Trying to understand how operational amplifiers work

[Femme_physics]

So I'm looking at this non-inverting Op-Amp

http://img140.imageshack.us/img140/6660/shomermofa.jpg [Broken]

Trying to understand how Op-Amps work.

It seems rather strange they're saying no current flows from + to - because then how does the current from Vin spreads through?

After all, there is a infinite-resistance resistor there that blocks passage, from my understanding. As such:

http://img51.imageshack.us/img51/1152/opamps.jpg [Broken]

 

[yungman]

The model looks reasonable except the other side of $Gv_{in}\;$ don't come out. It is only generalizing that no current flow in or out the input. It is not exactly like $R_{in}\;$ in your drawing. It is an input stage of either a BJT or FETs. The input current range from 1uA down to 0.01pA depending of the technology.

The definition of op-amp is it has an inverted and a non inverted input. The amp has very high gain of whatever input is presented ACROSS the +ve and -ve input ( called differential input). Therefore in a closed loop feedback where the output is connected back to the -ve input with a resistor R2 in your case. any voltage at the +ve input will cause a differential voltage across the two inputs. The amp has infinite gain, so the output react immediately and swing to the direction of the +ve input. It will swing until the voltage at the -ve input equal back to the +ve input, then the output stop rising.

Let's use your drawing, the voltage at the input is the voltage divider voltage of the output by R1 and R2. so if R2 is twice the value of R1, then the voltage at the -ve input is 1/3 of the output voltage according to the divider, so to every 1v input, you get 3v output in the same direction. So your amp is a non inverting configuration with a gain of 3.

Hope this help.

Alan

 

[I like Serena]

Hey! Did you trace the links of my pictures back to their origin (which is wikipedia)?

It seems rather strange they're saying no current flows from + to - because then how does the current from Vin spreads through?

After all, there is a infinite-resistance resistor there that blocks passage, from my understanding.

Yep! An op-amp is a complex piece where the current is drawn from the additional power supplies.
In your picture the current is drawn from Vs+ and Vs-.
This is done in such a smart way that V+ and V- do not draw a noticeable current.

 

[waht]

An ideal op-amp is defined to have:

1) infinite input resistance: Rin -> infinity
2) zero output resistance: Ro -> 0
3) infinite gain: G -> infinity

$$V_{in} = V_{+} - V_{-}$$ And $$V_{out} = G\cdot V_{in}$$
This is the starting place. Every op-amp circuit can solved by applying these conditions.

Of course, in reality nothing is ideal. A real op-amp integrated circuit is a non-ideal approximation of an ideal op-amp. As an example, it may have parameters like Rin = 100K, Rout = 4 or G = 20,000.

But solving circuits such as the non-inverting amp above, it kind of assumes that you have to use the ideal op-amp model, unless it's stated otherwise and you are given op-amp parameters.

So what's really going with the non-inverting op-amp circuit, and why has the ideal op-amp been defined the way it is?

The answer to that comes from control theory which is a huge subject that deals with feedback systems. In such a system, some of the output is fed back to the input of some element with defined input and output.

In this case, the defined element is the op-amp. You can see, the voltage divider formed by R1 and R2 samples the output voltage and feeds a fraction of it to the input terminal V_.

From a voltage divider set up,
$$V_{-} = V_{out} \cdot \frac{R_1}{R_1+R_2}$$ And $$V_{+} = V_{in}$$
As to not confuse those variables used in the schematics above that use V_in in two different places for different things.

Let's call
$$V_{d} = V_{+} - V_{-}$$
$$V_{out} = G\cdot V_{d}$$

Having established this, you can follow this calculations to obtain the gain of the amplifier which is defined as

$$A = \frac{v_{out}}{v_{in}}$$ and it can be seen that as you take the limit of the op-amp gain G to infinity things nicely cancel and simplify where A converges to a finite value.

This is the long way of solving the problem, but it gives insight into what really happens. There are also shortcuts you can do that are derived from the op-amp definitions and control theory. From condition 1, infinite input resistance will block input currents so that I_ = 0 and I+ = 0. And from control theory under negative feedback conditions, the output will try to cancel the input, so that v_d = 0.

With
$$V_d = 0$$
It follows that $$V_{+} = V_{-}$$
And $$V_{in}=V_{+}=V_{-}$$
By applying KCL, Kirchhoff's Current Law at V_:
$$\frac{V_{in}}{R_1} = \frac{V_{out} - V_{in}}{R_2} + I_{-}$$
With I_ = 0, if you carry this out, the gain comes out to:
$$A = 1+\frac{R_2}{R_1}$$
The shortcut doesn't always work with op-amp circuits, so you have to be careful. But just in case, long way will always work.

 

[vk6kro]

Here is a diagram of the insides of an opamp. This one is a fairly old type and there are much better ones around now.

The complex arrangement of transistors is typical of opamps, although the input devices (Q1 and Q2 in this case) could be FETs in some other types of opamps.

http://circuitzoo.com/wp-content/uploads/2010/09/LM741CircuitDiagram.jpg

Fortunately, you don't have to know the exact internal circuit of each opamp as the external circuits used compensate for differences in the internal design.

Notice that this opamp has an internal capacitor of 30 pF. Capacitors inside integrated circuits are expensive to produce so you won't see this very often.

 

[technician]

3 essential things to know to sort out op amps
1) V at the - input = V at the + input ( this is a good assumption because the gain of an op amp is so high)
2) No current flows into or out of the +input or -input (this is a good assumption because the input resistance is so high)
3) The output voltage cannot be greater than the +supply voltage or less than the -supply voltage
In your circuit Vin connected to the +input must also be the voltage at the -input
This means that there is a voltage = Vin across R1
This means that a current is flowing through R1 and this current must come from Vout of the amplifier.
This current flows from the output through (R1+R2) in series.
So Vout = I x(R1 + R2) and Vin = I x R1
The voltage gain is Vout/Vin = (R1+R2)/R1
You see how this is subtly different to the gain of an inverting amplifier (-R2/R1)

 

[Femme_physics]

So the inner workings of an op-amp is beyond the scope of understanding to a beginning electronics student who just started with op-amps? I appreciate the replies, everyone. It does seem too complex.

 

[technician]

Don't be discouraged!!
You do not need to know all the details of the inner workings of an op amp to be able to use them and design circuits using them.
The applications you have met recently can be sorted out in a fairly straight forward way

 

[jim hardy]

find this old document from Texas Instruments

"Op Amps for Everyone"

it has a document number starting with SLOD
yu should be able to find it from google.
it is a good text. I think Bassalisk found it mqybe he still has link

opamps sound crazy at first

the secret is to realize this:::
because opamps have such preposterous parameters (gain of millions, input resistance of giga-ohms) it is incumbent on designer to surround them with a circuit that keeps them operating within limits of power supply.

so you as designer are in control.
Opamp is just a triangle with Vout = 10^6 X (Vin+ - Vin-)

around 1967 i had a difficult time transitioning from tubes, where gain of 40 might be good, to Fairchild ua709 opamp with its gain of a hundred thousand.....

 

[Femme_physics]

Don't be discouraged!!
You do not need to know all the details of the inner workings of an op amp to be able to use them and design circuits using them.
The applications you have met recently can be sorted out in a fairly straight forward way

I know I am a little late, but I shall take your advice into considerations now

Hey! Did you trace the links of my pictures back to their origin (which is wikipedia)?

Yes sir!

Yep! An op-amp is a complex piece where the current is drawn from the additional power supplies.
In your picture the current is drawn from Vs+ and Vs-.
This is done in such a smart way that V+ and V- do not draw a noticeable current

.

What smart way? What additional power supplies?

so you as designer are in control.
Opamp is just a triangle with Vout = 10^6 X (Vin+ - Vin-)

I don't get where does this 10^6 comes from. It appears like massive energy made out of the blue, and we all know that's impossible.

Here is a diagram of the insides of an opamp. This one is a fairly old type and there are much better ones around now.

The complex arrangement of transistors is typical of opamps, although the input devices (Q1 and Q2 in this case) could be FETs in some other types of opamps

Fair enough, it's complex. But nowhere in all this do I see how it pumps up energy out of the blue.

 

[technician]

FP
We get hung up on all the details of what an op amp does and how it does it.
Don't forget the energy/power ultimately comes from the power supply. It does not come from the op amp....this controls how the energy is delivered.
As you have seen we are so concerned about the details that most of the time the power supplies are not even shown on the circuit diagram. We take it for granted.
Think about a car, the engine and controls are complicated but the energy comes from the fuel....hope that analogy does not sound too trivial.

 

[Femme_physics]

Fair enough, it doesn't really matter where the energy comes from. It's the amplification process that forever befuddles me. How can you amplify a given source of energy? Going with the fuel example, it's like putting fuel in a machine that eventually through a complicated process creates more fuel than was originally put , just by having gone through devices, without actually adding extra fuel.

You see what I mean?

 

[technician]

I would say that the energy of the fuel is converted to a different form of energy but not more energy.
In a similar way an op amp enables the voltage of a battery and the charge contained in the battery to be manipulated to produce what you want.
Small electrical signals can be used to control large signals. That is what amplification means

 

[pantaz]

... Going with the fuel example, it's like putting fuel in a machine that eventually through a complicated process creates more fuel than was originally put , just by having gone through devices, without actually adding extra fuel.

But you are adding extra fuel. It's just coming via a different path.

Amplifiers still obey Ohm's Law. A basic NPN transistor amp is a good way to envision it -- a small signal into the base picks up additional power from the collector, thus appearing amplified at the emitter. It's a similar mechanism with any amplifier.

 

[I like Serena]

Yes sir!
.

What smart way? What additional power supplies?

Vs+ and Vs-.

Your Vin signal itself is not amplified.
It only opens a kind of gate for the large current to flow, drawn from Vs+ and Vs-.

It is similar to the fuel valve in your block diagram in your other thread.
It only takes a little energy to open the fuel valve, after which a lot of fuel can start flowing.

 

[Femme_physics]

I would say that the energy of the fuel is converted to a different form of energy but not more energy.
In a similar way an op amp enables the voltage of a battery and the charge contained in the battery to be manipulated to produce what you want.
Small electrical signals can be used to control large signals. That is what amplification means

So it doesn't really amplify the current? And can you give an example of "Small electrical signals can be used to control large signals", please?

Your Vin signal itself is not amplified.
It only opens a kind of gate for the large current to flow, drawn from Vs+ and Vs-.

Ahhhhhhhhhhhhhh........so Vs+ and Vs- are the REAL big current, whereas Vin is just the gate current, like "base current" of transistors, yes?

It is similar to the fuel valve in your block diagram in your other thread.
It only takes a little energy to open the fuel valve, after which a lot of fuel can start flowing.

Ah. But now, why do you even need this analogous electrical valve? I thought amplifiers are meant to amplify current, not to CONTROL it. That's what resistors and switches are for.

 

[I like Serena]

Ahhhhhhhhhhhhhh........so Vs+ and Vs- are the REAL big current, whereas Vin is just the gate current, like "base current" of transistors, yes?

Yep!

Ah. But now, why do you even need this analogous electrical valve? I thought amplifiers are meant to amplify current, not to CONTROL it. That's what resistors and switches are for.

Huh?
An amplifier does control current (like you said, you can't create energy out of nothing).
Just like a switch (but an advanced switch that is controlled electrically).

 

[sophiecentaur]

Ah. But now, why do you even need this analogous electrical valve? I thought amplifiers are meant to amplify current, not to CONTROL it. That's what resistors and switches are for.

Not many electronic devices work without an external Power Supply!
An amplifier 'amplifies' the low power of an input signal by controlling power from the power supply to produce a vastly increased version of the signal power. The analogy with a valve or a tap is a good one. The crudest way of controlling power is an ON/OFF switch. It is, in effect, a very simple non-linear power amplifier. The power from your finger is enough to control the multi kW power feeding a massive heating system. Also, the low power involved in moving the slider on a rheostat (big variable resistor) can control, in a linear fashion, the high power delivered to a motor; this is also a form of amplification. An electronic amplifier is achieving the same sort of thing only in a more sophisticated way.
With an OP Amp, the Power Gain is so huge that you can equate the input power to zero for some purposes. I know that sounds daft but, in comparison with the output power, it is negligible during calculations. You always 'tame' this enormous gain using Negative Feedback. At the negative input to the amplifiers in your diagrams, the current flowing 'in' through the series input resistor is almost exactly the same as the current flowing 'out' through the feedback resistor. A miniscule current flows into the Op Amp input and it can be treated as virtually zero. (Electronics is Engineering - in which assumptions like that are constantly being made, in one way or another)

The overall performance of the Amplifier is defined by that feedback and depends very little on the Op Amp characteristics (as long as the device gain is high enough at the highest operating frequency).

 

[technician]

FP
If you take an audio system as an example of amplification the input could be from a microphone or a record player pickup. These are usually very low power and only provide small voltages (mV) and small currents (μA or mA).
We want to reproduce this small signal through a loudspeaker and require POWER of maybe 10Watts. This could be achieved by a voltage of 10V and a current of 1A (or some other combination!) through the loudspeaker.
This would be called power amplification and needs amplification of voltage and current.
Amplifiers are usually designed in 2 sections: A Voltage amplifier followed by a current amplifier to give the required power output. The current amplification tends to be fairly simple ...a single transistor could be enough.
The voltage amplification is a little more involved because signal processing is usually incorporate in the voltage amplification.
Tone controls, volume control etc.
If you start with maybe a small signal of 10mV from a microphone and you want an output of 10V to drive the loudspeaker you can see that a voltage gain of 1000 is needed.
This is the area we are in with op amps.
I hope this gives you some background that lies behind the implementation of amplification!!!
The tiny input power controls the large output power which comes from the power supply.

 

[Femme_physics]

An amplifier does control current (like you said, you can't create energy out of nothing).
Just like a switch (but an advanced switch that is controlled electrically).

So why is it called an amplifier and not a controller?

In fact, I'd go so far as to say an amplifier doesn't really "AMPLIFY" the original current but rather REDUCE to control it. It should be called a regulating-reducer if anything!

Not many electronic devices work without an external Power Supply!
An amplifier 'amplifies' the low power of an input signal by controlling power from the power supply to produce a vastly increased version of the signal power.

Is that because it stores current and then relaunches it, or something like that? Like a capacitor with a regulator?

The analogy with a valve or a tap is a good one.

Again, flow "REGULATION", not flow "amplification".

We want to reproduce this small signal through a loudspeaker and require POWER of maybe 10Watts. This could be achieved by a voltage of 10V and a current of 1A (or some other combination!) through the loudspeaker.

So a very small current flows FROM to mike to the loudspeaker, and suddenly turns into a big current due to the op-amp? But how? But we go back to the same point-- if op-amps are like valve, how can it increase the current?

 

[technician]

I would say the small signal (voltage) from the mike is connected to the input of an op amp.
The op amp consists of many transistors that amplify the voltage/current. If I was interested in the inner workings of the op amp I would want to know more about their internal structure.
I am happy to accept that someone has designed these things and they can be treated as a 'black box' Given the main characteristics it is possible to use them in a very simple way to achieve voltage and current amplification. I think that 'amplification' is a good term because what you want is a copy of the input from the mike but larger in size!!
We must not forget that none of this would happen without a power supply so ultimately the op amp controls what comes from the batteries.....phew

 

[Jiggy-Ninja]

So a very small current flows FROM to mike to the loudspeaker, and suddenly turns into a big current due to the op-amp? But how? But we go back to the same point-- if op-amps are like valve, how can it increase the current?

Power amplifiers usually aren't op amps, but that's not an important distinction at this point. As for where the power comes from, I'll repeat what sophiecentaur said because it's the answer to your question:

Not many electronic devices work without an external Power Supply!

How does a valve increase water flow? It gets that from an external pump.

How does the loudspeaker increase current? It gets it from the power supply.

If you're thinking in terms of energy conversion, like a transformer, that's wrong. Op amps don't work that way. Op amps control the signal level, and ideally, there would be no current flowing into the inputs, so no energy going in. All it is doing is controlling the voltage level of the output based on the voltage level of the input.

I'll repeat this from sophiecentaur too, since it's easily the most brilliant thing in his post:

Also, the low power involved in moving the slider on a rheostat (big variable resistor) can control, in a linear fashion, the high power delivered to a motor; this is also a form of amplification. An electronic amplifier is achieving the same sort of thing only in a more sophisticated way.

The energy used to move the rheostat slider is not the energy used to power the motor; it's far far too little. Moving the slider produces a signal that is used to control the motor's power.

It's the same with an op amp. The energy put into the input is not used to drive the output; it too is far far too little. The input signal is used to control the level of the output, which produces an amplified version of the original signal.

 

[technician]

FP here is all you need to know to tackle practically every op amp circuit you have had to deal with !!!

main characteristics of op amp

1) No current flows into or out of the +/- inputs
2) V at the + input = V at the - input
3) Vout cannot be greater than +Vs or less than -Vs

Because of condition 1) any current flowing towards the - input must flow through Rf to the output. This is the link between input and output known as feedback. I think that sophiecentaur said something along these lines in his last post.

Print those 3 conditions out and pin them on your wall !!
Do you only cover theory of op amps in your course? Have you ever built an amplifier using op amps?

 

[sophiecentaur]

So why is it called an amplifier and not a controller?

A fair enough question for which the answer is only based on word usage. But there is some logic in the choice of words:
Firstly - it's not called a 'controller' because that term is already reserved for something else.
Secondly, it 'Amplifies' a signal and not the Volts or the Current, as such. There is nothing of the electrical Energy in the original signal appearing at the output. It's an entirely new lot of energy - supplied from the power supply - just varying by more than the input energy varies.
You could complain about many words that appear in Science and Engineering. For instance, where did the name 'transistor' come from? It came from the notion of Transfer Resistance - varying a resistance according to some input signal. That has been lost in the mists of time and we just use the word without thinking about it. The thermionic 'Valve' was an early (the earliest ?) amplifying device and it is easy to picture the grid volts 'controlling' the current through the device just like a gas valve. An FET works very similarly to a thermionic valve but it's still not called a valve.

And those rules 1,2,3 of technician look absolutely loopy until you start to think about the practicalities of Op Amps. They then make excellent sense. Learn and don't forget! Write in lipstick all over your bedroom mirror!!

 

[Femme_physics]

We must not forget that none of this would happen without a power supply so ultimately the op amp controls what comes from the batteries.....phew

Aaaaaaaaaaaaha!

Gotcha!

That's all I needed to hear

Secondly, it 'Amplifies' a signal and not the Volts or the Current, as such.

AHA!

There is nothing of the electrical Energy in the original signal appearing at the output.

Of course! I see it now!

It's an entirely new lot of energy - supplied from the power supply - just varying by more than the input energy varies.

Aha! Aha!

You could complain about many words that appear in Science and Engineering. For instance, where did the name 'transistor' come from? It came from the notion of Transfer Resistance - varying a resistance according to some input signal.

Heh, that reminds of a joke.

What's a transistite? A transistor that doesn't know whether it's NPN or PNP




Thanks to you and to Jiggy-Ninja and technician for helping me fully understand it now!

The thermionic 'Valve' was an early (the earliest ?) amplifying device and it is easy to picture the grid volts 'controlling' the current through the device just like a gas valve. An FET works very similarly to a thermionic valve but it's still not called a valve.

Is it really possible to analogize op-amp and (hydraulic) valves in that fashion? Electronics is one of a kind in that sense. You can't amplify a "water flow" signal AFAIK.

FP here is all you need to know to tackle practically every op amp circuit you have had to deal with !!!

main characteristics of op amp

1) No current flows into or out of the +/- inputs
2) V at the + input = V at the - input
3) Vout cannot be greater than +Vs or less than -Vs

Because of condition 1) any current flowing towards the - input must flow through Rf to the output. This is the link between input and output known as feedback. I think that sophiecentaur said something along these lines in his last post.

Print those 3 conditions out and pin them on your wall !!
Do you only cover theory of op amps in your course? Have you ever built an amplifier using op amps?

So far, theory, but thanks a bunch, I'll remember and will add it to my formulas page......but only one problem ->

2) V at the + input = V at the - input

What about this case->


http://img217.imageshack.us/img217/6305/thisagain.jpg [Broken]

V- = 0
V+ = 7.8V

Explain that

 

[Jiggy-Ninja]

V- = 0
V+ = 7.8V

Explain that

Technician's 2nd characteristic is only applicable to op amp circuits using negative feedback to connect the output to the inverting input. For comparator applications or other circuits not using feedback, it is not applicable.

If the + and - are significantly different from each other, the output will be saturated (according to technician's 3rd characteristic).

 

[gnurf]

What about this case->


http://img217.imageshack.us/img217/6305/thisagain.jpg [Broken]

V- = 0
V+ = 7.8V

Explain that

Are you confusing the supply voltage (V_s = 7.8V) with the V+ and V- inputs? The circuit compares one of 6.3V or 4.8V to 5.53V, depending on the position of the switch. From the wikipedia comparator article:

When the non-inverting input (V+) is at a higher voltage than the inverting input (V-), the high gain of the op-amp causes the output to saturate at the highest positive voltage it can output. When the non-inverting input (V+) drops below the inverting input (V-), the output saturates at the most negative voltage it can output. The op-amp's output voltage is limited by the supply voltage.

In your case that means the output is either 7.8V or 0V, depending on the position of the switch.

 

[sophiecentaur]

That circuit is not a good example of the use of an OpAmp. There is no feedback so the gain is millions so it will basically 'slice' any waveform that is applied to the input as it crosses from just below to just above the reference input. Even that is a poor way of doing the job because if there is any low level noise or interference on one of the inputs, if the signal crosses slowly over the 'decision level' there will be a burst of terrible peak-to peak output noise. It is normal, in a 'comparator' type of circuit like that, to use a small amount of positive feedback (Schmitt trigger / hysteresis) to flip and hold the output one way or another until the input goes back past the triggering point. So I would say that it's not a good circuit to base any useful learning on.

As with all information on the Internet. Published circuits can be of variable quality.

Also, as gnurf has said, the 7.8V you see is a power supply voltage and not the voltage on the + or - inputs. You need to get used to reading circuit diagrams accurately - it's a language of its own and a very pernickity one too.

 

[technician]

I agree with gnurf and sophiecentaur. This is a switching circuit and as jiggy ninja points out the rule about V+ being equal to V- does not apply !!!!
gnurfs numbers are spot on.
If you want to get anything from this circuit look at the power supply.... it is not the usual symmetrical +Vs and -Vs. It is possible to use just one battery (+Vs) but in an amplifier of the sort we have been meeting a way is usually found to have a 'half way' voltage which would do the same job as our usual 0 Volts !!! (a potential divider across the power supply with the junction of the divider connected to the + input so that V+ = 0.5Vs)
I will be a little disappointed if you cannot see into this :rofl:

 

[I like Serena]

Heh, that reminds of a joke.

What's a transistite? A transistor that doesn't know whether it's NPN or PNP

 

[Femme_physics]

Thanks for all the help getting me to understand

 

[jim hardy]

and indeed, (one of your earlier posts posed this question) a hydraulic system CAN amplify a signal. a device called a 'pilot operated valve' will do the trick.

consider for a moment the power steering in your automobile. it amplifies the small force you apply to steering wheel into enough to turn the car's front wheels, even if car is not moving and they must be forced. test that yourself next time you go out- turn the wheel slightly while parked and listen to engine slow down as power steering pump demands power.
your car's power steering box is a hydraulic powered mechanical amplifier changing one finger-power at steering wheel into several muscleman-power at the front wheels, power supply is that belt driven pump.
That's an amplifier you can see feel and hear!

 

[Jiggy-Ninja]

and indeed, (one of your earlier posts posed this question) a hydraulic system CAN amplify a signal. a device called a 'pilot operated valve' will do the trick.

consider for a moment the power steering in your automobile. it amplifies the small force you apply to steering wheel into enough to turn the car's front wheels, even if car is not moving and they must be forced. test that yourself next time you go out- turn the wheel slightly while parked and listen to engine slow down as power steering pump demands power.
your car's power steering box is a hydraulic powered mechanical amplifier changing one finger-power at steering wheel into several muscleman-power at the front wheels, power supply is that belt driven pump.
That's an amplifier you can see feel and hear!

There is electric power steering as well, not just hydraulic. (It's what I test at work)

 

[pairofstrings]

Hi everyone. Sorry to jump in here and ask this following question..
We know that there are two types of Integrated Circuits
1. Digital ICs
2. Linear ICs
Can anyone please tell me if Operational Amplifier(Op-Amp) is a Digital IC or Linear IC? Can you tell me reasons?

 

[vk6kro]

The way that opamps are usually used is in a linear mode.

This means that they have many levels of input signal that will produce many levels of output signal.

A digital circuit generally has two levels of input signal (possibly 5 volts and zero volts) and the output is one of two possible levels too.

There are exceptions to this and some opamp circuits can be digital in output. For example, the Schmitt trigger circuit or the opamp used as a comparator produce just two levels of output signal.