# Expressing output voltage in a circuit with zener diodes without any value given

I have this circuit and I need to find $u_{o}(u_{i})$ if $ZD_{2}$ and $ZD_{1}$ are ideal zener diodes with $U_{z}$ zener voltage and the op-amp is also ideal. My problem is that I have no values, I just need to find the expression for $u_{o}$, but how do I know if a diode is on or off or reverse-biased? I don't know where to start.

• Start by noticing that the op-amp inverting input is a virtual ground. – The Photon Dec 1 '17 at 15:46
• If the zeners and R3 are removed what would be the gain? – JIm Dearden Dec 1 '17 at 15:51
• So because it is ground, if $u_{i}$ is a positive voltage, then $u_{o}$ will be negative voltage. That leads me to the conclusion that ZD1 would be forward-biased and on, but how do I know that the reverse voltage on ZD2 is larger than Uz? That is what confuses me. – Nebeski Dec 1 '17 at 16:04
• If the zeners and R3 are removed then $u_{o}=-(u_{i}*R_{2})/R_{1}$ – Nebeski Dec 1 '17 at 16:06
• Now consider that if the (ideal) zeners are connected back to back (as they are shown) and if the output voltage is less than the zener voltage would the zeners have any effect? What would happen if the output tries to exceed the zener voltage? – JIm Dearden Dec 1 '17 at 16:12

I do not want to simply provide the solution, but I can point you in the right starting direction.

I am assuming that you have already seen the fundamentals of an Inverting Op-Amp:

Inverting Op-Amp Fundamental Properties

Your circuit is of the same nature, but has a complex feedback resistor $R_{f}$. Since you mentioned that these are ideal zener diodes, we can ignore their resistance, and only need to pay attention to the conditions when they are conducting current.

If at least one zener is not conducting, then the effective $R_{f}$ is equal to your $R_{2}$.
If both zeners are conducting, then the effective $R_{f}$ is equal to $R_{2}\parallel$$R_{3}$.

The final piece of the puzzle is to identify what circuit scenario is required in order for both zener's to conduct.

Each Zener will conduct current when either:
- The forward voltage is more positive than its Zener Forward Voltage ($V_{F}$)
- The reverse voltage more negative than its Zener_Breakdown_Voltage ($V_{Z}$)
(Fundamentals of Zener Diodes)

There are two scenarios when both Zeners are conducting:
- Scenario 1: ($V_{ZD1}\geq$ $V_{F\_ZD1}$) && ($V_{ZD2}\leq$ $V_{Z\_ZD2}$)
- Scenario 2: ($V_{ZD1}\leq$ $V_{Z\_ZD1}$) && ($V_{ZD2}\geq$ $V_{F\_ZD2}$)

Thus, the resulting $u_{o}(u_{i})$ will be comprised of two discrete equations:
- One equation for the voltage states where $R_{f}$ == $R_{2}$
- One equation for the voltage states where both zeners to be conducting

• But how do I find when will Scenario 1 and Scenario 2 happen in respect to $u_{i}$? That is what I can't figure out. – Nebeski Dec 1 '17 at 17:09
• Set up a system of equations. One equation for scenario 0 (nonconducting), the second equation for scenario 1+2. Then you have to write the solution in the form "When V_negative_terminal>=some_voltage, the solution looks like so and so, When V_negative_terminal<=some_voltage, the solution looks like this, etc." – Miron V Dec 1 '17 at 17:26