# Load resistance in voltage regulator

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\$$. simulate this circuit – Schematic created using CircuitLab

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

First method:

I take off the diode as the load and find thevenin voltage : $$V_{th}=\frac{R_L}{R_l+R_s}V_s$$

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 method:

If $$\R_L\$$ is minimal then $$\I_L\$$ passing through it would be maximal and $$\I_z\$$ minimal ($$\= I_{z_0}\$$) 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}\$$)

• Maybe you could rearange the text in more lines, so it would be better readable. For examle 1. and 2. ,...Altrough nice use of latex equations, so +1. A good staritng point would be $V_{R_L}=V_Z$ – Marko Buršič Mar 8 at 10:00
• Will do! That's what I did, look at "First". – Luyw Mar 8 at 10:02
• +1 from me, this is a good example of how the 'I'm confused with this circuit' questions should be asked. OP explains what they are trying to so/understand/solve, then explains their thinking behind it, and shows how they attempted to tackle the problem. – MCG Mar 8 at 10:07

A typical 300mW Zener uses a test current Izt=5mA with an incremental resistance Zzt and a threshold current, at 1/10 Izt or It @ 0.5mA which results in a knee resistance of Zzk approx. 10x Zzt.

The same is true for LEDs.

Make a Thevenin equivalent circuit for 5mA at a low power.
This gives the low rated resistance, Zzt and is important to include in the equation to solve for R1. simulate this circuit – Schematic created using CircuitLab

Now you can easily solve for R1

$$\ V_S = V_{R_S} + V_Z\$$

$$\ V_S = I_S\cdot R_S + V_Z\$$

$$\ I_S = I_Z + I_L\$$

$$\ V_S = (I_Z + I_L)\cdot R_S + V_Z\$$

$$\ I_L = \dfrac{V_Z}{R_L}\$$

$$\ V_S = (I_Z + \dfrac{V_Z}{R_L})\cdot R_S + V_Z\$$

$$\ R_S =\dfrac{V_S - V_Z}{I_Z + \dfrac{V_Z}{R_L}} \$$

$$\ R_S =\dfrac{V_S - V_Z}{I_{Z_0}+ \dfrac{V_Z}{R_{L_{min}}}} \$$

$$\R_{L_{min}}=\dfrac{R_S\cdot V_Z}{V_S-V_Z-R_S\cdot I_{Z_0}}\$$