For The Circuit Shown Assume The Opamp Is Ideal. What Kind Of Filter Is Shown?

Ultimate Electronics: Practical Excursion Design and Analysis

7.1
The Ideal Op-Amp (Operational Amplifier)

The platonic op-amp model is a key building block of designing analog filters, amplifiers, oscillators, sources, and more. 13 min read

Operational amplifiers, usually shortened to simply "op-amps", are an essential building block of analog electronic systems. In different configurations with a few other components, op-amps can be used to process and manipulate an analog voltage indicate in many dissimilar ways. This includes many kinds of filters (low-pass, loftier-pass, band-laissez passer, integrator, differentiator), amplifiers (buffer, inverting, non-inverting, differential, summing, instrumentation), oscillators, comparators, sources (voltage, current), converters (voltage-to-electric current, electric current-to-voltage), and even some nonlinear applications.

These applications are tremendously useful, and nosotros'll expect at ieach one individually in the upcoming sections, only first allow's understand the ideal op-amp on its own.

Today, an op-amp is an integrated excursion (IC) containing a few dozen individual transistors and passive components. Historically, before the age of ICs (1960s-1970s), about amplifiers or analog signal processing stages would be purpose-designed for a specific awarding to avoid the op-amp's relatively high complexity and cost. But now that IC op-amps have only a few pins and cost just a few pennies, it unremarkably makes sense to have advantage of their enormous potential for making analog designs simpler.

Most op-amps aspire to perform similar the ideal op-amp, a theoretical model that both works well in simulation and makes it like shooting fish in a barrel to solve circuits by hand. Every bit a result, virtually design and analysis volition treat the op-amp as being ideal, and that's how we'll begin.

Later, we'll discuss the ways in which this ideality breaks downwards in real-world non-platonic op-amps. These limitations are crucial for knowing when you can estimate your analysis every bit an ideal op-amp, and when you can't. They tin can likewise help y'all cull the correct op-amp to implement your blueprint.

The ideal op-amp is a voltage amplifier with 2 inputs and one output:

Ideal Op-Amp Symbol

The 2 inputs are called the not-inverting input (+) and the inverting input (-).

Go along a shut eye on the + and - signs labeled within the triangle! The op-amp is commonly drawn either manner, with + on top or on bottom, whatever makes the rest of the schematic easiest to describe. (In CircuitLab, select the op-amp and press "V" to flip the symbol vertically.) If y'all unintentionally bandy the two inputs, your design won't work, both on paper and in the real globe!

Conceptually, the platonic op-amp subtracts the 2 inputs, and and so multiplies that difference by a huge number called the open-loop proceeds :

Every bit signal processing steps, this subtraction and multiplication looks similar:

Ideal Op-Amp Subtraction and Multiplication

Alternatively, the ideal op-amp tin exist modeled equally a Voltage Controlled Voltage Source (VCVS):

Ideal Op-Amp as VCVS

If you await advisedly, the VCVS model higher up raises a new question: why did a ground of a sudden announced within the op-amp? Since voltages are always relative, this is implying that in the more complete and correct equation:

If we accept an op-amp and we short together the input terminals so that , the output volition exist . In the real world, in a real op-amp with the inputs shorted together, the output will not necessarily be any detail voltage, and whatever voltage it is will certainly be relative to whatever else we're measuring. All the same, in ideal op-amp excursion analysis, we normally assume as a simplifying assumption because either:

The op-amp is being used in a closed-loop feedback configuration, where a static get-go becomes irrelevant subsequently applying feedback rules (especially since the gain is so large), or
The op-amp is being used in an open-loop configuration with no feedback, in which case nosotros saturate the output into not-linear, non-ideal behavior quickly anyway.

How big is the gain? In existent-globe non-ideal op-amps, typical values of the open-loop proceeds are from the hundreds of thousands to the tens of millions:

That's really big! A millivolt departure in the inputs becomes hundreds or thousands of volts at the output! It'southward so big that in ideal op-amp assay, we make another simplifying assumption, taking the limit assuming that the proceeds goes to infinity:

That'due south the algebraic model of the ideal op-amp: it subtracts the voltage at the inverting input from the non-inverting input, and then multiplies the difference past a very big proceeds that approaches infinity.

Even in real op-amps, the datasheet often guarantees merely a minimum open-loop gain, but not a maximum. You tin can't and shouldn't blueprint a circuit relying on knowing the exact value of the open-loop gain of an op-amp.

Information technology can exist hard to recollect about infinities! One helpful mental trick is to intermission fourth dimension and imagine what'southward happening dynamically: instead of jumping immediately to infinity, imagine that when given a slight difference in inputs, the ideal op-amp'due south output voltage simply starts rising, ascent, ascension toward infinity! As we introduce different closed-loop feedback configurations later on, you lot'll see that this rapid rise of the output voltage eventually finds its way back to touch i or both of the same op-amp's inputs, so don't be alarmed: the infinities won't last very long.

It can be difficult to practise algebra with infinities, besides. A suggestion is to keep in place as a variable, and only at the end, have the limit .

The platonic op-amp continuously measures the voltages at its inputs, and adjusts its output voltage:

If the not-inverting (+) input is at a higher voltage than the inverting (-) input, the op-amp volition increment its output voltage.
If the non-inverting (+) input is at a lower voltage than the inverting (-) input, the op-amp will decrease its output voltage.

In equation course:

If feedback is present and in the correct direction, then the op-amp will continuously make adjustments to its output voltage until the two input voltages are the same.

At that place are a number of other assumptions engineers brand about platonic op-amps. All of these assumptions will break for real (non-ideal) op-amps, so go on an centre out for how they might affect your circuit.

By learning about these ideality assumptions, we can decide when we tin can design a circuit assuming the op-amp is platonic (and thus much easier to analyze), and when this simplified model is probable to collide with reality. We'll explore these issues in more than depth in later sections.

No electric current can flow into or out of the input terminals of an ideal op-amp. The input terminals can simply measure their voltages. From Thevenin Equivalent Circuits, this is like saying that the input impedance looking into the input terminals is infinite:

The output of an ideal op-amp can concur its and supply any amount of current, in or out, without that voltage changing. In the Thevenin equivalent model looking into the output concluding (and ground), it appears like a voltage source with zilch resistance – therefore zero output impedance:

In ideal op-amps we assume that the non-inverting and inverting inputs are perfectly balanced so that . In the real world, due to manufacturing processes, in that location'south some input offset voltage such that . You tin can think of this conceptually by simply adding a small voltage source in series with one of the inputs. If DC accurateness matters, this input outset (even just a few millivolts!) can be a big bargain, especially because it can drift while the circuit is operating. Only in an ideal op-amp, we presume:

The schematic symbol for the ideal op-amp omits connections to the ability supply, simply a real op-amp has to get power from somewhere and deliver ability to the schematic. On a datasheet, this starts with the op-amp's quiescent electric current . (Run across Power for a discussion of power and energy accounting in circuits.) In platonic op-amps, we treat this like a VCVS: it's an active source and tin supply power to the circuit.

The charge per unit at which an op-amp tin can change its output voltage is chosen the slew charge per unit. In real op-amps, there'due south a limit to how fast the output can rise or fall, measured in . (This is similar to the mental pull a fast one on about thinking nigh infinite open up-loop gain discussed above.) In ideal op-amps, we permit an infinite slew charge per unit: the output tin can move infinitely fast.

In add-on to the slew rate limit (which is a nonlinear limit), there'due south also a bandwidth limit in real op-amps: they are not responsive to all frequencies. Real op-amps accept an open-loop proceeds which is a part of frequency, , and it declines at high frequencies. In detail, the proceeds-bandwidth product (GBW) is the frequency at which the op-amp's open-loop gain drops to ane. Notably, the proceeds starts declining far before that frequency. But in ideal op-amps, we assume the open-loop gain is constant and large (budgeted infinity) for all frequencies.

Every bit discussed extensively to a higher place, we assume platonic op-amps have proceeds approaching infinity. Real op-amps have finite open-loop gain, which can limit the amount of amplification nosotros can get from a unmarried op-amp phase.

In ideal op-amps, we assume that if we double the input voltage difference, we'll double the output voltage. Real op-amps are made of nonlinear components and this isn't true. Notwithstanding, because op-amps are used in closed-loop feedback configurations, the feedback keeps the input voltage difference extremely small, within the range where we do see basically linear behavior. It's safety to assume linearity in the ideal op-amp.

An ideal op-amp tin can have inputs of whatever value; only their difference matters. Only in a real op-amp, there will exist limits on the allowed input voltages to prevent damaging the input transistors. The subtraction won't work properly if your inputs exceed these limits, and your circuit won't work as designed. (More subtly, you'll become nonlinear baloney before you reach the hard limits.) In near cases, the limits are correct around the positive and negative ability supply voltages, but you should check the datasheet to be sure.

An ideal op-amp can output any voltage. Merely in a real op-amp, you're limited to what the output transistors can deliver. These limits are usually correct around the positive and negative power supply voltages, simply yous should check the datasheet.

An platonic op-amp just responds to changing voltages on its non-inverting and inverting input pins. Just a existent op-amp may "leak" some of the variation from its ability supply pins into the output. (This is captured as the Power Supply Rejection Ratio [PSRR] spec on a datasheet.) This lets a noisy power supply contaminate a signal.

An ideal op-amp doesn't add whatever racket to the signal. But in a real op-amp, dissonance is added and possibly even amplified.

The ideal op-amp is pretty fantastic! Unfortunately, they're all sold out. The real IC op-amps you tin can buy are not-ideal in all of the ways described above, and semiconductor manufacturers have to make their own tradeoffs to hit their target specs and cost indicate.

Equally a effect, if the analog design problem you're trying to solve is particularly demanding in whatsoever direction, you might non want to use an op-amp. For example, if you need to design an amplifier stage with the admittedly highest frequency performance, or one with the admittedly lowest power consumption, you probably aren't going to utilize an op-amp.

Fortunately, at that place are thousands of unlike models of op-amps available for sale, and they all make different tradeoffs amidst these non-idealities. In many cases, past agreement your design problem and how information technology maps to these non-idealities, y'all'll be able to find one that meets your needs out of the box!

It's ofttimes useful to relax the "Unlimited Output Voltage Range" assumption above and instead, model an platonic op-amp with voltage rails, where the output is contrained to prevarication within the provided range.

Ideal Op-Amp with Voltage Rails Symbol

It's useful to run a DC Sweep simulation to come across what the output of the platonic op-amp looks like, open up-loop, with and without voltage rails. The two output curves overlap in the centre, when the limits aren't exceeded. Only with voltage track, the V(Output_with) line is chopped off to be flat and horizontal once the limits are exceeded:

Op-Amp With and Without Voltage Rails DC Sweep Comparison

Practice Click to open and simulate the excursion above and observe how ane output appears clipped equally the input varies.

(Note that for many real op-amps, its output can't swing all the way to the positive supply rails, and can't swing all the mode downward to the negative one.)

Now that we accept an ideal op-amp with voltage rail, we can apply it open-loop every bit a voltage comparator. The ideal op-amp's infinite gain is effectively overruled by having voltage output limits, so in outcome:

for some very pocket-sized .

This can be demonstrated past connecting 2 sinusoidal part generators with different frequencies to the op-amp's 2 inputs:

Op-Amp with Voltage Rails as Analog Comparator

Exercise Click to open and simulate the circuit above. Scout how the output swings to either extreme as the inputs cross.

In the real world, an op-amp is not a peachy analog voltage comparator: there are far amend purpose-congenital parts. However, it'due south one of the few applications of op-amps without feedback, so yous can actually build and test this in your lab.

Information technology's useful to model an op-amp circuits in the Laplace domain because we can solve feedback systems algebraically. In particular, a useful model for the ideal op-amp involves having a finite open-loop gain :

Ideal Op-Amp with Finite Gain: Laplace Block Model

An even more than useful model involves having a finite proceeds-bandwidth production . This is modeled as having a finite gain at DC, with a one-pole depression-pass filter with a corner frequency . The depression-pass component has transfer function , where . Combining the gain and depression-pass gives us:

and tin exist implemented in CircuitLab as shown:

Ideal Op-Amp with Finite Gain and Gain-Bandwidth Product: Laplace Block Model

We'll apply this model in the upcoming application sections to algebraically solve airtight-loop feedback examples.

How useful is it to take an amplifier with really enormous (ideally space!) proceeds? By itself, not so much. In this department, we've examined the open up-loop behavior, and the nearly useful outcome is a mediocre analog voltage comparator.

Just once nosotros build a circuit effectually the ideal op-amp, nosotros tin "shut the loop" and tame the wildly huge distension into something we can design and control with closed-loop feedback. It turns out that having a subtract-and-multiply-by-infinity component is an nigh magically useful building block for a wide range of analog signal processing needs. Nosotros'll explore these in the next several sections, starting with one of the simplest: the Op-Amp Voltage Buffer.