![[op-amp-basic-schematic-symbol.svg]]

The operational amplifier has a very high input impedance which makes it very good for amplifying low voltage signals.


Basically the OpAmp is a function like this:

$\displaystyle Y = A_v (X_1 - X_2)$

Where:
$$
\begin{flalign}
&Y = \text{Output Voltage}&\\\
&A_v = \text{Open Loop Gain}\\
&X_1 = \text{Input V1 (Non Inverting Input)}\\
&X_2 = \text{Input V2 (Inverting Input)}\\
\end{flalign}
$$
# Rules

**1. No Current flows in or out of the outputs**
**2. The op-amp tries to keep the input voltages the same**
The second rule only applies when the op-amp is in closed loop configuration

# Regions
Op Amps functions in different regions, just like diodes, and transistors.

![[op-amp-regions.png|400]]

# Regions

**Linear Region**
This is how the Op-Amp normally functions.

**Saturation Region**
When the output of the op-amp would be higher than $+V_{CC}$ or lower than $-V_{CC}$ the output value is clamped to those values.

In real life OpAmps have $A_V$ values as high as $10^8$ or $10^9$  due to this even very small input voltages would quickly leave the linear region. That is why we need 

# Negative Feedback
To use negative feedback we connect the output of the OpAmp to one of its inputs. This connection is modified by a *feedback factor* ($\beta$) which can be in the range $0 \le \beta \le 1$. 

Due to this feedback the new formula for the output $V_O$ is now:

$$
\begin{flalign}
&V_o = A_V * V_\Delta&\\\
\\
&V_- = \beta * V_o\\
&\text{Now we can say that }V_\Delta \text{is equal to:}\\
&V_\Delta = V_+ - \beta *V_o &| \textit{ Solve for }V_o \\
&V_o = \frac{V_+ - V_\Delta}{\beta}
\end{flalign}
$$


# Configurations

**Open Loop**
When the output of the Op Amp is not connected to any of its inputs, it is in the so called "open loop configurations"

**Closed Loop**
When we connect the output of the OpAmp to either $V_+$ or $V_-$ the OpAmp is in the "closed loop configuration".

# Bandwidth Limitations

Real op-amps behave differently depending on the input signals frequency. Usually the internal open-loop gain gets lower as the input frequency gets higher like this.

The op-amps bandwidth is the frequency range in which the voltage gain is above 70.7% ($3dB$) of its maximum output. The point at which it is below that gain, is called the **breakpoint**.

![[op-amp-bandwidth.png|400]]


This is also one of the reason we use op-amps in closed loop configuration. Because it allows is to trade maximum gain for a larger bandwidth.

![[op-amp-bandwidth-closed-loop.png|400]]

If we want to find out the bandwidth of an op-amp, we can check the datasheet. The *LM741 OpAmp* for example:

![[lm741-datasheet-bandwidth.png]]

The thing is that this frequency only applies when the op-amp has a gain of 1, this frequency point is also called **unity gain**. It is called the **Gain Bandwidth Product**, which is calculated as follows:

$GBP = A_V * f_c$

Where $A_V$ is the voltage gain, and $f_c$ is the cutoff frequency.

With this equation we can also solve for $f_c$ like so:

$\displaystyle f_c = \frac{GBP}{A_V}$


![[lm741.pdf]]