The I-V characteristics in the breakdown region of a Zener diode is almost vertical. That is, the current I_Z can rapidly increase at constant V_Z . To prevent damage due to excessive heating, the Zener current should not exceed the rated maximum current, I_ZM . Hence, a current-limiting resistor R_S is connected in series with the diode.

I_Z and the power dissipated in the Zener diode will be large for I_L = 0 (no-load condition) or when I_L is less than the rated maximum (when R_S is small and R_L is large). The currentlimiting resistor R_S is so chosen that the Zener current does not exceed the rated maximum reverse current, I_ZM when there is no load or when the load is very high. 

The rated maximum power of a Zener diode is P_ZM = I_ZM = V_Z

At n-load condition, the current through R is I_ZM = I and the voltage drop across it is V – V_Zr , where V is the unregulated source voltage. The diode current will be maximum when V is maximum at V_max and I = I_ZM . Then, the minimum value of the series resistance should be

R_S, min = \(\cfrac{V_{max}-V_Z}{I_{ZM}}\)