EE 230 gain-bandwidth – 1

Bandwidth of op ampsAn 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.

The frequency response shows a very clear low-pass type of behavior. Yet, there is no capacitor in sight. What is going on?

–+

R1 R2

YRYV

250 k!1 k!

EE 230 gain-bandwidth – 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. At 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 – 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 $ (V) =$R

�+ V�E

where Ao is the gain at very low frequencies and ωb 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 time. 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 – 4

We can use feedback theory to see what is happening.

G =A

1+ Aβ

G (s) =A (s)

1+ A (s) β A (s) = Aoωb

s + ωb

G (s) =Ao

ωbs+ωb

1+ Aoβ ωbs+ωb

=Aoωb

s+ ωb (Aoβ + 1)

= Goωcl

s + ωcl

Go =Ao

1+ Aoβωcl = ωb (1+ Aoβ)

Feedback reduces (“controls”) gain and widens the bandwidth!

EE 230 gain-bandwidth – 5

R2

R1

+

–vo

+

–vf

+

–vo

+

–vd +

– Avd+–

vS

We can utilize the above expression, but now we replace A with its low-pass frequency-dependent form.

A

We can also go at it with straight-forward circuit analysis.

G =vovi

=1+ R2

R11A

�1+ R2

R1

�+ 1

(Done previously.)

A (s) = Aoωb

s + ωb=

Ao1+ s

ωb

EE 230 gain-bandwidth – 6

Wow.

Re-arrange a bit.

* =�+ 5�

5��$R

��+ 5�

5�

� ��+ V

�E

�+ �

* =�+ 5�

5�

�+ �$R

��+ 5�

5�

�+ V

$R�E

��+ 5�

5�

�

Note that 1 + R2/R1 is the gain that we would have expected for non-inverting amp without all of the frequency-dependent messiness.

* =*R

�+ *R$R + V *R

$R�E

*R = �+5�5�

call it:

then:

EE 230 gain-bandwidth – 7

Next, we note that Ao is very big and Go probably not so big for most feedback circuits, so that Go/Ao << 1.

Also, the quantity Aoωb/Go is also a frequency. Give it a new symbol:

With these two modifications, the closed-loop gain function can be written as:

* (V) =*R

�+ V�FO

�FO =$R�E*R

Look at that! The low-pass behavior of the op-amp manifests itself by causing the closed-loop amplifier to have a low-pass function. At low frequencies, the non-inverting circuit behaves just like it did in the ideal case, but at higher frequencies (above ωcl), the closed-loop gain also rolls off with a low-pass response.

EE 230 gain-bandwidth – 8

The quantity Aoωb 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, Ao = 200,000 and Aofb = 1.5 Mhz. This implies fb ≈ 7.5 Hz.

For 660, Ao = 2,000,000 and Aofb = 1.4 Mhz. This implies fb ≈ 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 – 9

Open-loop gain as function of frequency for 741 op amp (Aofb = 1.5 MHz).

At high frequency (f >> fb),

$ =$R

�+ V�E

=$R

�+ M��E

=$R

�+ MIIE

$ � $RIEMI

We note that magnitude of A will drop to 1 (0 dB) when f = Aofb. Therefore, the Aofb 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: ft = Aofb.

fb|A|= 1

ft = Aofb

The key thing to remember is that the unity-gain frequency (or gain-bandwidth product – take your pick) is a limiting factor for a amplifier. In your application, you cannot have Go· fcl be bigger than ft.

EE 230 gain-bandwidth – 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

*R�FO = $R�E or *RIFO = $RIE = IW

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

*R =$RIEIFO

=�0+]���N+] = �� Easy.

IFO =$RIE*R

=�0+]��� = ��N+]

EE 230 gain-bandwidth – 11

The gain-bandwidth limit may change how we design an amplifier circuit.

ExampleIn 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. Go· fcl = (25)(250 kHz) = 6.25 MHz. This is significantly bigger than ft.

Instead, try using two stages, each with a low-frequency gain of 5 and the other with a nominal gain of 6. With Go1 = 5, fcl1 = (3 MHz)/5 = 600 kHz. and with Go2 = 6, fcl2 = 500 kHz.

G (s) =

�Go1

1+ sωcl1

�·�

Go21+ s

ωcl2

�For the cascaded pair:

EE 230 gain-bandwidth – 12

��G�� =

�

� 5�1+

� 250600

�2

�

� ·

�

� 6�1+

� 250500

�2

�

� = 24.8

Plugging in the numbers to find the gain at 250 kHz (using real frequency instead of radial frequency):

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 fcl2 = 500 kHz). In that case, the cascaded gain is 28.8, which should be good enough. (Check it for yourself.)

G (jω) =

�

� Go1

1+ j�

ωωcl1

�

�

� ·

�

� Go2

1+ j�

ωωcl2

�

�

�