# voltage transfer function for Zener diode clipper

#### tbaumeister

Joined Mar 13, 2011
3
Hello everybody,
I'm new to this forum. I need a formula for the voltage transfer function of a simple voltage limiting circuit consisting of a Zener diode and a resistor, i. e. a mathematical expression for the clipped output voltage (across the Zener diode) as a function of the input voltage. Does anyone know where to find this?

#### Papabravo

Joined Feb 24, 2006
21,225
Transfer functions have a really tough time with non-linear devices. But you knew that -- Right?

Oh...maybe you meant the I-V characteristic. The Shockley Diode Equation models the forward and reverse bias regions but not the breakdown region. Actually I have no idea what you're looking for.

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#### t_n_k

Joined Mar 6, 2009
5,455
One option would be to assume a simple model for the zener diode comprising a series zener dynamic resistance rz and zener voltage Vz.

From this and the series ballast resistor R, you could develop a relationship between Vout, Vin, R, Vz and rz

i.e

Vout=f(Vin,R,Vz,rz)

#### Papabravo

Joined Feb 24, 2006
21,225
Is that really a transfer function in the sense that it describes a linear differential equation?

#### Ron H

Joined Apr 14, 2005
7,063
Differential?

#### Papabravo

Joined Feb 24, 2006
21,225
@Ron, I'm really confused by your posts.

I don't disagree that linear differential equations include the differential operator. Do you have a problem with my statement that a transfer function describes a linear differential equation? Do you also have a problem with the assertion that only a linear differential equation can have a transfer function. Lastly, since the zener diode has a non-linear characteristic there can be no transfer function for it.

#### Ron H

Joined Apr 14, 2005
7,063
Is that really a transfer function in the sense that it describes a linear differential equation?
OK, I finally get your point. I was looking at the OP's request for a "transfer function" as meaning a DC transfer function, which, of course, has no time derivatives.
Sorry for the confusion I caused.

#### t_n_k

Joined Mar 6, 2009
5,455
Is that really a transfer function in the sense that it describes a linear differential equation?
I was thinking in terms of a broader definition in which the transfer function provides a means of formulating the output of a device for a known input or range of inputs.

For instance, an ideal (linear) voltage amplifier simply multiplies the input voltage by a known factor.

Control systems frequently include non-linear components (e.g. dynamic limiters) and these can usually be accommodated in the overall analytical process on the basis of their input/output relationship.

#### Papabravo

Joined Feb 24, 2006
21,225
IMHO the "broader" definition is not appropriate to the situation. I think if you google "transfer function" you get the narrow definition. I have heard of the curves which describe a semiconductor device called "transfer characteristics" or "I-V" curves or something else, but nobody I know refers to them as "transfer functions". If you can cite such a usage in the literature I will be surprised and of course yield the floor.

#### Ron H

Joined Apr 14, 2005
7,063
IMHO the "broader" definition is not appropriate to the situation. I think if you google "transfer function" you get the narrow definition. I have heard of the curves which describe a semiconductor device called "transfer characteristics" or "I-V" curves or something else, but nobody I know refers to them as "transfer functions". If you can cite such a usage in the literature I will be surprised and of course yield the floor.
Yeah, I agree that a nonlinear circuit will not have a transfer function, but a voltage divider, or an ideal amplifier, does, IMHO, have a transfer function. I guess you could think of it as H(s), where there are no terms containing s in the transfer function. If applying the term "transfer function" to such circuits offends your sensibilities, then I guess you could call it gain, or attenuation, but I have heard the term "transfer function" applied to DC circuits all my life, and I'm OLD.

#### Vahe

Joined Mar 3, 2011
75
Hello everybody,
I'm new to this forum. I need a formula for the voltage transfer function of a simple voltage limiting circuit consisting of a Zener diode and a resistor, i. e. a mathematical expression for the clipped output voltage (across the Zener diode) as a function of the input voltage. Does anyone know where to find this?
I think the original poster meant a "voltage transfer characteristic" which is just Vout vs. Vin. A transfer function would only make sense in the frequency domain and it would have to be given as a ratio Vout/Vin. However, the original poster said that he was looking for a "mathematical expression for the clipped output voltage as a function of the input voltage" so I think this would just be a piecewise linear function.

The voltage transfer characteristic would be
$$V_{out} = \begin{cases} V_z, & \text{for } V_{in} \ge V_z \\ V_{in}, & \text{for } -V_f < V_{in} < V_z \\ -V_f & \text{for } V_{in} \le -V_f \end{cases}$$
where $$V_z$$ is the zener voltage of the zener diode and $$V_f$$ is the forward voltage of the zener diode. I am assuming that the cathode of the zener diode is at the positive polarity of the output voltage and the anode is tied to the negative side of the input voltage.

Cheers,
Vahe

#### Ron H

Joined Apr 14, 2005
7,063
Maybe I should refer to voltage dividers and ideal amplifiers as having a "voltage transfer characteristic" rather than a "transfer function". I never realized that "transfer function" had such a restrictive definition, with such a generic-sounding name.

#### t_n_k

Joined Mar 6, 2009
5,455
IMHO the "broader" definition is not appropriate to the situation. I think if you google "transfer function" you get the narrow definition. I have heard of the curves which describe a semiconductor device called "transfer characteristics" or "I-V" curves or something else, but nobody I know refers to them as "transfer functions". If you can cite such a usage in the literature I will be surprised and of course yield the floor.
Point taken.

I did find this which may be relevant .... See the 4th "definition"

#### t_n_k

Joined Mar 6, 2009
5,455
Or perhaps even this .....

http://www.its.bldrdoc.gov/fs-1037/dir-037/_5543.htm

The key point I guess is that the transfer function need only define the output in terms of the input.

I think Vahe offered the best interpretation of how one would describe the "voltage transfer function" for the particular question. My suggestion didn't allow for inputs below the zener voltage.

I'm not particularly wedded to the idea of the transfer function being anything other than what is generally accepted by most informed folk such as the good Papabravo. I have only posted the links since the notion of transfer function might possibly be open to other interpretations. It's not a big issue for me.

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#### Papabravo

Joined Feb 24, 2006
21,225
Maybe I should refer to voltage dividers and ideal amplifiers as having a "voltage transfer characteristic" rather than a "transfer function". I never realized that "transfer function" had such a restrictive definition, with such a generic-sounding name.
A voltage divider, a one pole RC low pass, and the ideal amplifier all have transfer functions because they are linear components. The zener diode does not have a transfer function because it is not a linear component.

In particular the voltage divider has a constant transfer function which is not a function of frequency. At least not until the lead inductance becomes significant.

#### tbaumeister

Joined Mar 13, 2011
3
Hello Papabravo, t_n_k, Ron H, Vahe,

thank you all for your quick replies. They show me that I have to clarify my problem:

- Yes, I know that transfer functions for non-linear devices are not easy!

- I didn't mean the I-V characteristic of a Zener diode. This can be found in many places; I measured it and derived formulas for the particular Zener diode I measured. They work very well for both forward and backward voltages. But the I-V characteristic doesn't solve my problem: it just tells you the voltage drop across the Zener diode _IF_ you know the current passing through it, or vice versa, but it does _NOT_ tell you the current that will flow when you apply a known input voltage to the series of resistor and Zener diode.
The same argument holds if you know the dynamic (i. e. voltage dependent) resistance of the Zener diode. No matter what you try, you always end up with 2 equations for 3 unknown variables.

>> One option would be to assume a simple model for the zener diode comprising a series zener dynamic resistance rz and zener voltage Vz.
The model I am looking for must not be too simple (e. g. rz passing from infinity to zero at vz as in Vahe's contribution) since I want to use it for the linearization of a data logger where the input has to be protected from overshoot.

- The voltage transfer function that I am looking for could be the solution of a (not necessarily linear) differential equation - as long as this is solvable under the appropriate boundary conditions. I am indeed interested in a DC transfer function, so time should play no role in it.
But time derivatives are not the only derivatives that characterize a possible differential equation for this problem.

- Like t_n_k, what I had in mind was a "broader definition in which the transfer function provides a means of formulating the output of a device for a known input". This is what I learned from quadrupole theory.

- If you restrict a voltage transfer function to a _linear_ differential equation, you certainly have a problem describing many circuits.

- I don't think that "a transfer function would only make sense in the frequency domain". Perhaps we all could find an agreement of terms by using the term "voltage transfer characteristic" for what I am looking for. Please forgive me if I caused a confusion; this may be due to the fact that I am not a native English speaker.

- I checked the link "http://www.google.com.au/search?hl=e...H&ved=0CBUQkAE" given by t_n_k. Unfortunately, I couldn't access the further link "wwwatnf.atnf.csiro.au/aips%2B%2B/docs/glossary/t.html" which sounded promising.

I wonder why such an appearantly common problem hasn't been solved by anyone.

#### Ron H

Joined Apr 14, 2005
7,063
Is the data you are logging really wideband? If not, you could consider an active limiter, such as an ideal diode (diode and op amp in a feedback loop). Then you don't have to worry about linearizing your input.

#### tbaumeister

Joined Mar 13, 2011
3
Hello Papabravo, t_n_k, Ron H, Vahe,

please forget what I wrote yesterday, please forget the whole thread: I found the solution of my problem. It is really simple: the known current/voltage characteristic of the Zener diode leads to 2 equations for 3 unknown variables, and I was just too stupid to see that this is enough to establish a relationship between input and output voltage of the clipping circuit. I derived a formula for the input voltage as a function of the output voltage. This is what I need, and it fits the measured data very well.
This equation is somewhat lengthy and cannot be solved for the output voltage, but that doesn't bother me right now.
Ron H is perfectly right with his suggestion to use a diode/op amp combination in order to avoid the linearizing problem, but I prefer to use the Zener diode approach for two reasons: first, it is easier for me to do the linearizing by software than to blow up the circuit, and second, the Zener diode method implies some amplitude compression which allows for a wider amplitude range given the limited A/D converter resolution.

Thanks anyway to all contributors! I excuse myself for bothering you about this simple problem. I'm glad to have found this forum where real experts are at work, and I'm ready to help other forum members in compensation.