# Superposition principle and Small signal analysis

first of all the superposition principle states that
$$[ f(ax_1 + bx_2) = af(x_1) + bf(x_2) ]$$ and this apply to all linear element like resistor and capacitor because if we considered the current to be the output it will match perfectly with the above equation $$\f\$$ is the ouput and $$\x\$$ is the input

non-linear elements like MOSFET doesn't apply to the above equation due to that $$\i-v\$$ characteristics is given by: $$i = \frac{1}{2} k_{n} (V_{ov})^{2}$$ we simply make linearization around a Q-point such that by small signal model : $$i_{D} = I + g_{m} (v - v_{0})$$

such that the $$\v=vgs+v_{0}\$$ is the total voltage and $$\v_{0}\$$ is the bias voltage

thinking about that linearized model this model doesn't apply to the superposition principle why is that ? consider $$\i_{D}\$$ to be the function $$\f\$$ and the $$\v\$$ to be $$\x\$$

if we apply $$\v\$$ =$$\v_{0}\$$ the current $$\i_{D}\$$=$$\I\$$ , this done by shorting the small signal voltage and applying only large signal voltage

going to the second voltage : if we only apply $$\vgs\$$ the result is equal:

$$i_{D}=I+g_{m}(vgs-v_{0})$$

1-which is totally wrong because that linearized model works only around the $$\v_{0}\$$ by shorting the $$\v_{0}\$$ to apply superposition it wont produce any current due to that model works only around that point if we short dc bias volt the dc is translated to zero and that small voltage is oscillating around zero dc which is far from linear curve which exist only around the dc bias point

2-also we cant apply superposition because superposition states that the sources must apply independtly and the $$\g_{m}\$$ depends on the dc bias voltage by shorting dc bias the gm will be have no definition

3- also the model for dc bias voltage is not the one for the small signal model which contradicts with the superposition assumption which states that the voltages apply to the same function $$\f\$$

so i think we don't really apply the superposition but rather we just calculating changes that came from the small signal voltage by translation of the axis around the dc point and makes a model for the curve by translation like taylor series we know that $$f=\bar{f} + \hat{f}$$ $$\ \hat{f}=k\hat{x}\$$ this is done by the translation of the axis around the dc-operating point to just calculate that changes alone and we previously knowing that $$\f=\bar{f} + \hat{f} \$$

• DC current "I" is the biasing operating point, not the small signal part.
– hana
Commented Jul 22 at 13:26
• @hana , I've stated that clearly when I said by applying Dc bias voltage the result current is $$I$$ Commented Jul 22 at 13:28
• I don't quite understand your question. Are you asking if superposition applies to the linearized small signal model? If that is the case, then yes, it does apply. If you're asking if the small signal model is obtained from the Taylor expansion, then it's correct. We make approximations by omitting higher-order terms and only keeping the linear part.
– hana
Commented Jul 22 at 13:46
• @hana, superposition States that I take every source independently and short the other one, if I took the small signal voltage and Short the Dc bias voltage, the circuit won't operate, that's why I've stated that the reasons why superposition won't be applicable Commented Jul 22 at 13:51
• Then I already answered in the second part above. It's not superposition, just an approximation which can be seen from the Taylor expansion.
– hana
Commented Jul 22 at 13:52

Quoting from this comment by OP Abdelrahman:

superposition States that I take every source independently and short the other one, if I took the small signal voltage and Short the Dc bias voltage, the circuit won't operate, that's why I've stated that the reasons why superposition won't be applicable

When we say "every source" in this context, we mean every small signal source. We don't mean the bias source can be changed or shorted.

If we have a source like

$$V(t) = V_0 + A_1 \cos(\omega_1 t) + A_2 \cos(\omega_2 t)$$

and $$\A_1\$$ and $$\A_2\$$ are both "small enough", then we can solve independently for the perturbations due to the signal at frequency $$\\omega_1\$$ and the signal at $$\\omega_2\$$, and recombine those solutions using superposition.

This opens up the possibility of using Fourier analysis to solve for the small-signal components of the response.

When we "remove" the DC bias sources (short the voltage sources or open the current sources) from an AC circuit model, we're doing that after linearizing the rest of the circuit (a step that depends very much on the DC sources), and with the understanding that their effect must be considered separately and re-combined in the final result if we want the complete time-domain solution for the circuit and not just the response to the perturbing signals.

• Why then most of electronics textbooks says that after linearizing the device around the dc bias , we make Dc-analysis then by we make Ac-analysis and short Dc bias and by superposition between DC and AC we add them together , isn't superposition does not hold by this claim because also after linearizing the circuit we can't short DC sources because if we do so the small signal source wont be able to turn on the circuit even in the linearized model because the linearized model is applicable around the dc bias point by shorting it , it does not hold any more (the linearized model) Commented Jul 22 at 15:21
• @Abdelrahman, The DC and AC solutions aren't combined by superposition. They're combined by perturbation analysis. Can you cite a specific text that claims that the combination of the AC and DC solutions is an example of superposition? Commented Jul 22 at 15:23
• After linearizing the circuit we can remove the DC sources because the nonlinear elements have been replaced by linear ones (that's what "linearizing" means). So they no longer require the DC source to be present in order to "turn on". Commented Jul 22 at 15:24
• Can you illustrate more how perturbation analysis allows us to combine the solutions ? also regarding the your last comment after linearizing element like MOSFET , the small signal current is equal to $g_{m}vgs$ and $g_{m}$ depend on the dc bias volt Commented Jul 22 at 15:29
• after linearizing the circuit if Dc and Ac solutions aren't combined by superposition , why then we short the Dc bias while calculating the small signal analysis , this procedure is very likely to being superposition , i'm not convinced with the superposition but i just want to know how is this done because short dc bias affects the $g_{m}$ , you asked me about the text , prof Behzad Razavi said we are doing superposition in his lectures explicitly Commented Jul 22 at 17:06

From your comments, it seems you misunderstood my explanation. I said that you can apply superposition to the small signal obtained by linearizing the circuit around the operating point.

However, your question seems to ask if the output can be calculated by the superposition of the DC biasing and the small signal around the DC biasing point.

So the answer is no, that is not true.

If the circuit is nonlinear, then you cannot apply superposition. We obtain the total result at the output as the sum of the DC operating point and the small signal through an approximation. We are neglecting all nonlinear parts by ignoring higher-order terms from the Taylor series. This result is just an APPROXIMATION and not superposition.

The image is taken from this.

Taylor series and the approximation:

• But in the textbook you linked it's written after linearizing we solve the circuit in 2 part first part to calculate the dc bias and the second part we short the DC bias and calculate the small signal response , isn't that a superposition ? Commented Jul 22 at 15:43
• No, superposition only applies to linear circuits, where it provides the exact result. In contrast, the approximation method mentioned above is used for non-linear circuits. This method approximates the result by ignoring the second, third, and higher-order terms in the Taylor series expansion.
– hana
Commented Jul 22 at 15:48
• Can you kindly explain to me what does the textbook means by saying that after making the MOSFET Linear we separate the analysis by 2 parts the first is the Dc analysis and the second is the small signal analysis and then combine them together , isn't that a superposition ? I know you replied to me that it is not superposition that separating the 2 analyses after linearizing the element , but can you explain how it is not superposition ? because that part is what confusing me i'm not convinced that it's superposition but it seems like superposition Commented Jul 22 at 15:59
• I explained this already, but you repeated the same question again, so I gave up. What the textbook mentions is just the Taylor series approximation, which I highlighted in pink in the image above.
– hana
Commented Jul 22 at 16:02
• to make sure that I understand your reasoning correctly , we calculate first the dc bias by dc sources and then change the axes around the dc bias point to calculate the linear term by Taylor expansion which is $\frac{df}{dx} \, \Delta x$ and while calculating the this term we short the DC sources because they do not contribute ? and the plus sign between them is not a superposition but indeed the plus from taylor does I understand correctly ? sorry for repeating the question but I did that because I did not really understand your explanation, Commented Jul 22 at 16:11

The bias point V-I curve is nonlinear so cannot be used with superposition. It can be added to the small signal ac voltage to obtain a result. But is is not superposition. The bias point can influence the small signal linearization.

If: $$I=kV^2_{ov}$$

then the small signal linear approximation at $$\Vbias_{ov}\$$ is:

$$i_{ac}=(2kVbias_{ov})v_{ac\_ov}$$

So the small signal gain depends on the bias point.

Superposition requires linear independence. Just being able to add the numbers together is insufficient.

• but in electronics textbooks we separate the DC analysis and small signal analysis after linearization and then add them together , isn't that a superposition ? Commented Jul 22 at 16:19
• It could be. But only if the ac and dc sources are linearly independent. The test is if you change the dc bias without the resulting ac amplitude changing or if the ac amplitude is changed without moving the bias point, then yes. But other wise no as I showed in my answer. Just adding them together is not sufficient to be considered superposition. Commented Jul 22 at 16:34
• but in elements Like MOSFET and BJT changing Dc bias point will results in change of the transconductance and accordingly the ac amplitude on the MOSFET will also change because it depends on the bias point , so this means adding analysis not a superposition , is that correct ? Commented Jul 22 at 16:50
• that is correct. Stated better, superposition cannot be applied.@Abdelrahman Commented Jul 22 at 17:09
• I get it ,even after linearizing MOSFET, the separation of analysis between dc and small signal and then add the results doesn't mean it's a superposition as you said but instead it means at this bias point we expand around it by Taylor series and since the total is equal to dc bias + ac( small signal) we just separate the circuit by it's small signal model to be able to just calculate the ac(small signal) in other words we change the axis to that dc point and just calculate the change by the new axis thats why we short DC by translation of axis the current we only see is due to small signal Commented Jul 22 at 17:40