Contents
Introduction
There are few audio designs today that do not use operational amplifiers
(opamps). Over the years, the poor opamp has been much maligned, with mainly
spurious claims about 'audibility', distortion, and other so-called defects.
Although some of the more basic opamps do have limitations which make them less
than desirable in some cases, most of the new breed are unsurpassed for
linearity, with total harmonic distortion figures as low as 0.00008%. Even the
basic types still have their uses in control circuits and other non-demanding
applications. The operational amplifier was first used in the 1940's, as the
basis of analogue computers. Since integrated circuits were unknown at the time
(this was before the invention of the transistor), the earliest versions were
made using valves. The basic concept is to have an amplifier with differential
inputs, whose operation is controlled only by the external feedback components.
By rearrangement of the feedback circuit, different "operations"
could be performed. Typically, these early opamps could add and subtract, and
these are essential functions to this day (even in audio). With the advent of
the IC and mass production techniques, the opamp became very popular and
remains so - with considerable justification. This article will concentrate
mainly on audio applications, but there are some configurations that are just
so wonderful that I cannot resist the temptation to include them. For the most
part, any of the configurations shown can use the simplest (and cheapest) opamp
you can get (especially for testing), unless extremely wide bandwidth or low
noise is a prime consideration. For any of the test circuits this is not an
issue. I also suggest that you build up the Opamp Design and Test Board
(Project 41), which is ideal for the experimenter. Most of the circuits shown
can be built using this test board, and will function perfectly, although there
will be limitations as to bandwidth and noise because of the LM1458 dual opamps
recommended for the project.
Essential Formulae
To understand this article, you need to know Ohm's law and its
derivatives. Ohm's law is fundamental to electronics, and with little more it
is possible to derive most of the other resistance based formulae. Ohm's law
states that a potential of 1 Volt through a resistance of 1 Ohm will cause a
current of 1 Ampere to flow. This is expressed as:
R = V / I ... where R is resistance, V is voltage and I is
current, or ...
I = V / R ... or ...
V = I * R
I = V / R ... or ...
V = I * R
Later on, we will also use the formulae for inductive reactance and
capacitive reactance, as well as calculating frequency response and some filter
design. These will be presented as needed.
You will also see references to 'an instantaneous level of "x"
volts AC'. At any point in time, an AC voltage has an instantaneous voltage -
this is the voltage that is present at that moment, and for analysis can be
treated as DC. This is valid only when we consider this 'DC' level as a
transient thing, since many of the circuits do not operate down to DC at all
(many others do, but this is beside the point :-)
Basic Rules of Opamps
Many years ago I used to teach electronics, and I devised what I called
the 'Basic Rules of Opamps' for the purposes of explanation. There are two
Rules, and although real life is never like theory (I could fill the page with
suitable examples, but shall refrain), they describe the operation of all opamp
circuits very accurately:
An opamp will attempt to make both inputs exactly the same voltage (via
the feedback path)
If it cannot do so, the output will assume the polarity of the most
positive input
Needless to say, this requires some explanation. So let's look at Rule
#1 ("No Poo....", oops, sorry, wrong Monty Python sketch).
An opamp will attempt to make both inputs exactly the same voltage
When an opamp is operated in its linear mode (which is most of the time for audio), the feedback circuit will cause a voltage to appear on the inverting input (-in) that is almost exactly equal to that present on the non-inverting input (+in). Any change of voltage on the +in terminal is reflected by a change in the output that causes more or less current to flow in the feedback circuit to maintain equilibrium. If this is unclear to you, see the further explanations below - but remember the 1st Rule ! If Rule #1 cannot be satisfied, the output will assume the polarity of the most positive input
There are many circuits that use opamps in non-linear mode, and this can also happen if the output cannot swing its voltage fast enough. In these cases, should the +in terminal be the most positive, then the output will (or attempt to) swing positive. If the -in terminal is more positive, the output will swing negative. There is almost no opamp circuit that you cannot understand once these Rules are firmly established in your thinking. Even circuits that use external transistors in strange ways will obey the Rules. An opamp that does not perform as above is being used outside of its normal operating parameters, and the results will be unpredictable and almost always unsatisfactory. It is more often explained that an opamp reacts only to the difference between the two inputs, and not to their common voltage. The ability to ignore the common voltage is called the Common Mode Rejection Ratio (CMRR), and will be covered later in this article.
When an opamp is operated in its linear mode (which is most of the time for audio), the feedback circuit will cause a voltage to appear on the inverting input (-in) that is almost exactly equal to that present on the non-inverting input (+in). Any change of voltage on the +in terminal is reflected by a change in the output that causes more or less current to flow in the feedback circuit to maintain equilibrium. If this is unclear to you, see the further explanations below - but remember the 1st Rule ! If Rule #1 cannot be satisfied, the output will assume the polarity of the most positive input
There are many circuits that use opamps in non-linear mode, and this can also happen if the output cannot swing its voltage fast enough. In these cases, should the +in terminal be the most positive, then the output will (or attempt to) swing positive. If the -in terminal is more positive, the output will swing negative. There is almost no opamp circuit that you cannot understand once these Rules are firmly established in your thinking. Even circuits that use external transistors in strange ways will obey the Rules. An opamp that does not perform as above is being used outside of its normal operating parameters, and the results will be unpredictable and almost always unsatisfactory. It is more often explained that an opamp reacts only to the difference between the two inputs, and not to their common voltage. The ability to ignore the common voltage is called the Common Mode Rejection Ratio (CMRR), and will be covered later in this article.
Some Essential Opamp Information
Before we cover the circuits themselves, we need to look at some of the
parameters you will come across, how to apply power and bypass the supply rails
and so on. There are many parameters that you will see in data sheets, and
these are covered in more detail a little later. There is no point doing it
now, as the importance will be lost until you know more about the opamp itself.
Configurations
Opamps come in a variety of configurations, but the most common are:
Single - one opamp in an 8 pin DIL (Dual In Line) package
Dual - Two separate opamps, sharing only the power supply pins - commonly in an 8 pin DIL package
Quad - Four separate opamps, again sharing the power supply - most commonly in a 14 pin DIL package
Dual - Two separate opamps, sharing only the power supply pins - commonly in an 8 pin DIL package
Quad - Four separate opamps, again sharing the power supply - most commonly in a 14 pin DIL package
Amazingly, nearly all opamps use the same pinouts, and these were
established many years ago by the venerable uA741 for single opamps, and the
likes of the LM1458 dual opamps set the stage for the others that followed.
Many of the quads use the same pinouts as well, and this has enabled people to
swap opamps for 'better' ones for a very long time. However - Don't count on
it! There are some variations, and although uncommon, they do exist. I shall
not be concerned with any of the different devices - only the common pinout
versions will be shown.
Figure 1 - Common Opamp Pinouts
Figure 1 shows the standard connections for single, dual and quad
opamps, but be aware that the remaining pins on the common single devices can
occasionally have uses other than those shown. Single opamps have additional
connections available. These are most commonly:
Offset Null - used to adjust the amplifier to ensure that the input
transistors are perfectly balanced, so that with no input signal, there is zero
volts output. This is important for DC amplifier circuits.
Compensation - Some single opamps do not have inbuilt frequency
compensation capacitors, and these are connected externally instead. This
allows the designer more freedom, and the opamp's high frequency performance
can be optimised for the design objective.
You may at times see these connections used in unconventional ways. This
may be to obtain greater bandwidth than might normally be available, or perhaps
just so the designer can show how clever s/he is. Either way, I shall not be
delving into these aspects of the design process 
.
.
Applying Power
No (active) circuit works without power, so this has to be the first
step. Most opamps will operate with a maximum of 36V between the supply
terminals. As this is the absolute maximum, operation at a lower voltage is the
general rule, and the most common is to use +/-12V to +/-15V to power the
circuit. Some opamps are rated for higher voltages, and others for less, so
consult the spec sheet from the manufacturer. A dual supply is not required,
but it does simplify the design, and is recommended for most applications. A
dual supply has the advantage that all inputs and outputs are earth (ground)
referenced. This can eliminate a great many capacitors from a complex design,
and is the most common way to power most opamp circuits. Note that from a
commercial perspective, elimination (or reduction) of capacitors is done for
economic reasons rather than any great desire to simplify the signal path.
Since the pinouts are nearly always the same, Figure 1 will be applicable in
most cases, but as I said earlier "Don't count on it!". When in
doubt, get the specification sheet from the manufacturer. When not in doubt,
get the specification sheet anyway. Power Supply Rejection Ratio
The PSRR (Power Supply Rejection Ratio) of an opamp is a measure of the amount of power supply noise that finds its way into the output signal. Most specification sheets give the test conditions for this measurement, and this should be consulted if an unusual design is contemplated. Mostly it can be ignored. Bypassing
Although most opamps have a very good PSRR, it is always recommended that the supply be bypassed with capacitors - especially with high speed opamps. Bypassing should always use capacitors with good high frequency performance, and ceramics are probably the best in this regard. It is common for designs to use electrolytic capacitors, themselves bypassed by low value (100nF) capacitors. This ensures that all noise sources are minimised, and helps to prevent oscillation. When this occurs with a high speed (HS) opamp, it will commonly be in the MHz region, and is extremely hard to see on most oscilloscopes. A sure sign is inexplicable distortion, that mysteriously disappears (or appears) when you touch the opamp, or a component in its immediate vicinity.
The PSRR (Power Supply Rejection Ratio) of an opamp is a measure of the amount of power supply noise that finds its way into the output signal. Most specification sheets give the test conditions for this measurement, and this should be consulted if an unusual design is contemplated. Mostly it can be ignored. Bypassing
Although most opamps have a very good PSRR, it is always recommended that the supply be bypassed with capacitors - especially with high speed opamps. Bypassing should always use capacitors with good high frequency performance, and ceramics are probably the best in this regard. It is common for designs to use electrolytic capacitors, themselves bypassed by low value (100nF) capacitors. This ensures that all noise sources are minimised, and helps to prevent oscillation. When this occurs with a high speed (HS) opamp, it will commonly be in the MHz region, and is extremely hard to see on most oscilloscopes. A sure sign is inexplicable distortion, that mysteriously disappears (or appears) when you touch the opamp, or a component in its immediate vicinity.
Figure 2 - Bypassing The Opamp Supplies
Even with HS opamps, electrolytic capacitors are not needed for each
device (generally needed only on each board), but the use of ceramic bypass
caps between the supply pins of each device is highly recommended. Figure 2
shows the most common method of bypassing power supplies for opamp circuits
(A), but there are others. In some cases, the supplies may not be bypassed to
earth (ground), but just to each other. This has the advantage of not coupling
supply noise into the earth (ground) system (B). Claims have been made that
supply bypassing ruins the sound (rubbish), or that ceramic caps should never
be used in audio, even for bypassing (more rubbish), and even that high value
capacitors (> 1uF) slow down the sound (unmitigated drivel). These claims
are made by frauds and charlatans, and should be completely ignored - they have
no basis in fact whatsoever, and indeed, quite the reverse may be true in each
case.
The Ideal Opamp
An ideal opamp has an infinitely high input impedance, and therefore
needs no bias current. It is also capable of infinite gain without feedback, so
there are no errors between the two inputs (i.e. Rules 1 & 2 will hold for
all cases). The ideal opamp also has zero ohms output impedance, and is capable
of supplying as much current as you will ever need. The ideal opamp does not
exist :-( Although it does not exist, the ideal opamp is the common model for
nearly all opamp circuits, and few errors are encountered in practice as a
result of designing for the ideal, and actually using a real (non-ideal)
device. The tolerance of even the best resistors will ultimately limit the
accuracy of any opamp circuit at low frequencies (where gain is highest). The
primary practical limitations are:
Input Impedance - Typically from one to several hundred Megohms. FET
inputs are used for very high impedance inputs
Gain - 100dB at up to a few hundred Hertz is common
Common Mode Input Voltage - typically limited to the supply voltages,
but may be up to 0.6V greater with some designs
Bandwidth - opamps with a usable high frequency limit of 1MHz at unity
gain are now common
Output Current - most common opamps are limited to about 20mA of output
current.
There are others, such as input offset voltage and current, but we shall
not concern ourselves with these parameters just yet. Power opamps may be
capable of up to 10A, but these are outside the scope of this section of the
article. The use of ideal opamps is assumed for much of the following, but all
are designed to function properly with real world devices. In practice the
difference between an ideal opamp and the real thing are so small as to be
ignored, but with one major exception - bandwidth. This is the one area where
most opamps show their limitations, but once properly understood, it is quite
easy to maintain a more than adequate frequency response from even basic
opamps.
The common mode input voltage can be important in some applications.
Ideally, an opamp only reacts to the voltage difference between its inputs.
Provided this does not change, in theory, the actual voltage between the two
inputs and the common (zero volt line) may be anywhere within the specified
range with no change in the output voltage. In other words, the inputs
can assume any voltage between the negative and positive supplies, and there
will be (almost) no change at the output.
With a real (as opposed to ideal) opamp, there will be some change, and
this is specified as the common mode rejection ratio. An opamp with a CMRR of
100dB (not uncommon) will ensure that the change in output voltage is 100dB
less than the change of input voltage (as applied to both inputs
simultaneously. Any difference between the inputs is amplified
normally. CMRR is affected by the open loop gain of the opamp, so is usually
worse at high frequencies.
The Basic Opamp Circuits
The following collection shows the most common configurations for
amplifiers. These are intended as linear amplifiers, in that they are
essentially distortion free (within the capabilities of the opamp itself, of
course). As we progress, most of these original circuits will be seen over and
over again, since they are the very foundations of building an audio circuit
using opamps. In all cases, a dual power supply is assumed, and this is not
shown on the circuits. This partly for clarity, since the additional circuitry
makes the diagrams harder to understand, and partly because it is a convention
not to show all the supply connections anyway. We all know they have to be
there, so there is little point in showing the obvious over and over again.
Likewise, bypass capacitors and other support components are not shown - only
the basic opamp and its associated components. You will also see reference to
the "instantaneous value of the AC waveform". This is like a snapshot,
and we simply freeze time while we analyse the operation of the circuit. At any
point in an AC waveform, it can have only one value of voltage and current, regardless
of the complexity of the signal source. A sinewave is no different from any other
signal - provided its amplitude and frequency are within the capabilities of
the opamp.
The Non-Inverting Amplifier
The most common of all configurations is the non-inverting amplifier. I
will therefore use this as a starting point, because it is also the simplest to
understand. Figure 3 shows a completely conventional non-inverting opamp
voltage amplifier.
Figure 3 - Non-Inverting Opamp Amplifier
Rin is the input resistor, and is needed because an opamp needs a reference
voltage at the input. In this case the reference voltage is the zero volt
(earth) bus. Input impedance is equal to the value of Rin in parallel with the
opamp input impedance. Generally the latter can be ignored because it is so
high. The gain (Av - Amplification, Voltage) is set by the ratio of R1 and R2,
and is equal to:
Av = (R1 + R2) / R2
The gain of this stage cannot be less than unity, regardless of the
resistor values used. As shown in the diagram, the gain is 11 times, so a 100mV
input will become a 1.1V output. To re-examine Rule #1, it is obvious that if
100mV (instantaneous AC or DC) appears at +in, the amplifier must have 1.1
volts at the output, since the voltage divider R1/R2 will ensure that 100mV
also appears at -in. This is obtained from the simple voltage divider formula,
which is strangely familiar ...
Vd = (R1 + R2) / R2
This will hold for any gain and any output within the capabilities of
the power supply and the opamp's ability. A signal at 10MHz will not follow the
rule, since the opamp will almost certainly be incapable of amplifying such a
high frequency. An input voltage of 10V with a gain of 11 will also break the
rule, since the opamp has only +/-15V supplies, and the output cannot exceed
the supply voltage. Likewise, an 8 ohm load will break the rule, since the
opamp cannot supply the current needed to drive such a load.
To see how the opamp behaves in these abnormal conditions, I suggest
that the circuit be built, and run the tests if you have access to an
oscilloscope. Examine the inputs as well as the output, since the inputs are by
far the most interesting when the opamp is appearing to break the Rules.
Inverting Amplifier
Once, all amplifiers were inverting. A single valve or transistor stage
(other than a cathode or emitter follower buffer stage) always inverts the
signal, and this is how it must be (see Amplifier Basics - How Amps Work for more info). With the advent of the
opamp, all this changed, and the inverting amp is a very different beast from
the simple discrete designs. The gain ratio is again set by a pair of
resistors, but the +in terminal is earthed, either directly, or via a resistor.
This configuration is also called a virtual earth stage, and is common in
mixing consoles and many other signal processing circuits.
Figure 4 - Inverting Amplifier
Since +in is earthed and Rule #1 says that both inputs must be the same,
-in will "appear" to be at earth potential as well (i.e. zero volts).
Assume an input of 100mV DC. The output will be at -1V DC, a gain of -10 (the
minus indicates only that it is inverting, not that the circuit has
"negative gain" which is actually a loss). Input impedance is equal
to the value of R2, and voltage gain is R1/R2, or 10 as shown. Note that this
configuration is capable of negative gain (loss). If R2 is larger than R1 (say
20k), then the gain is equal to R1/R2 as before, so is now -0.5. With the input
at 100mV (again instantaneous AC or DC), the input current will be 100mV / 1000
(using Ohm's law), which is 100uA. The current through the feedback resistor
must be exactly equal and opposite to ensure that zero volts is at the -in
terminal (so we don't break Rule #1). As it happens, 1V / 10k equals 100uA, so
the requirement is satisfied, since the output is negative. As before with the
non-inverting amp, the limitations of the opamp and its supply may cause Rule
#1 to be broken, but the amp is now no longer operating in its linear mode, and
Rule #2 will take over. Observation of the -in terminal will show a distorted
waveform when the opamp can no longer operate in linear mode.
Inverting and Non-Inverting Buffers
A very common opamp application is the buffer stage, which (for the
non-inverting configuration) can have an extraordinarily high input impedance,
and a low output impedance. As with all opamp circuits, the output impedance
may be very low (typically < 10 ohms), but the output current capability
will not allow the circuit to drive such an impedance at more than the 20mA or
so that is typical of most opamps. This would limit the output voltage (before
clipping) to a maximum of +/-160mV, or about 113mV RMS into 8 ohms. Distortion
will be unacceptably high, and the end result is not worthy of further
consideration.
Figure 5 - A) Inverting and B) Non-Inverting Buffer
In many cases the non-inverting buffer can be replaced by an emitter
follower, but performance is nowhere near as good. Input impedance is lower,
output impedance is higher, and the gain is not quite unity. In addition there
is more distortion and lower output drive capability. The inverting buffer is
more of a convenience than anything else, and is simply a normal inverting
amplifier with unity gain. Input impedance is the same as R1, and very high
values are not possible without excessive circuit noise.
Some Interesting Variations On Basic Circuits
It is now time to look at a few of the many variations on the basic
circuits discussed above. It is not possible to cover all the different
circuits that have been made using opamps, since there are so many that I could
easily end up with the world's longest web page. I doubt that this would be
appreciated by most of you :-) I shall only cover the more common, or most
interesting, as this will give a better appreciation of how versatile these
building blocks really are. All of the circuits that follow will work - they
are not theoretical, but real designs, and can all be made on the opamp test
board.
High Impedance Amplifiers
The inverting buffer has been used in some very interesting ways. For
example, a standard low cost TL071 opamp has an input bias current of about
65pA, and a claimed input resistance of 1012 ohms. To put this into
perspective - assuming we have a way to supply the bias current without
affecting input resistance - the input impedance could be as high as
10,000,000,000,000 ohms. That is 10Tohms (1 Tera-Ohm is 1000 Gig-Ohms ). We will
be completely unable to achieve this in practice, since the insulation
resistance of a PCB is nowhere this figure, and the smallest amount of
contamination will reduce the impedance dramatically. In reality, we can easily
expect to be able to get an input impedance of 100M ohms or more, but care is
needed, since with high value resistors additional noise is produced. Since
noise in a resistor is proportional to the voltage across the resistor and its
resistance, it is easy to see how a simple circuit can become a real noise
generator. Figure 6 shows the circuit and PCB layout for a very high impedance
amplifier.
Figure 6 - High Impedance Amplifier
The bias resistor is 'bootstrapped' from the output, and this allows a
lower resistance (for lower noise), while maintaining an extraordinarily high
input impedance. A circuit such as this could be used for a capacitor
microphone (for example), which will typically have such a small capacitance
that any loading will reduce the low frequency performance to an unacceptable
degree. The guard track can be seen encircling the input and the input end of
R1. What on earth is a guard track? Read on .... To prevent the resistance of
the PCB from causing a problem, the input section may be 'guarded' with a
section of track connected back to the output. Bootstrapping and guarding work
in the same way. The guard track works by maintaining a voltage from a low
impedance source around the input circuit that is the same voltage as the
input. If they are the same voltage, no leakage current will flow. In reality
it is not quite that simple. Assume that the opamp has 100dB of gain at 1kHz
(our test frequency). This equates to 100,000 - a little shy of infinity! Since
the opamp has a finite gain, the 'unity gain' buffer will actually have a gain
of 0.99999 - not 1 as we had assumed. This error reduces the ability of the
opamp to bootstrap the circuit perfectly, so the 100k input resistance will
only be effectively increased to 10G ohms. But wait .... how does it increase
the effective resistance at all? This is very simple. Assume an instantaneous
AC voltage of 1 volt input to the amp. Normally, this would cause a current of
10uA into the 100k resistor of Figure 6. Because the bootstrapping action causes
the voltage at the junction of R1 and R2 (Fig 6B) to be 0.99999V, there is
actually only 1 - 0.99999 = 10uV across the resistor. The input current is now
10uV / 100k = 100pA (1 pico-amp is 10E-12A). We can now calculate
the equivalent resistance as R = 1V / 100pA = 10G ohms. This will fall at
increasing frequencies as the opamp starts to run out of gain. Oh yes, the term
"bootstrap" comes from the unlikely picture of a man "lifting
himself off the floor by his own bootstraps". As you might have guessed, the
term is somewhat antiquated, but there has never been any move to change it
(thank goodness). It is intended to show that the impossible can be done, but
it is not really impossible, and is just a very clever example of lateral
thinking.
 |
The bootstrapped circuit cannot be used at DC, since it requires a
capacitor for its operation. This is not as much of a limitation as may first
be thought, since DC is quite inaudible
|
Many common transducers use capacitance as their mechanism.
"Condenser" microphones are the most common example, but there are
many others. These are normally supplied from a high voltage (50-200V) via an
extremely high value resistor. One would expect noise, but they are usually
much quieter than expected. This is actually easily explained ... The
capacitance may only be small, but the resistor is such a high value (commonly
10M or more) that the transducer itself acts as a filter capacitor. For
example, even a 100pF capacitor makes an excellent low pass filter when fed
with an impedance of 100MΩ, having an upper -3dB frequency of 16Hz. Any noise
is effectively filtered out by the capacitance of the transducer. Remember too
that there should be no voltage across the resistor, as that implies that
something is drawing current (unacceptable for a capacitive transducer).
Simulated Inductor
This circuit has to be one of the all-time classics. Although it can
also be done with a single transistor, the performance of the opamp version is
so much better that the alternative is not really worth considering. Inductors
have always been a problem in electronics, as they are by nature relatively
large, and being made from a coil of wire, tend to pick up mains hum as well as
other noise in the electromagnetic spectrum. The idea of simulating an inductor
using an opamp has been about for a long time. The inventor was apparently a
Dutchman named Bernard Tellegen, see Wikipedia for more info.
Figure 7 - Simulated Inductor
The circuit is so much smaller than a real inductor (at least for the
larger values), and does not suffer from noise pickup. It does have a limited Q
(quality factor), but it is rare that very high Q circuits are needed in audio,
so this is not really a problem. It is also variable over a moderately wide
range, something that is very difficult with the wire wound genuine article.
So, how does it work? The idea is very simple, but operation is less easy to
understand. Essentially the circuit uses a capacitor, and 'reverses' its
operation, thus making an 'inductor'. The essential character of an inductor is
that it resists any change in its current, so if a DC voltage is applied to an
inductance, the current will rise slowly, and the voltage will fall as the
external resistance becomes more significant. An inductor also passes low
frequencies more readily than high frequencies - the opposite of a capacitor. An
ideal (that word again) inductor has zero resistance, so will pass DC with no
limitation, but will have an infinitely high impedance at infinite frequency.
These limits are generally considered to be outside the audio range. To
understand how the circuit works, remember that the output of the opamp is
(almost) exactly the same as the non-inverting input. Imagine a DC voltage of
1V is suddenly applied to the input, via resistor R1. The opamp will ignore the
sudden load because the change is coupled directly to the input via C1. The
opamp will represent a high impedance. Just as an inductor would do. With the
passage of time, C1 charges via R2, the voltage across R2 falls, so the opamp
sees less and less of the input signal, and starts to draw current from the
input via R1. This continues as the capacitor approaches full charge, and the
opamp has close to zero input, so the output is also close to zero volts.
Eventually resistor R1 becomes the only limiting factor to current flow, and
this appears as a series resistance within the inductor in the same way as the
resistance of the wire in a real inductor behaves. This series resistance
limits the available Q of both the simulated and real inductor, with the main
difference being the magnitude - real inductors generally have much less
resistance than the simulated variety. Inductance is measured in Henrys, and
for the simulated inductor is equal to ....
L = R1 * R2 * C1
A more accurate version of the formula (due to Siegfried Linkwitz) is
shown below, but normally the error from the simple version will be very low
with typical values - a ratio of 100:1 would normally be the lowest one would
use, and this will have an error of only 1%. Component tolerance will have more
effect, but for completeness, here is the accurate version ...
L = C1 * R1 * (R2 - R1)
so for the circuit shown is 1 Henry. This is a large inductance, and
would be very expensive and bulky if made conventionally. The real inductance
will have lower resistance and higher Q, but will need to have a large iron
core to be able to withstand even a small amount of DC, and the inductance will
change depending on how much DC is present. The simulated inductor is limited
by the current capability of the opamp, so can handle up to +/-20mA with no
change in performance.
There are some limitations to the simulated inductor ...
First (and most annoying) is that one end of the inductor is earthed.
Although simulated inductors have been made that are floating (can be connected
in any way you like), these are expensive and uncommon. Fortunately the
standard version is quite suitable for many audio applications, so this is not
too great a burden.
The simulated inductor cannot be made with high Q, since the value of R1
cannot be made low enough to allow a Q of more than about 10. This is due to
the limitations of the opamp - a minimum value of 100 ohms is usually specified
for R1, although lower values are sometimes used. This represents a series
resistance (equivalent to wire resistance in a real inductor.
There is also a resistance effectively in parallel with the simulated
inductor, equal to the value of R1 + R2. Although this can be measured, it is
not generally a hindrance to practical circuit design.
Although the simulated inductor acts in many ways like the real thing,
it does not have the same energy storage, and cannot respond like a proper
wound inductor. When the input voltage is suddenly removed from a real
inductor, the collapse of the magnetic field causes a large voltage pulse of
the opposite polarity - this does not happen properly with a simulated
inductor, since there is no magnetic field involved. The simulated inductor
will still try, but is limited to the voltage swing of the opamp, so the
flyback pulse is limited to this value.
Figure 8 shows two simple LC filters. One is using a real inductor, and
the lower circuit has a simulated inductor. They are both series resonant
circuits, and are tuned to the same frequency (159Hz). The reference level
(near the top of the graph) is 0dB, and neither circuit exhibits any
appreciable loss outside the stop band.
Figure 8 - LC Filters, Real And Simulated
The performance of both is almost identical as a simple filter as seen
from the response plot, except the thing you don't see (mainly because I didn't
include it) is that the simulated inductor has a slightly shallower notch, at
about 37dB instead of 40dB. The frequency is calculated from ...
F = 1 / (2 * Ï€ * √(LC)) Hz
A series resonant circuit has minimum impedance at resonance, and in the
configuration shown will act as a notch filter, reducing the level at the
resonant frequency. Because of the relatively low Q, the notch is not very sharp,
but the simulated inductor is an important building block for equalisers and
spectrum displays, and is quite common in audio. Note that at the junction of
Cin and the inductor, the voltage is higher than the input voltage. This is
normal behaviour for a series resonant circuit.
All-Pass Filter
The all-pass filter is one of the strange ones. It passes all
frequencies perfectly, with no attenuation at all within the capabilities of
the opamp used. All it does is change the phase of the signal, and this circuit
is used in everything from phase correction circuits for sub-woofers to guitar
effect pedals. It is a versatile and useful building block, and the circuit is
shown in Figure 9.
Figure 9 - All-Pass Filter
The circuit shown will have a 90 degree phase shift at 159Hz. At DC,
phase shift is 180°, and at high frequencies it is 360° (note that 360° phase
shift is almost the same as 0° - there is a subtle difference for
transient signals, so the two can be considered identical only for steady state
signal conditions). The shift of phase about the centre frequency is completely
inaudible, but if a pot is substituted for R2, the phase can be swept back and
forth. This is audible, and by cascading a number of these circuits
"phaser" or vibrato (frequency modulation) effects pedals can be
made. One of the latter is described in my projects pages. The input signal is
effectively applied to both opamp inputs, but there is always a small phase
difference except at DC or infinite frequency. The value of C1 and R2 determine
the frequency at which there is a 90° (or 270°) shift, and the frequency is
determined with the formula ...
Fo = 1 / (2 * Ï€ * R2 * C1) ... where Fo is the 90° phase shift
frequency
A quick analysis will show how this works. Assume a DC input of 1V; at
DC the cap has no effect, so the circuit operates just like an inverting
buffer. The output is therefore -1V, so there is a 180° phase shift. At high
frequencies, the reactance of C1 is negligible, and the full input is supplied
to the opamp's +in terminal. Remembering Rule #1, the opamp output will be such
that both inputs will have the same voltage, and in order to do this, the
output must be equal to the input at high frequencies. At intermediate frequencies,
the combination of C1 and R2, along with R1 and R3 will ensure that the output
amplitude remains constant, but the phase will change. The relative positions
of C1 and R2 may be reversed, which will modify the characteristics of the
circuit.
Phase Shift Oscillator
There are many things in life I do not understand, but one of the
simpler ones is the phase shift oscillator implemented using an opamp. Don't
get me wrong - the circuit I understand perfectly. The bit I don't
understand is how come (up until recently) I have never seen this circuit
published - anywhere ??? In its heyday, this circuit was used almost anywhere a
simple sine wave oscillator was needed, and I have seen it made with valves,
transistors and even FETs. What I had not seen until I designed one was
a phase shift oscillator using an opamp. As it transpires, although I
had not seen this done, my trusty editor in the UK has. He has seen it in a
number of publications including John Linsley-Hood's "The Art of Linear
Electronics". JLH also supplies the equation for frequency calculation ...
Fo = √ 6 / (2 * Ï€ * C * R)
Loop gain must be 29.25 dB according to Mr Linsley-Hood, and lacking
further information I must assume that the formula only applies if all
resistors and capacitors are equal, and gain would be the minimum required for
the circuit to oscillate.
Figure 10 - Phase Shift Oscillator
The frequency stability of this circuit is quite good, but as with all
phase shift oscillators the amplitude varies when the frequency is changed. Any
resistor can be varied to change the frequency, and the use of a pot allows
continuous variation over a 5:1 range (or more if you experiment with the
component values). This is a perfect example of an opamp being unable to obey
Rule #1, and its operation is governed by Rule #2. The circuit is essentially
unstable, and the opamp is always trying to play catch-up (without success, or
the circuit would stop oscillating). The frequency is a cow to determine if
different values are used for R or C, and although I believe there is a
formula, it is apparently a very tedious process (I've not seen it myself). The
circuit shown above will run at about 420Hz, with a sinewave output of around
500mV (with +/-15V supplies) - although JLH's formula seems to indicate that it
should oscillate at 390Hz. If you really want to know, you will have to
build one. Changing the value of any resistor or capacitor will change both
frequency and amplitude. The square wave at the output is at almost the full
+/-15V supply voltage (limited only by the output circuit of the opamp). The
sinewave shown on the oscilloscope trace is obtained from the Vout terminal,
and the square wave is obtained from the opamp's output (these are not to
scale). The string of resistors and caps acts as a phase shift network, and
oscillation takes place at that frequency where there is an exact 180 degree
shift, converting negative feedback into positive feedback. The circuit is
stable at DC, since it has negative feedback through the string of resistors.
Let's have a look at how it works. Remember Rule #2? Now have a look at the
signal at the inverting input. As you can see, the output takes the polarity of
the most positive input, so when the -in terminal is positive, the output is
negative. Over a period of time based on the resistance and capacitance, the
voltage on the -in terminal will fall towards zero volts, and will eventually
become negative - the output promptly swings positive, and the cycle repeats.
Like all filter circuits, the resistor / capacitor (R/C) network introduces a
time delay, and it is this (plus the simple low-pass filter formed) that produces
a sinewave with less than 1% distortion. By no means wonderful, but quite
adequate for a number of simple applications. The sinewave output is at
relatively high impedance, and should be buffered with an opamp before use. Any
loading will alter both amplitude and frequency.
Schmitt Trigger Oscillator
Also known as a free-running multivibrator, the Schmitt trigger
oscillator is one that is much more conventional in terms of opamp designs.
Like the phase shift oscillator (indeed, like all oscillators) it is an
inherently unstable circuit. Also like the preceding example, this circuit
cannot obey Rule #1 (since that would make it stable), so follows Rule #2
instead.
Figure 11 - Schmitt Trigger Oscillator
This circuit is very common where an oscillator is needed, but as shown
produces a triangular waveform that is quite high in harmonic content. Note the
use of positive feedback, via R2 and R3. This particular connection creates a
Schmitt trigger, a useful but fairly inscrutable circuit for the beginner.
Although this is a simple circuit, understanding how it works is not. Assume a
supply voltage of +/-15V, and ignore the losses in the opamp output stage. We
shall start at the point where the opamp's output is at +15V. The +in terminal
will be at 7.5V, since there is a voltage divider from the output to earth. C1
will therefore charge to a positive voltage, until such time as the voltage is
very slightly greater than 7.5V. Since Rule #2 must be obeyed, the -ve input is
now the more positive, so the output will swing negative. The cap now must
discharge its positive voltage and start charging to a negative voltage. Once
the negative voltage is less than (more negative than) -7.5V, the output will
swing positive and the cycle repeats. At least it is possible to determine the
frequency of this oscillator, and it is approximately equal to ....
X = R3 / (R2 + R3)
Fo = 1 / (2 * R1 * C1 * loge ((1 + X) / (1 - X))
Fo = 1 / (2 * R1 * C1 * loge ((1 + X) / (1 - X))
For the example above (after noting that it is the natural log not
base 10 :-) frequency is ...
X = 100 / 200 = 0.5
Fo = 1 / (2 * 100k * 10nF * loge (1.5 / 0.5))
Fo = 1 / (0.002 * loge (3))
Fo = 455 Hz
Fo = 1 / (2 * 100k * 10nF * loge (1.5 / 0.5))
Fo = 1 / (0.002 * loge (3))
Fo = 455 Hz
The triangle wave output is at relatively high impedance, and should be
buffered with an opamp before use. Any loading will alter frequency, but not
amplitude (this is fixed by the voltage divider of R2 and R3).
References
I have used various references while compiling this article, with most
coming from my own accumulated knowledge. Some of this accumulated knowledge is
directly due to the following publications: National Semiconductor Linear
Applications (I and II), published by National Semiconductor
National Semiconductor Audio Handbook, published by National Semiconductor
IC Op-Amp Cookbook - Walter G Jung (1974), published by Howard W Sams & Co., Inc. ISBN 0-672-20969-1
Data sheets from National Semiconductor, Texas Instruments, Burr-Brown, Analog Devices, Philips and many others.
National Semiconductor Audio Handbook, published by National Semiconductor
IC Op-Amp Cookbook - Walter G Jung (1974), published by Howard W Sams & Co., Inc. ISBN 0-672-20969-1
Data sheets from National Semiconductor, Texas Instruments, Burr-Brown, Analog Devices, Philips and many others.
OPERATION AMPLIFIERS DESIGN
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