Voltage Followers#

Definition: Voltage Follower

A voltage follower is a differential voltage amplifier whose voltage gain \(A\) is \(1\):

\[ A = 1 \]

Example: Voltage Follower

A voltage follower can be constructed using an ideal operational amplifier by connecting the output directly to the inverting input:

Voltage Follower via Op-Amp

As long as the ideal op-amp is operated in its linear region, the above circuit behaves like a voltage follower with the following voltage gain:

\[ A = \frac{v_{\text{out}}}{v_{\text{in}}} = 1 \]

To ensure that the ideal op-amp is indeed operated in its linear region, we need to have \(v_{\text{in}} \in \left[-V_{\text{sat}}; +V_{\text{sat}}\right]\). Moreover, the resulting voltage follower is itself an operational amplifier, since \(i_{-} = i_{+} = 0\).

We can see this by analyzing the network.

Linear region:

When the ideal op-amp is operated in its linear region, we know that \(v_d = 0\), i.e. \(v_{-} = v_{+}\).

From the circuit diagram, the input and output voltages are connected to the amplifier terminals as follows:

\[ \left\vert\begin{aligned}v_{+} &= v_{\text{in}} \\ v_{-} &= v_{\text{out}}\end{aligned}\right. \]

Since \(v_{-} = v_{+}\), we can equate the two expressions to obtain the following:

\[ v_{\text{out}} = v_{\text{in}} \implies \frac{v_{\text{out}}}{v_{\text{in}}} = 1 \]

Saturation regions:

When the ideal op-amp is operated outside its linear region, we know that \(v_d \ne 0\).

By definition of the differential voltage, we have the following expression for \(v_d\):

\[ v_d = v_{+} - v_{-} \]

Substituting the connections from the network (\(v_{+} = v_{\text{in}}\) and \(v_{-} = v_{\text{out}}\)), we obtain:

\[ v_d = v_{\text{in}} - v_{\text{out}} \]

When the ideal op-amp is operated in its negative saturation region, we have \(v_d \lt 0\) and \(v_{\text{out}} = -V_{\text{sat}}\):

\[ v_d = v_{\text{in}} - (-V_{\text{sat}}) = v_{\text{in}} + V_{\text{sat}} \lt 0 \]

By performing some equivalent transformations, we get the following:

\[ v_{\text{in}} \lt -V_{\text{sat}} \]

Therefore, we know that the ideal op-amp is operated in its negative saturation region whenever \(v_{\text{in}} \lt -V_{\text{sat}}\).

By contrast, when the ideal op-amp is operated in its positive saturation region, we have \(v_d \gt 0\) and \(v_{\text{out}} = +V_{\text{sat}}\):

\[ v_d = v_{\text{in}} - V_{\text{sat}} \gt 0 \]

By performing some equivalent transformations, we get the following:

\[ v_{\text{in}} \gt +V_{\text{sat}} \]

Therefore, we know that the ideal op-amp is operated in its positive saturation region whenever \(v_{\text{in}} \gt +V_{\text{sat}}\).