5 Derive Equation for Optimal Isolation Resistance

To reduce errors while maintaining closed-loop stability, the optimal R_ISO resistance must be determined. The optimal isolation resistance is the smallest series resistance that produces an acceptable transient response when driving the capacitive load. While a phase margin of 45° is generally considered stable, this may not be sufficient for applications that demand low overshoot. In certain cases, it is typical for a phase margin of 60° to be considered optimally damped. If voltage error is of primary concern, a phase margin of 45° can be targeted as this allows the smallest isolation resistance to be used.

Figure 5-1 Small-Signal Step Response and Phase Margin

The closure frequency (f_cl) is the frequency in which the A_OL and 1/β curves intersect. Selecting an R_ISO resistance that places f_z1 exactly at the closure frequency, targets a phase margin between 50°-60°, and is a good tradeoff between voltage error and transient performance. In this case, the optimal R_ISO results in f_z1 such that,

For a typical amplifier in a buffer configuration, the closure frequency is simply the gain bandwidth (f_gbw) of the op amp. However, in the case of the loaded A_OL curve in which f_p2 causes an additional 20 dB decrease in A_OL per decade, the closure frequency is always less than the gain bandwidth. The exact closure frequency must be determined to accurately select the optimal f_z1 frequency.

Figure 5-2 Amplifier Open-Loop Gain with Capacitive Loading

Figure 5-2 plots an amplifier’s loaded A_OL curve in relation to the frequencies f_p2 and f_gbw. Looking at this bode plot geometrically reveals two triangles with known slopes and vertices that intercept at f_cl. Assuming a second order system with consistent slopes of -20 dB/decade and -40 dB/decade, f_cl is always located at the midpoint between f_p2 and f_gbw on the logarithmic axis. With a known load capacitance, and data sheet extracted values for R_O and f_gbw, the following equations can be used to solve for f_cl.

Setting f_z1 equal to f_cl, and considering the shift in f_p2 due to R_ISO results in Equation 10.

Combining Equation 10 with Equation 5 produces Equation 11.

Simplification results in the following quadratic equation for R_ISO in standard form.

Using the quadratic formula to solve for R_ISO with the assumption that R_ISO must be positive results in the equation that was used to determine the optimal R_ISO in Section 3.

Where,

• f_gbw is the gain bandwidth of the amplifier in Hz, as defined in the data sheet
• R_O is the amplifier’s open-loop output impedance in Ohms, as defined in the data sheet
• C_LOAD is the load capacitance in Farads

For an amplifier driving a large capacitive load, Equation 13 can be used to quickly derive the isolation resistance required to target 50°-60° of phase margin. Spice simulation tools such as TINA-TI must be used to verify the resultant phase margin, as this can vary considerable depending on the initial phase margin and other circuit conditions. TI Precision Labs online training videos show how to perform stability analysis for amplifier circuits in a variety of conditions.

Equation 13 is derived for amplifiers in a buffer configuration, but this equation can also be used for gain stages by adjusting the frequency term to account for the gain bandwidth, as in Equation 14.

Where A_V equals the non-inverting gain of the amplifier stage in V/V.

The feedback network in gain configurations can produce additional poles and zeros in the amplifier’s 1/Beta and loaded A_OL curves that can contribute to instability. In these cases, further analysis is required and additional stabilization techniques such as feedback compensation can be implemented. Always use spice simulation tools such as TINA-TI to verify that the circuit is stable and operating properly.