Summary 

The input circuit is shown below. The general purpose of the circuit is to transform an input signal U_E to an output signal U_A. Ideal is a linear coherence between U_E and U_A. The circuit as shown below limits U_A to 0 - 4.7 V and produces a nearly linear coherence for |U_E| ≤ U_E' < 4.7 V, where U_E' is supposed to be as large as possible. It's actual value depends on the slew rate of the used zener diods D_Z1 and D_Z2 and has to be detected by experiment.

In order to achieve a symmetric coherence of U_A and U_E for positive and negative values of U_E we need to use zener diodes of an equal zener voltage for D_Z1 and D_Z2. A zener voltage of 4.3 V and ~0.4 V forward voltage is a good choice. Below we will call the sum of them U_Z. In this case we have U_E' ≈ 4.7 V for R_M = 0 Ω. But it seems like zener diodes never have the zener voltage they are declared to have, it seems like their real zener voltage is almost always lower. As long as the zener voltage of the both diodes is nearly equal, this should not be a problem.

Annotation: Of course the serial connection of the two zener diodes can be replaced by a contrary parallel connection of respectively one regular diode and one zener diode (contrary to the regular one). This might be helpful if you cannot manage to find two zener diodes of correct dimensions.

Analysis

1. Understanding 

What we are going to do here is to point out a linear coherence of U_A and U_E for certain circumstances. Potentiometer R_M is meant to extend the linear coherence to higher voltages as we will see:

(1.1) I_E = U_E / ( R_i + R_M )

(1.2) I_E = I_D + I_i + I_S

Due to R_S1 + R_S2 » R_i and (even more) R_S1 + R_S2 » R_i || R_G we can idealize (1.2) to:

(1.3) I_E = I_D + I_i

(1.4) | I_E · R_i | < U_Z ↔ ( I_D = 0 ↔ I_i = I_E )

This means, the potential drop over D_Z1 and D_Z2 (called U_D below) is equal I_E · R_i for I_E · R_i < U_Z. With (1.1) we have:

(1.5) U_D ≡ I_E · R_i = U_E · R_i / ( R_i + R_M ) for | U_D | < U_Z

Which is a linear coherence. With R_M = 0 it says:

(1.6) U_D = U_E for U_E < U_Z

With greater U_E we won't have a linear coherence anymore but U_D = U_E for all U_E ≥ U_Z with ideal diodes. In reality the coherence for U_E ≥ U_Z is logarithmic. But as (1.5) says we can increase the scope of linear coherence between U_D and U_E by increasing R_M to those U_E which correspond U_E ≥ U_Z:

The purpose of R_S1, R_S2 and the 4.7 V voltage source, which actually form up a simple voltage divider, is to linearly transform U_D from [-4.7V, +4.7V] to [0, 4.7 V]. So finally we have for | U_D | < U_Z :

(1.7) U_A = ( U_E · R_i / ( R_i + R_M ) + 4.7 V ) / 2

This implies:

1. U_A < 4.7 V / 2 →  U_E < 0
2. U_A > 4.7 V / 2 →  U_E > 0
3. U_A = 4.7 V / 2 →  U_E = 0

The ADC of the ATmega8 is connected to the terminals of U_A. According to ATmega8 datasheet (page 248) it's ADC has an input resistance of 55 to 100 MΩ. This means the voltage divider R_S1 / R_S2 can be assumed as not stressed. Nevertheless it could be useful to use an impedance converter here.

2. Exact coherence

The coherence as pointed out in (1.7) is an approximation. Now we are looking for the exact relationship between U_A and U_E. In order to do that we will limit our self to the case I_D = 0 (expecting linear coherence).

(2.01) I_D = 0 →  I_E = I_S + I_i

Utilizing the princip of superposition.

1. We keep U_E and bypass the 4.7 V voltage source:

(2.02) I_S' = I_E' · ( [ R_S1 + R_S2 + R_G ] || R_i ) / ( R_S1 + R_S2 + R_G )

(2.03) → U_A' = I_S' · R_S1 = I_E' · ( RS1 / [ R_S1 + R_S2 + R_G ] ) · ( [ R_S1 + R_S2 + R_G ] || R_i )

(2.04) U_E = I_E' · R_M + I_E' · ( [ R_S1 + R_S2 + R_G ] || R_i )

(2.05) → I_E' = U_E / ( R_M + [ R_S1 + R_S2 + R_G ] || R_i )

(2.06) → U_A' = U_E · ( R₀ / [ R₀ + R_M ] ) · ( R_S1 / [ R_S1 + R_S2 + R_G ] ) ,  with R₀ := ( R_S1 + R_S2 + R_G ) || R_i

2. We keep the 4.7 V voltage source and bypass U_E:

(2.07) U_A'' = 4.7 V + I_S'' · R_S1

(2.08) 4.7 V + I_S'' · ( R_S1 + R_S2 + R_G + R_i || R_M )

(2.09) → I_S'' = -4.7 V / ( R_S1 + R_S2 + R_G + R_i || R_M )

(2.10) → U_A'' = 4.7 V · ( 1 - R_S1 / [ R_S1 + R_S2 + R_G + R_i || R_M ] )

According to the princip of superposition we have U_A = U_A' + U_A'':

(2.11) U_A = U_E · ( R₀ / [ R₀ + R_M ] ) · ( R_S1 / [ R_S1 + R_S2 + R_G ] ) + 4.7 V · ( 1 - R_S1 / [ R_S1 + R_S2 + R_G + R_i || R_M ] )

When comparing (2.11) to (1.7) now in order to verfiy our results we notice:

1. 4.7 V · ( 1 - R_S1 / [ R_S1 + R_S2 + R_G + R_i || R_M ] ) ≈ 4.7 / 2 V because of R_{S1 =} R_S2 » R_G assuming R_M = 0
2. R₀ / [ R₀ + R_M ] ≈ R_i / ( R_i + R_M ) because of R_S1 + R_S2 + R_G » R_i
   
3. R_S1 / [ R_S1 + R_S2 + R_G ] ≈ ½ because of R_{S1 =} R_S2 » R_G