Load resistance in voltage regulator

[archaic]

Homework Statement

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I want to find the minimum resistance $R_L$ so as to maintain $V_z$(voltage of the zener corresponding to the minimum current $I_{z_0}$) across the same resistor $R_L$.

Homework Equations

$V_z$(voltage of the zener corresponding to the minimum current $I_{z_0}$)

The Attempt at a Solution

I have two ways of looking at this, one of them is wrong and I need your help to figure it out!

First, I take off the diode as the load and find thevenin voltage, $V_{th}=\frac{R_L}{R_l+R_s}V_s$, and since $V_{th}=V_{R_L}$ I'll equate it with $V_z$ which gives $\frac{R_L}{R_L+R_S}V_S=V_z$ thus $R_L=\frac{R_S.V_z}{V_S-V_z}$

Second, If $R_L$is minimal then $I_L$ passing through it would be maximal and $I_{z_0}$ minimal thus we'd have $I_S = I_L+I_{z_0} \Leftrightarrow \frac{V_S-V_z}{R_S}=\frac{V_z}{R_L}+I_{z_0}\Leftrightarrow R_L=\frac{R_S.V_Z}{V_S-V_Z-R_S.I_{z_0}}$
(I'm skeptical concerning the implication $I_L$ maximal $\Rightarrow$ $I_z$ minimal, or in other words $I_z = I_{z_0}$)

Thank you for your time!

 

[Merlin3189]

The premises of your second attempt seem ok, but I can't understand how you can ignore the zener in your first attempt. There must always be a current at least I_Z0 in the diode, so that corresponds to a resistance in parallel with R_L equal to V_Z/I_Z0. I'd have thought you need to take that into account either as part of the load, or as part of the source.

 

[archaic]

The premises of your second attempt seem ok, but I can't understand how you can ignore the zener in your first attempt. There must always be a current at least I_Z0 in the diode, so that corresponds to a resistance in parallel with R_L equal to V_Z/I_Z0. I'd have thought you need to take that into account either as part of the load, or as part of the source.

Hm I guess instead of $V_z$ it should be $V_z+r_zI_{z_0}$ right? i.e I am considering the minimum Thevenin voltage for the diode to let current flow.

 

[Merlin3189]

I assumed we were supposed to consider V_Z fixed, since it looked like a constant
But if you wish to use a more accurate model of the zener, you have the right form.

The calculation with method 2 also seems to work out right.

I'm not familiar with the notation of the zener model, so whether the closer expression should be $V_Z + r_Z I_{Z0} \ \ or \ \ V_{Z0} + r_Z I_{Z0}$ I don't know. This would be in the Data and Relevant formulae, I think.it won't make much difference