OPAMP gain

Figure 1 shows two flavors of OPAMP amplification modes.

Figure 1: Two flavors of amplifications using OPAMP. … [+] An ideal OPAMP has infinite gain and draws no input current. The analysis is done by setting \(v_-=v_+\) and \(i=0\) and using Kirchhoff’s current and voltage laws.

The gain analysis is done by setting \(v_-=v_+\) and \(i=0\) to get the input output relations: \[\begin{eqnarray} v^{non-inv}_{out}=v_{in}\frac{R_F+R_1}{R_1},\, \text{and}\quad v^{inv}_{out}=-v_{in}\frac{R_F}{R_1} \tag{1}, \end{eqnarray}\]

A larger amplification can be reached by selecting large \(\frac{R_F}{R_1}\). An OPAMP has inherent voltage and current noise. The current noise couples to the input impedance.
The output noise can be calculated by projecting the noise items into the input terminals[1]. Consider the non-inverting amp in Fig. 2 for noise analysis.

Figure 2: The circuit for noise analysis in non-inverting circuit. … [+] Resistors have thermal noise. OPAMP has noise in the input voltage and current. The input current noise passes through \(Z_{in}\)..

The voltages at the OPAMP inputs:

\[\begin{eqnarray} v_+=v_s+v_{ts}+v_n+i_n R_S = v_-=(v_o+v_{tf})\frac{R_1}{R_1+R_F}+v_{t1}\frac{R_F}{R_1+R_F} -i_n\frac{R_1 R_F}{R_1+R_F} \tag{2} \end{eqnarray}\]

Solving for the output noise:

\[\begin{eqnarray} v_o=(1+R_F/R_1)\left(v_s+v_{ts}+v_n+ \color{red}{i_n \left[R_S+\frac{R_1 R_F}{R_1+R_F}\right]}-\frac{v_{tf}R_1 +v_{t1}R_F}{R_1+R_F} \right) \tag{3} \end{eqnarray}\]

OPAMPs typically use BJTs in the input stages, which generate much greater noise currents at the input. These noise currents flowing into high impedances, the red term in Eq. (3) , create large equivalent input noise.

Let us now consider the inverting amp in Fig. 3 which shows the circuit diagram for noise analysis.

Figure 3: The circuit for noise analysis in inverting circuit. … [+] Resistors have thermal noise. OPAMP has noise in the input voltage and current. The input current noise passes through \(Z_{in}\)..

An alternative analysis method uses Norton equivalent elements. Thermal noise voltages can be converted to currents when convenient.

The voltages at the OPAMP inputs:

\[\begin{eqnarray} v_+=v_n+(i_s+i_{ts}+i_n) R_S||R_F +(v_{tf}+v_o)\frac{R_S}{R_S+R_F} = v_-=0 \tag{4}. \end{eqnarray}\] Solve for the output noise: \[\begin{eqnarray} v_o=-\frac{R_F}{R_S}\left( R_S \left[i_s +i_{ts} +\frac{v_{tf}}{R_F}\right]+v_n \frac{R_F+R_S}{R_F} +\color{red}{i_n R_S}\right) \tag{5} \end{eqnarray}\]

We see again the noise current, the red term in Eq. (5) , flowing into high impedances \(R_S\), creating large equivalent noise.