# Kirchhoff current loop in Resistor Diode Ladder network

I am looking for an approach on how to apply Kirchhoff current / voltage law in the infinitely long diode ladder network. Can anyone help me with this ?

I am looking for 1D differential equation or an implicit equation.

The admittance looking into the network "$Y_{net}$" can be expresed as:

$$Y_{net} = \dfrac{1}{R + \dfrac{1}{Y_{net} + Y_{diode}}}$$

Rearranging as a quadratic and solving:

$$Y_{net}^2R + Y_{net}Y_{diode} - Y_{diode} = 0$$

$$Y_{net} = \dfrac{\sqrt{Y_{diode}}\sqrt{4R +Y_{diode}}}{2R}$$

Where

$$Y_{diode} = \frac{nV_t}{I_s}e^{-\frac{V_f - I_fR}{V_t}}$$

Then

$$I_f = V_f\dfrac{\sqrt{Y_{diode}}\sqrt{4R +Y_{diode}}}{2R}.$$

I think this implicit equation could then be solved for $I_f$. Once you know the first forward current you can plug $V_f - I_fR$ into the diode equation to find the current in the first diode in the string, then iterate using the above and the implicit equation to find the current in any diode.