One of my readers posted the following questions in the comment section of MasteringElectronicsDesign.com: An Op Amp Gain Bandwidth Product.
I am doing a work on fully differential Negative feedback op-amp with capacitive divider configuration. I have some questions and confusions, can you please clarify?
What is the difference between closed loop gain and open loop gain, and are they dependent to each other?
How can we calculate the unity gain frequency if I have a 3-dB frequency of 100Hz and closed loop gain of 40dB?
Does the feedback factor (BETA) has importance with respect to any other parameters?
How will it help in finding the closed transfer function of the system assuming the op-amp as a single pole system?
The answers needed some space, more than the comment section could offer, so here is a post on the topics of op amp open-loop, closed-loop and feedback.
Q1: What is the difference between closed loop gain and open loop gain, and are they dependent to each other?
A1: In general terms, an amplifier in has gain, which represents the ratio between the output signal amplitude versus the input signal amplitude. If the gain is frequency dependent, we note it with A(ω) to show that dependence. Figure 1 shows the amplifier, represented as a black-box, with two input signals, V1 and V2, and an output signal Vo. The signals are shown with respect to ground.
This is the basic op amp. The output Vo depends on the difference between the two inputs as follows:
If we bring negative feedback from output to input around this amplifier, in other words, close the loop, the entire system gain changes and its value depends on feedback. As such, we call A(ω) open-loop gain, and the gain of the op amp with negative feedback, closed-loop gain, noted ACL(ω). Figure 2 shows the block diagram of an amplifier with negative feedback, where the F box shows the feedback network.
When the loop is closed, equation (1) becomes
F is called the feedback coefficient. ACL(ω) depends on A(ω) with the following formula:
Of course, F can be dependent on frequency as well, but I want to keep this simple for now.
Note: If you want to know how this formula can be derived, here are a few quick steps:
Q2: How can we calculate the unity gain frequency if I have a 3-dB frequency of 100Hz and closed loop gain of 40dB?
A2: Compensated op amps have one pole. The gain drops at 20 dB per decade after that pole. (see Figure 3).
In a closed loop system, the gain is set by the feedback network, provided that the open loop gain is high (see answer 3 as well). No matter the closed loop gain level, the product between gain and bandwidth, or the gain bandwidth product (GBW) is constant. Therefore, the GBW in this case is
We can apply this value to calculate the unity gain frequency:
Q3: Does the feedback factor (BETA) has importance with respect to any other parameters?
A3: Yes it does. The first answer shows that the feedback factor is used in the closed loop gain calculation. Also, if the open loop gain is high, the feedback factor determines the closed loop gain at DC and in band. Indeed, let’s show this by rewriting equation (3) at DC.
where with ACLO I noted the closed loop gain and with Ao the open loop gain, both at DC. If Ao is high enough so that 1 in the denominator can be neglected, ACLO becomes
Besides determining the gain, if F depends on frequency, it will also modify the amplifier bandwidth. Active analog filters can be designed by simply designing the correct feedback network.
Q4: How will it help in finding the closed transfer function of the system assuming the op-amp as a single pole system?
A4: The answer to this question is given by equation (3). Assuming that the feedback coefficient is not frequency dependent, the closed loop transfer function is
If the network feedback is made with two resistors, as in the non-inverting amplifier shown in Figure 4,
The transfer function becomes
An operational amplifier open loop gain can be written as
where with Ao I noted the op amp open loop gain at DC and with ωo the op amp cutoff frequency in radians per second.
Replacing (11) in (10), and after calculations, the closed-loop gain becomes
Based on (7), the closed-loop gain or transfer function can be written as
where with ωCLO I noted the cutoff frequency in closed loop, in radians per second.
It should be clear now that the feedback coefficient modifies both the amplifier gain and bandwidth. The following figure shows an example. ADA4004 open-loop gain starts rolling off at 24Hz. The red trace shows the open-loop gain. If we close the loop as in Figure 4, with R1 = 100 koms and R2 = 1 kohm, the new transfer function is represented by the blue trace which is a plot of the transfer function shown in (13).