To simplify the analysis of the Op-Amp circuits, we further make the following assumptions:

- The huge input resistance can be treated as infinity .
- The input current drawn by an op-amp is samll ( ), and could be approximated to be zero .
- The small output impedance can be treated as zero , i.e., the output is not affected by the load (so long as it is much greater than ).
- Based on the fact that , we could assume , i.e., the virtual ground assumption.
- The bandwidth is large ().

**Voltage follower (buffer)**

As the output is the same as the input, why can't we replace this op-amp circuit by a piece of wire?**Inverter**Current into the op-amp is negligible, and . Applying KCL to the node of , we have

In general, and in the inverter can be replaced by two networks (with impedances and respectively) containing resistors and capacitors and the analysis of the circuit can be carried out easily in frequency domain:

This is a convenient way to design filters of various frequency characteristics.**Non-Inverting Amplifier**

**Summer-inverter**Apply KCL to :

**Differential amplifier**Define , we get:

But as

therefore

Consider some special cases:

- If and , we get

- If (open circuit, and can be any value), then and
we get

This is a combination of inverter and a non-inverter amplifiers. - If , then , this is the follower.
- If (), then we get the inverter

- If () and , we get the non-inverter:

**Note 1:**It is likely that both inputs are subjected to some common noise (such as interference of 60Hz power supply):

In this case the output is

not affected by the common noise at all, i.e., the differential amplifier can suppress*common-mode signal*(such as the noise signal ) while amplify the*differential-mode signal*(such as and ).**Note 2:**If one of the two inputs, e.g., is connected to a constant voltage treated as a reference , then the differential amplifier can also be used as a level shifter. As

we get

But

we have

where

In other words, the output is times the input , shifted by a constant value . This level-shifter circuit can be used to change the DC level of the signal (e.g., removal of DC component) as well as amplifying it.- If and , we get
**Instrumentation Amplifier**One drawback of the differential amplifier is that its input impedance () may not be high enough if the output impedance of the previous stage is not low enough. To overcome this problem, two non-inverters with high input resistance can be used each for one of the two inputs to the differential amplifier. The resulting circuit is shown below:

The analysis of this circuit is very simple. As the output impedance of the non-inverter is low, the three op-amp circuit can be considered as three independent circuits. The outputs of the two non-inverters are:

The output voltage of the differential amplifier is:

Of course the two resistors can be combined to become , i.e., , then the output can be written as:

Alternatively, we consider the current going from to :

From the equation of the first two terms we get:

From the equation of the second two terms we get:

Using the equation of the differential amplifier above, we get the same result as above:

**Algebraic summer (inputs of different signs)**Define . Apply KCL to and we get:

Solving the 2nd equation for we get:

and substitute it into the first equation to get

**A/D converter**Without feedback, the output of an op-amp is . As is large, is saturated, equal to either the positive or the negative voltage supply, depending on whether or not is greater than . These two possible outputs, positive and negative, can be treated as ``1'' and ``0'' of the binary system. The figure shows an A/D converter built by three op-amps to measure voltage from 0 to 3 volts with resolution 1 V.

Due to the voltage divider, the input voltages to the three op-amps are, respectively, 2.5V, 1.5V and 0.5V. The output of these op-amps are listed below for each of the input voltage levels. A digital logic circuit is then needed to convert the 3-bit output of the op-amps to the two-bit binary representation.

Voltage (volts) 0 1 2 3 Op-amps Outputs 000 001 011 111 Binary Representation 00 01 10 11 **First order system -- integrator and differentiator****Integrator**In time domain, as and , we have (KCL)

where . In frequency domain, we have:

**Differentiator**If we swap the resistor and the capacitor, we get in time domain:

In frequency domain, we have:

**First order systems - low-pass and high-pass filters****Low-pass filter:**

where , . Intuitively, when frequency is high, is small and the effect of negative feedback is strong, therefore the output is low.For example, when , , the Bode plots are shown below:

**High-pass filter:**

where , . Intuitively, when frequency is low is large and the signal is difficult to pass, therefore the output is low.For example, when , , the Bode plots are shown below:

**Band-pass filter:**

where , , .For example, when , , , the Bode plots are shown below:

**Higher order systems**Higher than first order systems can be built with multiple integrators, as shown here for a third order system:

From the diagram, we can get

But we also have

i.e.,

we get the transfer function

**Second order system by 2 integrators**From the diagram, we can get

substituting the first two equations into the last one, we get

from which we obtain the transfer function as

which is a second order system. In particular, if , we have

Comparing this with the canonical 2nd order system transfer function

we see that we can let and . Moreover, , i.e., the feedback from the output should be negative. is a constant scalar which can take any value.**Sallen-Key Topology**The Sallen-Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity.

We represent the input and output in s-domain as and , respectively, and the voltage at node a as , and apply KCL to nodes a and b to get:

Solving the second equation for we get

Substituting this into the first equation we get

**Example 1**, , , , then we get a second order low-pass filter:

where

**Example 2**, , , , then we get a second order high-pass filter:

where