# Voltage Regulators - Problem 3

#### PsySc0rpi0n

Joined Mar 4, 2014
1,446
Problem 3 to solve.

Qestions are:

a) Vout at R_Load = 10 Ω

b) Range of values for Rz if the Zener Power Dissipation = 250 mW.

c) Power Dissipation at both T1 and T1.

a) wrote a net loop equation as:

-Vz + Vbe1 + Vbe2 + Vout = 0 V
Vout = 5.6 V - 1.4 V = 4.2 V

So, as Vout is not affect by Vin, this is a voltage regulator!

b)

Here I have a question... To be thorough, shouldn't I evaluate 4 values for Rz?

I mean, firstly 2 values for (a) Rz for Vin = 10 V - 2 V = 8 V and (b) for Vin = 10 V + 2 V = 12 V, considering the Iz_min???

Then 2 more values for (c) Rz for Vin = 10 V - 2 V = 8 V and (d) for Vin = 10 V + 2 V = 12 V, considering the Iz_max???

Edited:

I have evaluated for 2 situations, which are Rz for Vin = 8V and for Iz_min which I found to be Rz = 523 Ω and for Vin = 12 V also for Iz_min which found to be Rz = 1.4kΩ

Last edited:

#### WBahn

Joined Mar 31, 2012
26,398
Problem 3 to solve.

View attachment 97069

Qestions are:

a) Vout at R_Load = 10 Ω

b) Range of values for Rz if the Zener Power Dissipation = 250 mW.

c) Power Dissipation at both T1 and T1.

a) wrote a net loop equation as:

-Vz + Vbe1 + Vbe2 + Vout = 0 V
Vout = 5.6 V - 1.4 V = 4.2 V

So, as Vout is not affect by Vin, this is a voltage regulator!

b)

Here I have a question... To be thorough, shouldn't I evaluate 4 values for Rz?

I mean, firstly 2 values for (a) Rz for Vin = 10 V - 2 V = 8 V and (b) for Vin = 10 V + 2 V = 12 V, considering the Iz_min???

Then 2 more values for (c) Rz for Vin = 10 V - 2 V = 8 V and (d) for Vin = 10 V + 2 V = 12 V, considering the Iz_max???

Edited:

I have evaluated for 2 situations, which are Rz for Vin = 8V and for Iz_min which I found to be Rz = 523 Ω and for Vin = 12 V also for Iz_min which found to be Rz = 1.4kΩ
Why use Iz_min for both limits? Doesn't Iz_max come into play at all?

Remember, the resistor has to be big enough so that the max power dissipation of the zener is not exceed (which will occur for Vin at which limit?) while not being so large that the zener current falls below it minimum allowed value (which will occur for Vin at which limit).

You say that you came up with Rz = 523 Ω for 8V and Iz_min. What does this mean? It means that if Rz is ANY LARGER than 523 Ω that you can't guarantee that the zener current will remain above Iz_min.

Fine.

But then you immediately come up with another result saying that Rz can be as large at 1.4 kΩ, which is considerably LARGER than the upper limit you already established.

You will also be much better off if you start letting the math do the work for you. Set up your equations to mirror what you are trying to do.

For instance:

We want Iz to always be between Iz_min and Iz_max.

So set up an equation (or an inequality, in this case) that reflects that:

$$I_{z_{min}} \; \leq \; I_z \; \leq \; I_{z_{max}}$$

So now what is the expression for I_z?

$$I_z \; = \; \frac{\( V_{in} - V_{z}$$}{R_{z}}
\)

So plug this in and solve for Rz

$$I_{z_{min}} \; \leq \; \frac{\( V_{in} - V_{z}$$}{R_{z}} \; \leq \; I_{z_{max}}
\)

Remember that when to take the inverse (reciprocal) of an inequality, the direction of the inequality reverses.

$$\frac{1}{I_{z_{max}}} \; \leq \; \frac{R_{z}}{\( V_{in} - V_{z}$$} \; \leq \; \frac{1}{I_{z_{min}}}
\)

Multiply both sides by Vin-Vz

$$\frac{\( V_{in} - V_{z}$$}{I_{z_{max}}} \; \leq \; R_{z} \; \leq \; \frac{$$V_{in} - V_{z}$$}{I_{z_{min}}}
\)

At this point you can ponder which limit of Vin to put into each side, or you can find two ranges, one for Vin at its upper limit and one for Vin at its lower limit. Then just realize that you can only choose values of Rz that satisfy the inequalities for BOTH limits.

#### PsySc0rpi0n

Joined Mar 4, 2014
1,446
Morning...

To be honest, the advantage you have is that you already know what are you calculating when you plug in the 10 V - 2 V or the 10 V + 2 V value. I mean you already know that when you plug in the 8 V value, you'll be evaluating, let's say, Rz min... And vice-versa, when you use the 12 V value you already know that you're evaluating, let's say, the Rz max, therefore you also know, in the first place, that the other 2 values that I was thinking about to calculate are not needed because they are already "covered" by these 2 values... That is the detail I still cannot assess due to lack of intuition!

I mean, in this problem, I can't figure out, for instance, if I use the 8 V value, if I'm going to find Rz min or Rz max. So I always need to evaluate all possibilities and at the end choose the lowest and the highest values of Rz and look into the 4 situations to check which ones correspond to the 2 extreme values found!

#### Bordodynov

Joined May 20, 2015
2,733
Small revision:
(Vin_max-Vz)/Iz_max < Rz < (Vin_min-Vz)/Iz_min

#### WBahn

Joined Mar 31, 2012
26,398
But the highest and lowest values of Rz that you calculate will generally NOT be the correct limiting values.

You develop your intuition by asking yourself what it is that you are trying to determine and then asking yourself how changes in the parameters you have to work with affect that.

It is not unreasonable to be able to figure out that increasing the voltage drop across a resistor will increase the current through it while decreasing the voltage drop across it will decrease the current through it. Right?

Once you have chosen a value of Rz, then it should be fairly easy to see that using the supply voltage that puts the smallest voltage drop across Rz will result in the least amount of current through it. This then corresponds to the situation in which you are trying to ensure that at least some minimum current passes through the zener.

Similarly, for a give value of Rz, it should be fairly easy to see that using a supply voltage that puts the largest voltage drop across Rz will result in the greatest amount of current through it. This then corresponds to the situation in which you are trying to ensure that no more than some maximum current passes through the zener.

Couple this with careful, systematic development of your equations and inequalities and you will be well on your way.

#### WBahn

Joined Mar 31, 2012
26,398
Small revision:
(Vin_max-Vz)/Iz_max < Rz < (Vin_min-Vz)/Iz_min
I left it as it was because I didn't want to take that next step right now. So I left it as two dual inequalities that have to be evaluated at each limiting value of Vin. Then I was going to show him how to systematically go from where I left it to the result you have here.

#### PsySc0rpi0n

Joined Mar 4, 2014
1,446
Yes, I understand that... But more current through Rz means less current through the zener, no? And vice-versa, I guess!

If this is correct, then, the lower Rz, higher is I_Rz and higher is Iz, no??? because as it is coming more current into that node from Rz, more current there is to split into Iz and transistor base, right?

And the higher Rz, lower is I_Rz and lower will be Iz and lower will be the transistor base current.

And at the end, means that the highest Rz value is to set the minimum current that can flow through the zener which allows it to work as a zener and regulate voltage at it's terminals.

And the lowest Rz sets the maximum current flowing through the zener, avoiding it to blow up or burn!

But this is regarding currents and resistor values. And this is already 'mind tricking' because we need to be jiggling back and forth with the words high and low or max and min or whatever, and that leads, a lot of times, to confusion!

Now I still need to think about what input voltages 'creates' higher and lower Rz currents!

#### PsySc0rpi0n

Joined Mar 4, 2014
1,446
I left it as it was because I didn't want to take that next step right now. So I left it as two dual inequalities that have to be evaluated at each limiting value of Vin. Then I was going to show him how to systematically go from where I left it to the result you have here.
I can do the math... I already got that symbolic result! I was replying in the meantime so I have not evaluated the values yet!

#### WBahn

Joined Mar 31, 2012
26,398
I can do the math... I already got that symbolic result! I was replying in the meantime so I have not evaluated the values yet!
Ah, good.

So now show, systematically and rigorously, how to go from

$$\frac{\( V_{in} - V_{z}$$}{I_{z_{max}}} \; \leq \; R_{z} \; \leq \; \frac{$$V_{in} - V_{z}$$}{I_{z_{min}}}
\)

in which you have to evaluate two sets of limits and then combine them, to

$$\frac{\( V_{{in}_{max}} - V_{z}$$}{I_{z_{max}}} \; \leq \; R_{z} \; \leq \; \frac{$$V_{{in}_{min}} - V_{z}$$}{I_{z_{min}}}
\)

(Note that MimeTeX does a lousy job of typesetting double subscripts well)

#### PsySc0rpi0n

Joined Mar 4, 2014
1,446
Ah, good.

So now show, systematically and rigorously, how to go from

$$\frac{\( V_{in} - V_{z}$$}{I_{z_{max}}} \; \leq \; R_{z} \; \leq \; \frac{$$V_{in} - V_{z}$$}{I_{z_{min}}}
\)

in which you have to evaluate two sets of limits and then combine them, to

$$\frac{\( V_{{in}_{max}} - V_{z}$$}{I_{z_{max}}} \; \leq \; R_{z} \; \leq \; \frac{$$V_{{in}_{min}} - V_{z}$$}{I_{z_{min}}}
\)

(Note that MimeTeX does a lousy job of typesetting double subscripts well)
OFFTOPIC:

@WBahn do you use the tag \displaystyle {}, sometimes it helps the text formating and the visual outcome!

#### WBahn

Joined Mar 31, 2012
26,398
OFFTOPIC:

@WBahn do you use the tag \displaystyle {}, sometimes it helps the text formating and the visual outcome!
No, I don't. Does MimeTeX even support that tag?

I haven't been able to find a reference for just what subset of TeX that MimeTeX supports. So I end up having to do trial and error and there's a limit to what I'm will to play with. I also found, sadly, that the subset supported by vBulletin is not quite the same as that supported by XenForo, so I've see quite a few equations from before the transition that no longer render properly.

If you could post a couple of examples that render well here, then I would be happy to get them over into the LaTeX sticky so that we would have it for future reference.

#### PsySc0rpi0n

Joined Mar 4, 2014
1,446

$$\displaystyle {\frac{V_{in}-V_{z}}{I_{z_{max}}}\leq R_{z}\leq \frac{V_{in}-V_{z}}{I_{z_{min}}}}$$

I'm still not sure which Vin value to use on both sides of the inequation.

For instance, the left side, I think, is for the lowest Rz value, meaning that more current will flow through it. This said, it means that to have more current flowing through the resistor, I need to have more voltage applied to the Rz terminals... More voltage means that the difference between Vin and Vz must be greater, which results in 12 V - 5.6 V... Is this correct?

#### PsySc0rpi0n

Joined Mar 4, 2014
1,446
No, I don't. Does MimeTeX even support that tag?

I haven't been able to find a reference for just what subset of TeX that MimeTeX supports. So I end up having to do trial and error and there's a limit to what I'm will to play with. I also found, sadly, that the subset supported by vBulletin is not quite the same as that supported by XenForo, so I've see quite a few equations from before the transition that no longer render properly.

If you could post a couple of examples that render well here, then I would be happy to get them over into the LaTeX sticky so that we would have it for future reference.
Well, looks like the issue is deeper than what I thought... I'm more familiar with TEX/LATEX... And sometimes that tag helps to format integral symbols, fractions, superscripts, subscripts, matrix and probably more, in terms of size (height). But about the forum back OS, so to speak, limitations, I'm not that familiar. Not with vBulletin neither XenForo!

And apparently, my equation rendering is not that different from yours, using that tag!

#### WBahn

Joined Mar 31, 2012
26,398
Well, looks like the issue is deeper than what I thought... I'm more familiar with TEX/LATEX... And sometimes that tag helps to format integral symbols, fractions, superscripts, subscripts, matrix and probably more, in terms of size (height). But about the forum back OS, so to speak, limitations, I'm not that familiar. Not with vBulletin neither XenForo!

And apparently, my equation rendering is not that different from yours, using that tag!
Yeah, I've found a number of tags that MimeTeX will recognize but appears to ignore (which I guess is better than not rendering anything or rendering a bunch of question marks).

What's weird is that I ran across a really obscure LaTeX code that demonstrated doing graphic by drawing a smiley face and when I posted that here, it worked! So that got me stoked to dig into it deeper, but I quickly found that a lot of stuff doesn't work and, like I said before, I just decided it wasn't worth my time.

#### PsySc0rpi0n

Joined Mar 4, 2014
1,446
Yeah, I've found a number of tags that MimeTeX will recognize but appears to ignore (which I guess is better than not rendering anything or rendering a bunch of question marks).

What's weird is that I ran across a really obscure LaTeX code that demonstrated doing graphic by drawing a smiley face and when I posted that here, it worked! So that got me stoked to dig into it deeper, but I quickly found that a lot of stuff doesn't work and, like I said before, I just decided it wasn't worth my time.

Just an example:

With '\displaystyle {}' tag

$$\displaystyle {\Large\frac{dv^m}{ds}=-\Gamma^m_{oo}v^{o^2} =-g^{mn}\Gamma_{noo}v^{o^2}=\frac12g^{mn}g_{oo,n}v^{o^2}}$$

Without '\displaystyle {}':

$$\Large\frac{dv^m}{ds}=-\Gamma^m_{oo}v^{o^2} =-g^{mn}\Gamma_{noo}v^{o^2}=\frac12g^{mn}g_{oo,n}v^{o^2}$$

It's pretty much the same.
This code was taken from:
http://www.forkosh.com/mimetexmanual.html

#### PsySc0rpi0n

Joined Mar 4, 2014
1,446
Ok, getting back to the thread...

Have you read my post #12? To confirm or deny my thought?

#### WBahn

Joined Mar 31, 2012
26,398

$$\displaystyle {\frac{V_{in}-V_{z}}{I_{z_{max}}}\leq R_{z}\leq \frac{V_{in}-V_{z}}{I_{z_{min}}}}$$

I'm still not sure which Vin value to use on both sides of the inequation.

For instance, the left side, I think, is for the lowest Rz value, meaning that more current will flow through it. This said, it means that to have more current flowing through the resistor, I need to have more voltage applied to the Rz terminals... More voltage means that the difference between Vin and Vz must be greater, which results in 12 V - 5.6 V... Is this correct?

The rigorous way (well, there's more than one such way) is to say that since the range of values for Rz must satisfy the Iz limits for all values of Vin, our lower limit for Rz will be whichever value of Vin produces a higher floor for Rz. Similarly, the upper limit will be whichever value of Vin produces a lower ceiling for Rz.

Hence:
$$\text{MAX} \( \frac{V_{{in}_{max}} - V_{z}}{I_{z_{max}}} \; , \; \frac{V_{{in}_{min}} - V_{z}}{I_{z_{max}}}$$ \; \leq \; R_{z} \; \leq \; \text{MIN} $$\frac{V_{{in}_{max}} - V_{z}}{I_{z_{min}}} \; , \; \frac{V_{{in}_{min}} - V_{z}}{I_{z_{min}}}$$
\)

By inspection, we can see that the left side has to use V_in_max and that the right side has to use V_in_min.

If this wasn't obvious, we could determine it by using another inequality to determine which limit.

For instance, for the left side we might set it up as follows:

We use V_in_max if

$$\frac{V_{{in}_{max}} - V_{z}}{I_{z_{max}}} \; \geq \; \frac{V_{{in}_{min}} - V_{z}}{I_{z_{max}}}$$

which reduces to

$$V_{{in}_{max}} \; \geq \; V_{{in}_{min}}$$

Since this is a true statement, that means that we do use V_in_max for this side.

We would do something analogous for the right side.

Hence our MAX() and MIN() expressions reduce to

$$\frac{V_{{in}_{max}} - V_{z}}{I_{z_{max}}} \; \leq \; R_{z} \; \leq \; \frac{V_{{in}_{min}} - V_{z}}{I_{z_{min}}}$$

#### PsySc0rpi0n

Joined Mar 4, 2014
1,446
Ok, I like better the 1st approach. Is more intuitive and is also more into Electronic language! I guess.. The other approach is more mathematical!

#### WBahn

Joined Mar 31, 2012
26,398
Both approaches have there place and you need to be able to work with either or, in most cases, an appropriate mix of the two. As you get to more complex circuits and problems, you need to rely more and more on sound mathematical models of both the circuits and the problems.