How do we derive the limiting values of a series’ resistance for Zener diodes with varying load and varying input conditions?
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. The Zener voltage regulator.

The problem:


The Zener diode has a maximum power rating. e.g., 1 W. 
This in turn gives us the maximum current that the Zener can handle. \$ I_{Zmax} = \frac {P_Z}{V_Z} \$, where \$ P_Z \$ is the power rating of the Zener and \$ V_Z \$ is the reverse breakdown or 'Zener' voltage of the diode.
The higher Vin goes the more current the Zener has to shunt away from the load.
On the other hand, as Vin decreases the Zener current falls and at some point will reach zero. If Vin decreases below that point then the regulation is lost and Vo will decrease below the Zener voltage.


\$ R_{MIN} \$


The minimum value for R is that which limits the Zener current to its rated value at the maximum voltage. If the load current can reduce then the Zener has to divert more current so we need to calculate at \$ I_{Lmin} \$.
The voltage across the resistor will be \$ V_{INmax} - V_Z \$.
The current through the resistor will be the sum of the Zener current and RL current \$ I_{Lmin} + I_{Zmax} \$.
From Ohm's Law we can now calculate \$ R_{MIN} = \frac {V_{INmax} - V_Z }{I_{Lmin} + I_{Zmax}} \$.
Since \$ V_O = V_Z \$ the second equality is a simple substitution.


\$ R_{MAX} \$

You should be able to work out the other formula in the same manner. \$ I_{Zmin} \$ is the minimum current that will keep the Zener past its 'knee' voltage and into a region where the voltage is stable.