Inverting Amplifiers
An inverting amplifier uses feedback resistors to control the gain of an opamp.
The values of the feedback resistors determine the gain of the amplifier system.
An inverting amplifier uses negative feedback. This means that when the input changes, the output changes also. The output changes in such a way that tries to avoid saturation and counteract the change caused by the input. This makes the amplifier stable. The amplifer tries to resist change and so avoid saturation.
Recall that opamps:
 Have a very high open loop gain (A_{0})  typically 10^{6} for low frequencies
 A very high input impedance  typically Megaohms
 Amplify the difference between the two inputs
See the page about opamps for more details
The circuit diagram for an inverting amplifier is shown below.
The two resistors form a potential divider between V_{in} and V_{out}
The input resistor is called R_{i}
The feedback resistor is called R_{f} and connects the output to the "negative" input  hence negative feedback!
The diagram shows that the output is an amplified mirror image of the input. The output is the inverse of the input and hence we have an inverting amplifier. The gain shown here is A = 2
How it works
 The noninverting input is held at 0v
 The feedback will try to ensure that the inverting input is very close to 0v. This is because the difference between the inputs must be only µV if the output is not saturated. The inverting input is called a virtual earth.
 The resistors form a potential divider with the center at 0v
 Assume that V_{in} is 0v. Thus V_{out} must also be 0v
 If V_{in} rises then the inverting input is > noninverting and so V_{out} goes rapidly negative until the two inputs are once again equal (or atleast only µV's different)
 Similarly, if V_{in} goes negative then the inverting input is < noninverting input and so V_{out} rises rapidly to become positive
 For the amplifier to work properly the output must be able to change very quickly in order to react to the changes in the input. This limits the maximum frequency at which the amplifier can operate.
 The ratio of V_{in} and V_{out} depends on the ratio of the resistors in the potential divider and so, at low frequencies, the gain depends only on the values of R_{f} and R_{i}
 At higher frequencies the gain of the opamp is limited and so sets an upper limit on the gain of the amplifier circuit.


Amplifier gain
The inputs of the opamp take no current and so all the current flowing through R_{i} also flows through R_{f}.
Recall that generally V = IR and so I = V/R.
The potential at the inverting input is 0v (if the amplifier does not saturate)
and so voltage across R_{i} is V_{in} and voltage across R_{f} is V_{out}.
The assuming the input voltage is +Ve then the output voltage must be Ve
(to allow the current to flow the right way through R_{f}).
Thus current through R_{i} is I = V_{in} / R_{i}
and the current through R_{f} is I = V_{out} / R_{f}.
Thus V_{in} / R_{i} = V_{out} / R_{f}.
Amplifier gain is defined as A = V_{out} / V_{in} and so rearranging gives:
Gain, A = R_{f} / R_{i}
OpenLoop Gain & Frequency:
Unlike the ideal opamp (Fig. 51),
the opamp that is used in more realistic circuits today,
does not have infinite gain and bandwidth. Look at Openloop gain in Fig. 4 above, it is graphed for a type 741
opamp as a function of frequency. At very low frequencies, the openloop gain of an opamp is constant, but starts
to taper off at about 6Hz or so at a rate of 6dB/octave or 20dB/decade (an octave is a doubling in frequency,
and a decade is a tenfold increase in frequency). This decrease continues until the gain is unity, or 0 dB.
The frequency at which the gain is unity is called the unity gain frequency or fT.
Maybe the first factor in the consideration of a specific opamp is its "gainbandwidth product" or GBP.
For the response curve of Fig. 4, the product of the openloop gain and frequency is a constant at any point on
the curve, so that: GBP = A_{ol}BW
Graphically, the bandwidth is the point at which the closedloop gain curve intersects the openloop curve, as shown
in Fig. 5 for a family of closedloop gains. For a more practical design situation, the actual design of an opamp
circuit should be approximately 1/10 to 1/20 of the openloop gain at a given frequency. This ensures that the
opamp will function properly without distortion. As an example, using the response in Fig. 4, the closedloop gain
at 10Khz should be about 5 to 10, since the openloop gain is 100 (40dB).
One additional parameter is worth mentioning, the Transient Response, or rise time is the time that it
takes for the output signal to go from 10% to 90% of its final value when a stepfunction pulse is used as an input
signal, and is specified under closeloop conditions. From electronic circuit theory, the rise time is related to the
bandwidth of the opamp by the relation: BW = ^{0.35} / _{rise time}