

Figure 2-2 - Op-Amp Voltage Plot

The above image is a plot of the output voltage given the difference between V0 and V1. From the formula given previously we can see that the relationship between the difference in input voltages and the output voltage is going to be a straight line with a slope equal to the open-loop gain. [Note that the slope of the line shown in Figure 2-2 is much, much smaller than the real slope would be.]

So what keeps the output voltage from heading off to some huge value?  Well, the op-amp circuitry cannot generate an output voltage any larger than VCC or VEE.  Those voltages (VCC and VEE) are often called the “rail” voltages and most op-amps cannot generate an output voltage that is equal to a rail voltage.

What happens is that the op-amp circuitry “saturates” a little shy of the rail voltage.  The voltage difference between the saturation voltage and the rail voltage varies by op-amp but is typically a few tenths of a volt. [Note that some op-amps actually CAN generate an output voltage equal to the rail voltages. These are called ‘rail-to-rail’ op-amps.]

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