## What are Slew Rate, Open Loop and Bandwidth?

### Slew Rate:

imagine you are a good 100 m meter runner. Somebody applies you an blindfold and put you 10 meters in front of a concrete wall. You are in start position.

Ok, I care for you, the somebody starts instead of you. You stand beside the somebody and you are doing the starting shot. "Somebody" runs with full power, all bystanders: "go, go go". What happens? Full braking in front of the concrete wall. The somebody will get bad headache.

An amplifier behaviors like the runner. A runner starting with full power, blind and don't know where to go. A fast runner needs 10 seconds for the 100 m. A fast operational amplifier slews to about 4000 V per microsecond. But also an amplifier reach his concrete wall, the rails of the power supply. That's the maximum limit. Figure 1: an indicator for the speed of an amplifier - slew rate.

Slew Rate can be measured by checking the most steepest rise of an square wave response. It's common used in electronics to check the rise time between 10% and 90%. The runner need for the distance of 1m to 9 m about 1.5 seconds. So the runner achieve a slew rate of 8 m per 1.5 seconds. We normalize it to 1 second, resulting in 5.3m/s.

The same calculation can be done for an amplifier. We define 10% and 90% to 1.3V and 11.7V. Amplifier need for example 4 nanoseconds. Slew rate can be calculated as 10.4V/4ns. It's common used to normalize a slew rate to one microsecond. We get a high slew rate of 2600 Volt/microsecond.

An audio amplifier attain a dimension of 0.5 V/µs up to 100 V/µs. Of coures there are some other ciritical issues for the speed of an amplifier. For example the bandwidth - bandwidth and slew rate are closed combined, a high bandwidth is only possible if a high slew rate exists an vice versa.

### The Open Loop amplifier gain

is a very important characteristic to characterise an amplifier. Amplification mattered a small input signal should be amplified by a defined fix factor. Generally most amplifiers having a high open loop gain. There are some operational amplifiers, with a gain of 10 millions. This connotes: this amplifer needs only a little voltage of 100 nanovolt to reach an output voltage of 1 Volt.

Logarithm and decibel: it's not easy to calculate always with high numbers, therefore for gain calculations we use a transformation to decibels. for 10 millions resulting in 20*log (base of 10) of 10 mio = 140 dB.

An audio amplifier has a high open loop gain, let us consider he reach 100 dB. So an little voltage of 10µV on his input is quite enough to reach an output voltage of 1V. This gain is too much. Consider your CD player delivers 500mV, the amplifier with his gain of 100dB (100.000) = 50000 Volts at the output. No, that's not possible. It's the same situation like the 100m runner, the amplifier listen to the start signal, doing a maximum amplifiication until reaching the rails.

The first meters the runner start running with full power, but close before reaching the wall, somebody needs to tell him: "stop stop stop". Also the amplifier needs a "somebody" to tell him to stop, not to reaching the rails, for an amplifier this "somebody" is called "negative feedback".

It's not the task of the "somebody" to make the runner going slowly by giving bad food to him. No, the "somebody" must coach the runner to stop very fast. Same task for the negative feedback, helps to stop abrupt. The "somebody" and the "negative fedback" control the runner and the amplifier.

Reducing of gain by negative feedback for a wished fix gain:  an amplifier has two inputs, an inverting and a noninverting. It's best to do a mathematical model for a simple control loop.  Figure 2: already established, the input signal amplified with the high open loop gain A, the result is always the output signal. Figure 3: the negative feedback leds to a difference signal. In this case the input signal and the output signal forms a difference signal "diff" and only the small diff signal will be amplified by the open loop gain.
The right figure is very interesting. Remember the ideal amplifier: the output signal should have exactly the same shape as the input signal. An ideal amplifier has an infinite open loop gain. This matters the diff signal must be zero. It's a conflict between theorie and reality. An infinite open loop gain is never possible,  only a high open loop gain can be reached. The question is: "how much". A low open loop reach to a lower output signal than wished.

#### There is always a difference signal

Imagine, a input signal of 1V and a difference signal of 1mV. Thus the output signal calculated to 999mV. It's easy to calculate the open loop gain: 999mV divided by 1mV = 999. A gain of 999 = 59.99dB. Oh, the amplifier does an error! -yes- But I don't want an error, there is a gain error of 0.1 percent.

What helps?

we increase the open loop. the higher the open loop the lower the gain error on the output. The amplifier in figure 3 is a closed loop gain of "1" amplifier. The closed gain is the "wished" gain.

#### How to get for example a closed loop gain of "ten"?

we don't feedback the whole amount of the ouput voltage to the difference element, only a part of it. Figure 4: Feedback Factor Beta, makes it possible to adjust a higher gain than one in figure 3, up to the open loop gain of figure 2. The shown figure 4 is an elementary basic in the automatic control engineering. Also audio amplifier obey these rules. They don't have their own audio laws, like some people want.

On the left in figure 4 there are three equations, derived from the equivalent control diagram. Target, to derive with these three equations is the single equation describing the response (output/input) of the closed loop amplifier. One of the most important equation for an amplifier.

If we want a closed gain of 10, we need adjust Beta to 0.1. Possible by a resistor divider supplying the "inv." input by only 10% of the output voltage.

Consider 1V applied on input and an open loop of 1000. 10V on the output, the diff signal must be about 10mV. Let's calculate with the new equation: 1/(0,1 + (1/1000)) * input = 9,90099 Volt. Oh no - we want 10V, the error reaches already 1% percent error. With a gain of one it was only 0.1%.

Next step increasing gain to 100. 1/(0,01 + (1/1000)) * input = 90,90099 Volt, instead of 100V. Already 10% error.Next step increasing gain to 414: 1/((1/414) + (1/1000)) * input = 292 Volt anstatt 414 Volt. Decreased to 2/3. Bad. We reached the point of the bandwidth.

The Bandwidth of an amplifier, is the point in the amplitude response with an adjusted closed loop gain in the region of the open loop gain. Figure 5: the bandwidth is definded as a loss of  the outut voltage of 1/sqrt 2 to the wished output voltage. These relationship can be calculated by the open loop equation in figure 4. An amplifier with an open loop of 60 dB reach the bandwidth by a wished gain of 414. It's easy to see, for a high gain the open loop should be much higher than the closed loop gain.

The open loop is not constant over temperature and frequency. With an increasing frequency the open loop decrease. The equation in figure 5 is a model for further discussion under an acceptance open loop vs. frequency. The purpose is to demonstrate the close relationship between bandwidth, slew rate, open loop and closed loop.

### Problems with Open Loop?

many amplifier having a high open loop, but only at DC.

Main problem, Open Loop decrease with frequency.

There are many amplifier with a very high open loop gain at DC voltage, but with increasing frequency open loop decrease rapidly. Fast video operational amplifier having a lower open loop at DC but keeping it constant over a wide range of frequency compared to a DC precision operational amplifier.

Main problem, the open loop of an amplifier decrease with an increasing load.

a higher output current leds to higher collector currents of the transistors. The base currents increase too. And lower base currents reduce the open loop.

### Influence of open loop on sound?

#### Results of a Open Loop:

linear distortion, the outputsignal is not equal to the ideal amplified input signal. For higher frequencies the output amplitude decrease depending on bandwidth. The phase shift at higher signal frequencies leds to a incorrect signal.

none linear distortion, imagine you check a frequency of 1kHz and pump up the volume to a high level. It would sound horrible this 1kHz. But the existing 2kHz and 3kHz nonharmonics you can't hear very well, only measure.

### Reasons for distortions?

remember figure 4 and the problem of an decreasing Open Loop with increasing load. Imagine the 1kHz signal crossing the zero line, for this case the output current is low, thus open loop for example 60dB. When the voltage reach the peak, the output current is on the maximum and also the open loop decrease to the minimum in the period of the sine. As you see, the gain is nonlinear vs. input amplitude, this will results in non linear distortion.

#### What to do against?:

• ideally try to design for very constant open loop gains.

• reach very high levels for the open loop. Then it doesn't matter, if the open loop varied for example from 115dB to 110dB, it's still very high.

• best would be reaching both

#### This should be keep in mind:

a closed control loop always depends on figure 4. This equation can be widened. If you try put only most of the important parameters (Frequenz, Temperatue, adjusted gain, load, beta) in calculation, this complex transfer function converts figure 4 guaranteed to a big monster. Don't use too much time for too much calculation, the describing theorie is to poor for an exact calculation very close to reallity, better use your time for better circiut design.

### Some aspekts for amplifier without negative feedback

Amplifier without negative feedback having some disadvantages: their control path must be extremely linear to reach same low distortion results as closed loop amplifiers with high open loop gains. My opinion even best tubes are not linear enough to beat an closed loop amplifier in the categorie distortion. Another problem the low dynamic output resistance, they aren't dynamic, low as by neagative feedback, they can't. Listening to high levels leds to a higher distortion, that's probably.

An amplifier without neagative feedback in it's purest form is exactly figure 2. The designer tries to keep the open loop constant. The open loop is here equal to the wished frequency. Reactionless, no feedback from output signals to the applied inputsignal, no TIM.

But these amplifiers having some other problems. They can't control the output voltage for a varying load impedance. To reduce this effect, these amps needs to be large. Output current small compared to idle current.

some designer swear in amplifers without negative feedback. The say an amplifier with negative feedback will produce TIM, they are right, if they would add this: "a slow one". That's exactly the point of issue, slow amplifer having a low slew rate, a low open loop at high signal frequencies and a low bandwidth. These three parameters are essentiell for a good sound of an amplifier.