Hi Welcome to Fundamentals Friday Today we're going to take a look at how I got the increased bandwidth on my microcurrent design because a few people have asked about it. And it's really simple and it uses an old trick with opamps where you can Cascade opamps together in series to increase your bandwidth. So hence the title cascaded Op An Bandwidth Let's go now. let's start out with the simplest design, which is what I used in my previous microcurrent design.

Just what's called a single stage opamp. It's one stage because it only has one opamp in this case, a basic uh, non-inverting configuration. You're totally familiar with this or you should be Jelly Bean Uh. Circuit times 100 gain I've chosen the resistors here to give us, you know, amplify the signal by 100 times now.

Uh, Opamps have what's called a Gain Bandwidth product and what that is is the bandwidth of the Op amp at a gain of one. So if if you just short, uh, the output back to the input, you just have an OP out buffer like that. It's the minus 3db bandwidth. So you'll find this uh GB WP figure Gain Bandwidth product on your data sheets.

or it's also known as Unity Gain Bandwidth because you've got a gain of one Unity gain so it's the same thing they're equivalent. Now what we'll do is we'll assume that we've got an OP An with a 1 mahz Gain Bandwidth product here. And as the name says, the bandwidth is uh, the gain times the bandwidth p is product in multiplication. So we're multiplying the gain times the bandwidth So if you want to get the bandwidth, all you got to do is uh, divide that figure by the gain that you're using in your circuit Here, and all things being equal, it's pretty much a constant linear Factor So we can use this to calculate the bandwidth of our Opamp at higher gains so that Banner spec that you'll see on the opamps.

The Gain Bandwidth product might be a megahertz. You might think great I can use it at a mehz. Well, only at Unity gain if you start adding time 2 Gain Time 5 Time 100 Whatever, your bandwidth is going to drop accordingly in a linear relationship like that. So let's take the example: 1 MHz Game Bandwidth product A times 100 Amplifier Exactly the same Uh value as what I had in my original microcurrent Design 1 mahz Gain Bandwidth product divided by our gain of 100 Here, gain is also known as AV then our bandwidth has dropped to a lousy 10 khz.

That sucks. But let's say that we want more bandwidth out of our Op amp and for some reason we have to stick with this opamp. and in my microcurrent it was. There was a real legitimate reason why I had to because it was a very low offset Cho Ramp Etc There's not too many on the market that can do the job, so I pretty much had to stick with this Op amp.

So this is one of the reasons why I had to Cascade them. Ordinarily though, you should always try and do all of your gain in the one one opamp because it's going to be better in a couple of ways. But anyway, we're going to have a look at the cascaded configuration how you can use multiple opamps. So now we've got a twostage circuit here because we've got two opamps both identical opamps, both with a 1 MHz gain bandwidth product and uh, but now we still want our same times 100 gain out of here.

But of course we don't do it in one stage. We do times 10 and times 10. so we changed our resistor values there. we still got our times 100 gain.

but what does that do to our bandwidth? Now, actually, deriving this might be a little bit tricky, but in the end, there's a fairly simple formula, which, uh, can calculate the total system bandwidth here for as many stages as you want. On one big assumption, the assumption is that all of the opamps are identical bandwidth, product, and identical gain, but in our case, hey, that's perfect. So the formula here is the total bandwidth. The total system bandwidth is equal to the bandwidth of one stage here, which we'll take a look at in the formulas down here multiplied by the square < TK of 2 to the^ of 1 on n n being the number of stages.

In this case, it's two stages so it would be 2 the^ of a half and then minus one. so square root of all that Bingo We can pop this into the formula and determine our bandwidth. Let's have it go. So looking at our two- stage design here, we're just going to use this formula up here and plug the numbers in.

Now the bandwidth of one individual stage is now using this formula up here because we've now got a gain of times 10 instead of time 100 Here it's not a 10 Kilz Like this, it's 100 khz because it's the 1 MHz divided by our gain of 10 which gives us 100 khz here. and we multiply that by the square Ro T of 2^ of 1 on n n being two stages. So 2 ^ of a half, minus one I'll SP you the details of doing that on the calculator. The answer is a bandwidth out of here at times 100 gain of 64.3 khz.

Bingo! By adding this second stage and a second op amp and Cascade in them, we've now gone from a 10 khz bandwidth with a single stage for a times 100 gain for exactly the same gain. Our bandwidth is now look 6.4 times better. Oh awesome! So we got ourselves a real Bobby desler of an improvement going from one stage to two stage in terms of bandwidth. We increased it by an improvement of 6.4 times.

Fantastic. But do we get that same Improvement when we go to three stages? Well, let's take a look at it. now. instead of times 10 gain, we need our final times 100 gain.

Remember, our system gain has to stay exactly the same. So uh, a cube root of uh of 100 is 4.64 six. And yes, it does have to be that precise. especially if you're doing a Precision circuit like I was on the microcurrent.

You know, 05% You know you really have to get those gains pretty precise, which I'll talk about in a second. So for a three-stage one, we punch the numbers in the formula again. but our bandwidth is not 100 khz of each stage is not 100 khz. anymore.

it's improved. It's more than doubled. It's now 215. 44 cuz it's a 1 MHz divided 4.64 blah blah 200 115 khz.

So we've improved our bandwidth on each stage now by lowering the gain. obviously based on this formula, but does that have an overall net? Improvement Well, yes, it does punch the numbers In our total system bandwidth is now 109.5 khz or thereabouts, which is a decent Improvement on 64. but it's not the huge 6.4 times Improvement we got here. it's only an extra 1.7 times and if you go to to four stages and etc etc, you get just get diminishing returns on your bandwidth.

So I've calculated a fourth stage which has a gain of 3.16 blah blah and our bandwidth is only appro you know increasing to 137. So you know with an improvement of 1.25 times. So really, you know a practical limit. You're going to stop at three or four stages even though you could keep going on.

So with my microcurrent I stuck with a two-stage design here. Why? Well, you know? yeah, I could have increased the bandwidth by adding a third stage, but hey, there's extra cost. The chip isn't cheap, but the main reason why is because of this weird ass gain value here trying to select offthe shelf you know, E96? uh, value resistors or something like that to try and get an exact gain of exactly what I wanted. It's just ugly.

No, didn't want to go there. but with a times 10 gain that I had in the two- stage circuit, there times 10 is easy with E12 values I Just had 1K here and 2K K, 2 and 6K 8 in series and they gave me 9k there and 1K your standard non-inverting formula that was a times 10 gain precisely. So that's why I went with this two-stage one and didn't try and squeeze the extra bandwidth out of it. But hey, it was good enough when I uh changed from the max 4238 to the max 4239 which had a gain bandwidth product of not 1 MHz but 6.5 MHz So you can plug those numbers into the formula and see how it all goes.

but it was pretty decent bandwidth. much more a couple hundred khz. much more than the 100 khz I was aiming for. Now as I mentioned before, this magic formula over here is only valid for Uh stages that have exactly the same gain and exactly the same G gain Bandwidth Product: if you change the Op Amp used in here or you change the gain which you might want to do for example, on the front end because you might want to minimize your noise so you have as much gain on the front end here as possible, which we'll talk about in a second.

But yeah, it's only valid for that. If you change any of those parameters, the bets are off, eh. I'll leave that up to you to calculate using odd ball gains and odd ball. Uh, different opamps with different Uh game Bandwidth products Now I know what some of you might be thinking: Dave You've done a video on opamp noise before and yes, I have I'll Linked In down below if you want to check it out.

So what happens if we Cascade these Op amps? Aren't we just going to get increase our noise problem? Well, yes we are, but it's not nearly as bad as you might think. In fact, it's borderline trivial. So um, but uh, the noise will be dominated by this first Op amp as we'll see here. So really, you want the maximum gain on your front end here If noise is an issue for you, Now, let's take the case of one microvolt noise here.

Uh, RMS noise. Now I Won't go into the different types of noises, the current, noise, and input referred noise, and all sorts of stuff. H Let's just keep it simple: One microvolt of noise. We've gone back to time 10 amplifiers here.

So uh, our one microvolt noise obviously gets Amplified by the uh gain of there. and we're getting 10 microvolts noise out of here. No problem, right? But does that 10 microvolt noise get multiplied by 10 and a 10? again? Well, yes it does. So your noise would be exactly the same if you had a Time 1,000 amplifier here.

One, you're still going to get 1 molt noise out of th000 microv volts noise out of here with a single stage and it's going to be the same. almost the same, just a smidge more. Um, out of here like this. So let's take a look at it.

1 microvolt noise in 10 microvolts noise out of here. And then we've got to add in our extra 1 microvolts noise of this Op amp here. But of course that's not 11 microvolt. So it's not 11 * 10 which is 110 microvolts out of here.

No, you remember noise. Uh, when you add noise sources, it's the root sum of the squares like this. So noise number one here gets squared and then then added on to noise number two. and if you do the math there, well, it's only it's not 11.

it's only 10.05 so a little bit smaller than what you might think. so hopefully you can see that the noise just once again Vanishing returns here each time you Cascade So in this case, that's 100.5 + 1. Whack that into our uh root sum of the squares here and our total output noise is 1 105.0 5 microvolts we' just dropped because this only adds 1% again 1% each time. It's bugger all half a bees dick.

It doesn't really matter. if you had a single stage here, you'd still get a th000 microvolts of noise. What's the difference between a th000 and uh, 1,000 or five there? Well, .5% Eh, it's nothing. Forget about it.

So there you have it, that's cascaded Opamp bandwidth and as usual, we'll go over to the bench and we'll build this up thing up and see if we can verify that we get these multiplication factors here. when we increase our number number of stages and we're not going to measure noise, that's too tricky and really, uh, quite pointless. and we should be able to measure it at least to within the ballark of these multiplication uh factors that we get here based on our Um System bandwidth formula up here. But yeah, the game bandwidth product does vary with Uh Supply voltage, which we may be able to show and there's a big margin on it anyway.

but hey, should be able to get close and prove that this thing actually works all right. So what we've got here is is just a Uh Jelly Bean dual opamp I've got a TS 912 from Um St It doesn't really matter, just sort of pick one at random. We got a two stage uh time 10 uh amp here for a total gain oftimes 100 but I will change the configuration around. This will be the final Uh configuration.

We'll start out with a Unity gain amplifier just so just shorting input and output on a single stage, measuring that. And if we have a look at the data sheet here, the good thing about this one is that it has different parametric tape tables for different Supply voltages as I'll show you. So this is uh, what we're going to operate at as plusus 5 Vol which gives us a total uh rail of 10 volts. and if we go down here here we go.

Gain Bandwidth product: Uh, measured uh at a gain of 100 with under those uh at under those conditions there. but we're getting. you know it's telling us a typical figure is 1.4 MHz we may or may not get that, doesn't give us a minimum figure. You know it could be higher, it's likely to be that or higher it's likely to be under.

So let's hook up that as a Unity gain amp and see what we get. All right. What I've got here is two waveforms: yellow and green. You can see that they're different ones here and there.

Uh, the green one there Channel 2 is our input voltage which I'm going to set to 10 molts I have set to 10 molts RMS I'll have to uh, tweak that during the experiment to keep it at 10 molts due to the Uh nonlinearity of the well, the flatness of the built-in Uh function Gen of my agilant scope here. Anyway, that's just a little uh trap. If you're doing these sort of tests, don't assume that your function generator has a constant output amplitude over frequency. It may or may not.

So anyway, I'm using the build-in one and our output voltage there we go is exactly the same. Hey, it's Unity gain and we're measuring that at 100 khz. Not a problem. So what we want to do now is increase our frequency until our output voltage drops to- 3D b707 of that 10 ms there and you'll notice that it's actually going up.

And that's due to the Uh characteristic of this circuit which has a nonflat frequency response. You know how the frequency responses should ideally be flat and then uh, falls off? Of course at 6 DB per octave or whatever? then um, this one's not. It's got a little uh, you know it's It's got a bit of peaking at the Uh output there, but anyway, that's not going to concern us. We just increase our frequency until we get that.

707 I'll go into frequency fine here and you'll see that the input amplitudes Dro so I've got to, just you know, tweak that up a bit just to keep that at that 10 molt figure. And then we're looking at Uh getting the frequency at 707 of 10 MTS you know thereabouts near enough. Hey, look at that. We're getting 2.3 MHz much higher than the 1.4 MHz on the data sheet, but hey, that's to be expected.

Typically the Op 's going to Uh perform better than or At or better than its typical figure. and now I've got the time 10 amplifier configured there and as you can see, 10 molts in and we're getting you know a time 10 of that out. Not a problem. We're only down at Uh 7 odd KZ here.

So let's try and find the minus 3db point of that. And remember before we had a Unity gain bandwidth of measured not from the data sheet, but measured from this chip of 2.3 MHz So we expect this to drop to Uh 70.7 MTS at 1/10th of that uh uh frequency or 230 odd khz and aha, as you can see, look, we're down at 70.7 MTS Here Of course we get a phase shift at the higher frequency. now we're only at 145 khz, so it's not nearly as good as we thought it should be because this thing is not in this configuration is not going to have a constant gain bandwidth product, so some opamps may. this one in particular doesn't So you got to actually check this stuff out.

So uh, the data sheet value of a times 100 gain. You remember that it said it was specifi the gain bandwidth product at times 100. So let's increase this to 100 times gain and see if we get the typical data sheet figure. Anyway, we do want to note this because we're going to be using this is our measured bandwidth.

So when we Cascade them, then um, then we should, uh, get you know the formulas that we got on the Whiteboard So we need to note down that figure of 145 khz. that's our time. 10 bandwidth. So here we go: Configured As Time 100 our 10 molts in and we're getting 1 volt out and you know it's at a low frequency.

So let's find Theus 3db Uh, point of that and there it is. Uh, 707 molts out, 10 molts intimes 100 and we're getting a frequency of a bandwidth there of about 30 .4 khz and Bingo! That pretty much matches the data sheet of at the gain of 100 of our typical Uh bandwidth of unity gain bandwidth of 1.4 MHz. So we have to divide that by 100 of course. So we're expecting 14 khz there and we're getting very close to it.

Beautiful now. I Have a two-stage amplifier hooked up here. Uh, times 10 gain each. exactly like the original Uh data sheet value.

Now we will get in a 13.4 khz bandwidth before at times 100 game with a single stage opamp. Now, what bandwidth do we get with the T with the two-stage op in? So what we expect here: If we've got a measured Uh bandwidth at time 10 uh, gain which we've got two of those, then each one is 145 khz. Whack that into our Uh formula for our cascaded amps with a number at two, we should measure about 93.3 KH Let's see if we get close to that. Might be spot on, but should be near enough.

And there we go. that's not too far off. Uh, 77 MTS two stages there, 10 molts in 108 KZ there. So that's a little bit better than our 93.3 khz.

But eh, you know we didn't actually measure the gain of the other side opamp. but because it's on the same uh, die, you'd expect a similar Uh performance there. But hey, I'm going to call that close enough. now.

Another interesting thing to note is you want some margin in your game bandwidth product? Why? Because uh, your Opamp is typically going to distort. Now watch this. Okay, look at some Distortion in there. You can see some Distortion in that waveform starting to happen there.

If I Bring that over, look at that. We're starting to get little bits of distortion and that's you know, The input's still nice and clean. but yeah, look at that. so you just got to be careful there that you're are not trying to push the limits of your amplifier because they are going to distort just like that.

Even worse, this one's pretty mild actually. o ooh I Forgot to show you how the game Bandwidth product changes with Uh voltage or this one for this particular Op amp does. So this is at plusus 5 volts at the moment and in theory if it was, you know it didn't change it all with Supply voltage then we could wind it down and the output should stay the same. but if we lower the voltage to plus - 2.5 Vol or 5 Vols total, we'll notice the output voltage drop.

Look at that while the input stays the same. So the game bandwidth product is not the same against Supply voltage there. So there you go. I Hope you uh, enjoyed that quick little uh fundamental Friday on cascaded bandwidth there.

Sorry, we weren't able to measure the exact Uh numbers we got on the Whiteboard but uh, these things aren't exact in practice really. so you know, choose use another Op amp in here and we probably would. Got uh, different results again, but it's going to be near enough. And in theory on the Whiteboard that is actually how it works.

Um, but in practice different uh opamps? uh change their game Bandwidth: uh product across individual units across Supply voltage across uh, test frequency and all sorts of stuff. So and uh, input voltage so and output voltage. So really, you know it was unlikely that we were going to get spot-on values and we didn't but it showed how you can get increased bandwidth there. uh, cascading.

In this case we went from Uh 14 13.4 khz measured Uh with when on a times 100 gain using the single Op amp to up to 100 khz uh bandwidth on uh the when we just cascaded another one in our dual upamp here. Easy. So there you go. It's a nice little uh trick you can use if you have to.

but as I said right at the start, uh, really? you? if you need, you really should be picking the right Op amp for the job so you can do it in one Op amp. But if you have to like I did in the microcurrent for a Precision application, hey, this works a treat. Anyway, if you want to discuss it, jump on over to the Evev blog forum and if you like it, please give it a big thumbs up. Oh my thumb is too big.

Catch you next time.