# Inverting Amplifiers

An inverting amplifier uses feedback resistors to control the gain of an op-amp.
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 op-amps:
• Have a very high open loop gain (A0) - typically 106 for low frequencies
• A very high input impedance - typically Megaohms
• Amplify the difference between the two inputs

See the page about op-amps for more details

The circuit diagram for an inverting amplifier is shown below.
The two resistors form a potential divider between Vin and Vout
The input resistor is called Ri
The feedback resistor is called Rf 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 non-inverting 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 Vin is 0v. Thus Vout must also be 0v If Vin rises then the inverting input is > non-inverting and so Vout goes rapidly negative until the two inputs are once again equal (or atleast only µV's different) Similarly, if Vin goes negative then the inverting input is < non-inverting input and so Vout 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 Vin and Vout depends on the ratio of the resistors in the potential divider and so, at low frequencies, the gain depends only on the values of Rf and Ri At higher frequencies the gain of the op-amp is limited and so sets an upper limit on the gain of the amplifier circuit.

#### Amplifier gain

The inputs of the op-amp take no current and so all the current flowing through Ri also flows through Rf.
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 Ri is Vin and voltage across Rf is -Vout.
The assuming the input voltage is +Ve then the output voltage must be -Ve
(to allow the current to flow the right way through Rf).
Thus current through Ri is I = Vin / Ri
and the current through Rf is I = -Vout / Rf.
Thus Vin / Ri = -Vout / Rf.
Amplifier gain is defined as A = Vout / Vin and so rearranging gives:

### Gain, A = -Rf / Ri

#### The following section is "borrowed" from http://www.uoguelph.ca/~antoon/gadgets/741/741.html. Appologies to the author!

Open-Loop Gain & Frequency:

Unlike the ideal op-amp (Fig. 5-1), the op-amp that is used in more realistic circuits today, does not have infinite gain and bandwidth. Look at Open-loop gain in Fig. 4 above, it is graphed for a type 741 op-amp as a function of frequency. At very low frequencies, the open-loop gain of an op-amp 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 ten-fold 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 op-amp is its "gain-bandwidth product" or GBP. For the response curve of Fig. 4, the product of the open-loop gain and frequency is a constant at any point on the curve, so that:   GBP = AolBW
Graphically, the bandwidth is the point at which the closed-loop gain curve intersects the open-loop curve, as shown in Fig. 5 for a family of closed-loop gains. For a more practical design situation, the actual design of an op-amp circuit should be approximately 1/10 to 1/20 of the open-loop gain at a given frequency. This ensures that the op-amp will function properly without distortion. As an example, using the response in Fig. 4, the closed-loop gain at 10Khz should be about 5 to 10, since the open-loop 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 step-function pulse is used as an input signal, and is specified under close-loop conditions. From electronic circuit theory, the rise time is related to the bandwidth of the op-amp by the relation:   BW = 0.35 / rise time