# Help me calculate output impedance

#### GhostLoveScore

Nov 27, 2016
71
I've been reading on amplifier's output impedance. It's common collector amplifier. Take a look

Let me first warn you that circuit in red frame has not been calculated, meaning, I didn't calculate base resistor divider for some specific bias. I just put them there because for what I want to know they doesn't matter.

I want to know what is output impedance of circuit on the right, in red frame, at emitter. I read that it's Rsource/beta

But in this case what is Rsource? Is it output impedance of circuit on the left? If that's the case then Rsource is 330 ohms? Then if beta for 2n2222 is 100, output impedance of circuit on the right should be 3.3 ohms?

On the other hand, R1 could be interpreted as a load resistor on circuit on the left and then output impedance of circuit on the left would be Rc||Rl(R1)?

Moderator
Jan 21, 2010
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Nov 27, 2016
71

#### (*steve*)

##### ¡sǝpodᴉʇuɐ ǝɥʇ ɹɐǝɥd
Moderator
Jan 21, 2010
25,508
I scrolled through the first one I saw and got the answer.

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Jan 21, 2010
25,508

#### LvW

Apr 12, 2014
604
But in this case what is Rsource? Is it output impedance of circuit on the left? If that's the case then Rsource is 330 ohms? Then if beta for 2n2222 is 100, output impedance of circuit on the right should be 3.3 ohms?

At first, using approximations one should know if such a simplification is allowed or not. The explaining text (right from the transistor) gives an expression fo the output impedance which is a rough approximation.
The correct formula is:
r(out)=RE||[1/gm + Rsource/(1+beta)]
With: transconductance gm=Ic/Vt (Ic: collector current, Vt: temperature voltage).
The dynamic (differential) signal resistance 1/gm at the E node will be much larger than the 3.3 ohms you have calculated (dependent on the DC current).
It is correct that Rsource is the parallel combination of R1 and R2 and the signal output resistance of the previous stage.

Remark: In the document - linked by Steve - appears the base-emitter diff. resistance rbe. Please note that this value is in most cases unknown. However, the ratio rbe/beta is identical to the transconductance 1/gm as contained in the formula given above. This value can easily found using the known DC current Ic.

Last edited:

#### GhostLoveScore

Nov 27, 2016
71

Yes, that one. Again, they use Rs (Rsource) that's of known value, but that's what I don't know. I already knew that.

#### GhostLoveScore

Nov 27, 2016
71
It is correct that Rsource is the parallel combination of R1 and R2 and the signal output resistance of the previous stage.

Ah, thanks, that's around 320ohms.

#### Ratch

Mar 10, 2013
1,098
I've been reading on amplifier's output impedance. It's common collector amplifier. Take a look

View attachment 31338

Let me first warn you that circuit in red frame has not been calculated, meaning, I didn't calculate base resistor divider for some specific bias. I just put them there because for what I want to know they doesn't matter.

I want to know what is output impedance of circuit on the right, in red frame, at emitter. I read that it's Rsource/beta

But in this case what is Rsource? Is it output impedance of circuit on the left? If that's the case then Rsource is 330 ohms? Then if beta for 2n2222 is 100, output impedance of circuit on the right should be 3.3 ohms?

On the other hand, R1 could be interpreted as a load resistor on circuit on the left and then output impedance of circuit on the left would be Rc||Rl(R1)?

For your circuit, rs = source resistance, and rb = r1||r2 .

Applying the PIT below, rout is calculated.

Note that rout does not include the value of re.

Ratch

#### LvW

Apr 12, 2014
604
Ratch - I think, your equations are not easy to read.(and to understand)
* Why dont you distinguish between static (R) and differential (r) resistances ?
* In the transfer function, what is rs (with respect to the given circuit)?
* T=1 for rs=0 ?
* I doubt if your expression for rout is correct (re=RE ?).

Last edited:

#### GhostLoveScore

Nov 27, 2016
71
The correct formula is:
r(out)=RE||[1/gm + Rsource/(1+beta)]

Can you explain where does this formula comes from? I've seen few variations on my formula, but I've never seen this one.

#### LvW

Apr 12, 2014
604
In the link which was given by Steve (post #5), you will find the following expression:

This is the output resistance without the external ohmic resistor RE.
This can be writte as
rout=rbe/beta +Rs/beta.

We knbow that rbe/beta=1/gm=Vt/Ic.
Hence, rout=1/gm +Rs/beta.

(I have explained this already at the end of my post#6)

It is very advantageous to use this last form because we do not know the value of the base-emitter resistance rbe.
Also the value of beta is known only with very large tolerances. But the RATIO rbe/beta is known because it is the slope of the transfer function Ic=f(Vbe). This slope is the transconductance gm which can easily be found because of the exponential function. Remember that the BJT is - physically spoken - a voltage controlled device.

The approximation error (beta+1) instead of "beta" can be neglected.
OK?

#### Ratch

Mar 10, 2013
1,098
Ratch - I think, your equations are not easy to read.(and to understand)
* Why dont you distinguish between static (R) and differential (r) resistances ?
* In the transfer function, what is rs (with respect to the given circuit)?
* T=1 for rs=0 ?
* I doubt if your expression for rout is correct (re=RE ?).

You have a point. I should have used capital letters to designate the external base resistors (Rb), the external emitter resistor (Re), and source resistance (Rs). My analysis disregards the internal resistances of the transistor diodes (rb and re) because they are usually small compared to the external circuit resistances.

Ratch

#### Ratch

Mar 10, 2013
1,098
In the link which was given by Steve (post #5), you will find the following expression:

This is the output resistance without the external ohmic resistor RE.
This can be writte as
rout=rbe/beta +Rs/beta.

We knbow that rbe/beta=1/gm=Vt/Ic.
Hence, rout=1/gm +Rs/beta.

(I have explained this already at the end of my post#6)

It is very advantageous to use this last form because we do not know the value of the base-emitter resistance rbe.
Also the value of beta is known only with very large tolerances. But the RATIO rbe/beta is known because it is the slope of the transfer function Ic=f(Vbe). This slope is the transconductance gm which can easily be found because of the exponential function. Remember that the BJT is - physically spoken - a voltage controlled device.

The approximation error (beta+1) instead of "beta" can be neglected.
OK?

I believe that equation will have to be modified if an external base resistance (Rb) is present.

Ratch

#### LvW

Apr 12, 2014
604
I believe that equation will have to be modified if an external base resistance (Rb) is present.
Ratch
I think, everything is OK (except "beta+1" instead of "beta") if we consider Rs to be the effective (resulting) source resistance at the base node.

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