Easy method calculates comparator trip points

Virgil Lawrence, Micro Linear, San Jose, CA -- EDN, 6/10/1999

Using Millman's Theorem to calculate the resistor ratio reduces the time it takes to calculate the trip points on a comparator with hysteresis. This method eliminates lengthy computations and substitutions. Using this resistor ratio, you select two resistors, assign convenient values, and then calculate the third value.

Assume an inverting comparator with an upper limit of 4V and a lower trip voltage of 1.333V. The voltage on the inverting input needs to reach 1.333V for the output to switch to V_CC. Then, for the output to return to zero, the input voltage needs to reach 4V.

Millman's Theorem states that the sum of the products of the voltages times their respective conductances divided by the sum of the conductances gives the common junction-point voltage V_X (Figure 1). Or,

where G=1/R.

Using Millman's Theorem, with V_CC at the top of R₁ and R_F (Figure 2), set up the numerator with a Millman equation. Set up the denominator with another Millman equation when the output voltage is zero (with only one voltage source in the denominator). Assume that R_C is << R _F and therefore negligible in the calculations. The resultant equation for the voltage ratio is:

This large fraction equals the voltage ratio 4/1.333=3. First solve for R_F in terms of R₁, and select R₁ in relation to R_F, such as R₁=1 Mµ, and R_F=500 kµ. Then solve for R2. The resistance ratio always equals the voltage ratio minus one. (DI #2372)