Bandwidth of op amps. R 1 R 2 1 k! 250 k!

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1 Bandwidth of op amps An experiment - connect a simple non-inverting op amp and measure the frequency response. From the ideal op amp model, we expect the amp to work at any frequency. Is that what happens? Make a frequency response plot to check it out. R 1 R 2 1 k! 250 k! The frequency response shows a very clear low-pass type of behavior. Yet, there is no capacitor in sight. What is going on? EE 230 gain-bandwidth 1

2 The gain of the op-amp itself must have a low-pass type of frequency dependence. This is definitely a modification of our ideal view of an op-amp. We don t know the mechanism of the high-frequency roll off of the op-amp, but we can guess there must be capacitor internal to the amp. That is exactly what is happening. A certain amount of capacitance is purposely added to the op-amp circuitry by the designers to improve the stability of the amp. (We will examine the issue of amplifier stability soon enough.) The is the same capacitance that causes to the limited slew rate of the op amp, which we will also look at within the next few lectures. EE 230 gain-bandwidth 2

3 The corner frequency of the closed-loop amp changes with the gain in a predictable fashion. This suggests that the low-pass behavior of the op amp itself follows a simple model. It appears that the open-loop gain of the op-amp must have a low-pass type of behavior, suggesting the following modification: A A (s) =A o ω b s ω b = A o 1 s ω b where A o is the gain at very low frequencies and ω b is the corner frequency of the open-loop gain. The low-frequency gain value is generally listed in the op-amp data sheet (eg. 200,000 for the 741), but corner frequency is not listed directly. Instead it is given in terms of another quantity, the gain-bandwidth product. We will see what this means shortly. In order to see the effect of the frequency dependence of the op amp gain, we need to start with an op-amp model that has finite gain. Fortunately, we have already done this a couple of times. Most recently, we looked at the effect of finite gain when discussing op amp limitations (slide 3 of the notes on non-ideal op amps). EE 230 gain-bandwidth 3

4 We can use feedback theory to see what is happening. A G= 1 Aβ A ( s) G (s ) = 1 A (s ) β G ( s) = Ao sωωb b 1 Ao β sωωb b ωb A ( s) = A o s ωb Ao ω b = s ω b (A o β 1) ωcl Ao = Go ωcl = ωb (1 Ao β) Go = where and s ωcl 1 Ao β Here is something new: In addition to modifying the gain, feedback increases the bandwidth! If, as is typical in feedback situations where Aoβ >> 1, then Go 1/β and ωcl EE 230 ωb Ao ωb Ao β = Go ωb Ao = ωcl Go gain-bandwidth 4

5 We can also go at it with straight-forward circuit analysis. v S v d Av d v o G = v o v i = 1 R 2 R 1 1 A 1 R 2 R 1 1 R 2 (Done previously.) v f v o R 1 We can utilize the above expression, but now we replace A with its low-pass frequency-dependent form. A A (s) =A o ω b s ω b = A o 1 s ω b EE 230 gain-bandwidth 5

6 = Wow. Re-arrange a bit. = Note that 1 R 2 /R 1 is the gain that we would have expected for noninverting amp without all of the frequency-dependent messiness. call it: = then: = EE 230 gain-bandwidth 6

7 Next, we note that A o is very big and G o probably not so big for most feedback circuits, so that G o /A o << 1. The quantity A o ω b /G o is also a frequency. Give it a new symbol: = With these two modifications, the closed-loop gain function can be written as: ( ) = From the circuit, we see the same effect the bandwidth, as defined by the corner frequencies, of the closed-loop amplifier is wider than that of the open-loop amp. EE 230 gain-bandwidth 7

8 The quantity A o ω b is key parameter here. For obvious reasons, this is called the gain-bandwidth product of the op amp. This quantity is generally listed in the data sheets. (Usually in terms of Hz rather than rad/s.) For 741, A o = 200,000 and A o f b = 1.5 Mhz. This implies f b 7.5 Hz. For 660, A o = 2,000,000 and A o f b = 1.4 Mhz. This implies f b 0.7 Hz. (For a frequency response plot, see Fig. 10 in the 660 data sheet.) The open-loop corner frequency of the op amps is really low! Again, the amps are designed this way. Having this response improves the usability of op amp. Because there is so much gain to begin with, having it roll off in a low-pass fashion doesn t hurt in many (most) closed-loop applications. It is only at higher operating frequencies and with higher gains that gain-bandwidth limitation becomes an issue. EE 230 gain-bandwidth 8

9 Open-loop gain as function of frequency for 741 op amp (A o f b = 1.5 MHz). = = = f t = A o f b At high frequency (f >> f b ), f b A = 1 We note that magnitude of A will drop to 1 (0 db) when f = A o f b. Therefore, the A o f b is also known as the unity-gain frequency. At frequencies above this value, the amp no longer provides any gain it becomes an attenuator. Sometimes an alternate symbol used: f t = A o f b. The key thing to remember is that the unity-gain frequency (or gainbandwidth product take your pick) is a limiting factor for a amplifier. In your application, you cannot have G o f cl be bigger than f t. EE 230 gain-bandwidth 9

10 The gain-bandwidth of the op-amp makes it easy to determine the expected high-frequency limits of when using the amp. We saw earlier that = or = = Given an amp with gain-bandwidth of 3 MHz, we might want to make a closed-loop amp with gain of 100. The closed-loop gain will begin to roll off at = = = Using the same amp, if we want to provide flat gain out to 200 khz, we must keep the gain below = = = Easy. EE 230 gain-bandwidth 10

11 The gain-bandwidth limit may change how we design an amplifier circuit. Example In an application, you need to provide a gain of at least 25 at a frequency of 250 khz. The op-amps you have available have a unity-gain frequency of 3 MHz. (The gain can be slightly bigger, but it can be no less than 25 at 250 khz.) You immediately see that cannot achieve the goal using a single amplifier stage. G o f cl = (25)(250 khz) = 6.25 MHz. This is significantly bigger than f t. Instead, try using two stages, each with a low-frequency gain of 5 and the other with a nominal gain of 6. With G o1 = 5, f cl1 = (3 MHz)/5 = 600 khz. and with G o2 = 6, f cl2 = 500 khz. For the cascaded pair: G (s) = G o1 1 s ω cl1 G o2 1 s ω cl2 EE 230 gain-bandwidth 11

12 G (jω) = G o1 1 j ω ω cl1 G o2 1 j ω ω cl2 Plugging in the numbers to find the gain at 250 khz (using real frequency instead of radial frequency): G = = 24.8 Very close, but probably not good enough. Particularly if we allow for any variability in component values, etc. So try two amps, each with a gain of 6 (and f cl2 = 500 khz). In that case, the cascaded gain is 28.8, which should be good enough. (Check it for yourself.) EE 230 gain-bandwidth 12