I would like to discuss the "Miller effect" seen in amplifier circuits. I must say, I disagree with the terminology used to describe the Miller effect that is presented nearly universally. For example: "the Miller effect is the phenomenon wherein the equivalent feedback capacitance is increased by a factor of (gain+1) in an amplifier circuit." Or "the Miller effect is an increase in the apparent (gate to drain) capacitance compared to the real one due to a feedback effect (from the drain to the gate)" (from Microwaves101). Time and time again, I see explanations like this, and I wish we could use better terminology.

To be clear, my personal understanding of the Miller effect is that the capacitance for example between the base and collector in a CE BJT amplifier will always be the same. The voltage across the capacitance is what we observe to dramatically increase. While you can derive a capacitance from the voltage drop across the capacitance (Z = v/i) and derive a value for capacitance based on that, you're only comparing the impedance seen when the base-collector voltage is equal to the input voltage (collector held constant, capacitance is as expected) to the impedance seen when the base-collector voltage is vin + (-1 - gain)*vin i.e. when the collector is inverted and amplified with respect to the input voltage.

This is not a clear way to explain the effect, which is in fact quite simple. If the collector is the base voltage amplified and inverted, then the voltage across the collector-base capacitance will obviously be (gain + 1) times higher than if the collector was grounded. A helpful image:

In the top circuit, you see a transistor amplifier. A small input increase results in a large output decrease, hence the total voltage is the difference between these and the voltage across the capacitor is boosted by (gain + 1). And thus we have the Miller effect - boosted voltage across the same impedance causes more current to flow through the impedance. The bottom image shows a cascode, which circumvents the Miller effect by using a second transistor, with its gate grounded (at signal freqs). Since the emitter is tied to the base directly through re', it too is held constant i.e. signal ground and therefore so is the collector of the bottom transistor. So the bottom capacitance has only vin across it, and the top capacitance has vout across it. This clearly shows the Miller effect and the cascode amplifier without using the phrase "amplified capacitance" which there is not.

I do understand the temptation of calling it "effective capacitance" and from an engineer's perspective it's an interesting quantity. But it hides the simple fact that the Miller effect is when there is a higher voltage across the same impedance i.e. there is higher current which, when compared to a grounded collector, is (gain + 1) times greater. Mathematically speaking you may perfectly well use this terminology, but it is akin to saying that objects with mass m have an equivalent mass m' when it is accelerated (gain) times compared to the original. Yes, you could say that, thanks to F = ma, but it is misleading.

I am not expressing my opinion without expecting... feedback. (Get it?) If you disagree with this explanation then by all means point out the issues with it. But I know that the Miller effect is often very confusing for people, and it is because the idea of "equivalent capacitance increasing" is only used in rare cases, such as this.

Sam

 

If you prefer your explanation as to what is actually happening, that's fine.
I find that viewing it as an equivalent increase in capacitance, makes it easier to understand its effect on the amplifier response.

 

From what I can gather your discussion negates the need for the theory for the Miller effect since you can provide some different circuit where the Miller effect is not present.

Ummm, what is the sense in that? One would not be looking to negate the Miller effect if one is not aware of it in the first place.

Miller effect capacitors pop up in many surprising places. The intent of the equivalent capacitance is to show that a seemingly minor capacitance can have a major influence in a circuit since it may act as a capacitor hundreds of times the expected value.

John Milton Miller published the original paper on this effect in 1920 when he noticed the phenomena in a vacuum tube amplifier. That we are still discussing this nearly 100 years later shows how fundamental this concept is.

 

I would not say I'm "negating" the theory at all. Rather I'm saying that the way we describe the theory is misleading and needlessly so. The cascode is just there to show how simple the Miller effect really is, and how simple it can be to eliminate it if you know what's going on. Moreover, I like the idea that the tiny capacitance is amplified to a bigger capacitance, and that this is almost more intuitive when considering the effect it has on the amplifier. But the concept falls apart because the capacitance is not changing, and anyone who is unfamiliar with the Miller capacitance will have a difficult time understanding without a proper teacher to explain the fine details of the theory.

After all, with the "amplified capacitance" idea, it's not immediately obvious how to fix it. Sometimes it feels as though people teach the amplified capacitance idea because they like the ingenuity that came along with that idea, and they like thinking that way. But they pay little mind to whether or not that is the complete story, and they expect students to simply "keep up" or worse, just "accept it". The biggest crime in education is making simple things complicated, and making complicated things simple. If you leave something out when teaching someone, they will not know it. This applies particularly to things like the Miller effect where the same old explanation gets reused and students continue to be confused until they figure it out, and then they teach it the same way as they learned, confusing more students.

Students of electronics do not think in terms of "amplified capacitance" and so we should not describe the effect as such. The idea of boosted referred-to-input impedance is not found in many other places in all of analog electronics. If we are trying to make electronics and analog design accessible, there is no need to use clever explanations of simple effects. Please tell me if I am out of line, or if you still have reservations.

 

I will make a small observation. The cascode scheme does not remove the Miller effect. The gain of the first stage is 1. The effective increase in capacity in (1 + 1) = 2 times. There is one more harmful effect. With a fixed supply voltage, the voltage at the first transistor decreases, which increases the parasitic capacitance of the collector!

 

Again, I would recommend not using the terminology "effective increase in capacity" for understand-ability reasons. As for whether or not the cascode removes the Miller effect, I would say that you're right the Miller effect is not actually completely removed. The boost in current through the capacitance though is slightly less than 2.

The second point makes a strong argument for avoiding "increasing capacitance" talk when referring to the Miller effect: I cannot tell if you mean the capacitance Ccb literally increases due to junction biasing or if the input referred "Miller" capacitance is increasing i.e. if the current through the capacitor is increasing.

 

How would you define the cause, of the Miller effect?

 

I can do my best to write a definition.
The Miller effect: When a capacitance is shunting an input signal and its amplified output, causing a higher signal current to flow through the capacitance than if it was shunting the input and signal ground. The increase in current through the capacitance lowers the small signal input impedance of the amplifier as frequency increases.

Thoughts?

 

Again, I would recommend not using the terminology "effective increase in capacity" for understand-ability reasons. As for whether or not the cascode removes the Miller effect, I would say that you're right the Miller effect is not actually completely removed. The boost in current through the capacitance though is slightly less than 2.

How any particular circuit responds has no bearing on the underlying theory.

The second point makes a strong argument for avoiding "increasing capacitance" talk when referring to the Miller effect: I cannot tell if you mean the capacitance Ccb literally increases due to junction biasing or if the input referred "Miller" capacitance is increasing i.e. if the current through the capacitor is increasing.

Neither.

The Miller effect quite clearly describes an increase in the equivalent input capacitance when used in an inverting voltage amplifier configuration.

How would you define the cause, of the Miller effect?

Wikipedia has as good an explanation as any. I would refer you there, but do come back if you have more questions.

 

Yes but Wikipedia's definition is not as good as any. The fact is, the definition relies upon the word "effective"/"equivalent" depending on the wording and unless someone is aware of the importance of this word, you will find incorrect definitions such as:
"The Miller effect is the multiplication of the bandwidth robbing collector-base capacitance by voltage gain Av"
From our very own AAC online textbook. Can we replace this with:
"The Miller effect is the multiplication of the bandwidth robbing equivalent input capacitance of the amplifier relative to the input by voltage gain Av"?
Of course we could (and perhaps should if you would not like to use a different definition) but people will always tend to leave out the "equivalent input" part of the capacitance when explaining the Miller effect to someone who is unfamiliar. The best fix to the above statement is:
"The Miller effect is the increase in current through the collector-base capacitance caused by the increased voltage between the input and the inverted and amplified output ."
Any takers?

 

Wrong.

The increase in current has it's own existential existence.

The Miller effect describes an equivalent capacitance, connected to different points, that passes this same current.

Who are all these "people" you know using such sloppy terminology?

 

@ErnieM I disagree with you. Deriving the Miller effect:
An amplifier draws no input current and has gain Av. There is an impedance Z between the input and the output.
The only current drawn from the source is through impedance Z, which is given by:
I_in = dV/Z Note: "d" is "delta" is "change in", incremental not infinitesimal
= (V_in - V_out)/Z
= V_in (1 + Av) / Z
Therefore the input impedance of the circuit is:
V_in/I_in = V_in/ V_in(1+Av)/Z
=Z/(1+Av)
Thus the input impedance is much lower than we "expect" i.e. when the voltage across the impedance Z is only V_in.
If the voltage across the impedance Z is instead V_in (i.e. V_out = 0) then we have:
I_in = dV/Z = V_in/Z
And thus the input impedance of the circuit is:
V_in/I_in = V_in/(V_in/Z)
= Z
Using the reactive capacitance Xc = 1/(2*pi*f*C) = 1/(wC) (note: -j is left out, for those that are sticklers we could use the laplace transform's s parameter to have 1/(sC) as on Wikipedia where this derivation comes from) we can derive a formula for the input impedance (capacitive reactance) in terms of the capacitance Ccb. Thus we find that the input capacitance is the Ccb multiplied by the gain.

Yes this derivation makes mathematical sense, and it is simple to derive. But the whole "multiplied capacitance" part is hiding the true source of the problem: the current through the same impedance. The increase in current is the absolute source of the Miller effect, if you assume the Miller effect only applies to the input impedance being a multiple of the shunt capacitance.

But one might suggest taking this one step further, and consider the whole of the Miller effect not in terms of the "amplified capacitance" but in terms of the current through the capacitance and the way that it changes. The amplification of the capacitance is more of a novelty, a unique interpretation of a very simple effect: increased current draw that is dependent on frequency (i.e. decreased input impedance at higher frequencies).

To answer your question, the above quote was from the All About Circuits textbook. And do not be so quick to pronounce something as wrong without giving thought as to the validity of the statement.

 

I'm going to drop out of this thread as the content has droped to the level of trolling.

 

@ErnieM I can assure you, there is nothing "trolling" here. I'm sorry we didn't resolve the issue. Let me know if you'd like to continue another time, I suppose.

I'm dropping out as well, I have not handled myself well.

Sam

 

The Miller theorem is very useful because it help us simplify the analysis.
Take look at this example:



I hope that you see that is much easier to analyze the circuit without any feedback (right circuit).

And we can write transfer function by inspection

\( \frac{V_O}{V_{IN}}= \frac{\frac{1}{s C1 (1+A)}}{R1+\frac{1}{s C1 (1+A)}} *A \)

And because we are interested only in magnitude we can write:

\(\frac{\frac{1}{s C1 (1+A)}}{R1+\frac{1}{s C1 (1+A)}} = \frac{1}{\sqrt{1+(\frac{F}{Fh})^2}} \)

Where the F - is a signal frequency

Fh is -3dB frequency

\(F_H = \frac{1}{2\Pi * R_1 *C_1*(1+A)} \)

And finally we have:

\(\frac{V_O}{V_{IN}}=\frac{1}{\sqrt{1+(\frac{F}{Fh})^2}} * |A|\)

As you can see we can "transform" C1 into the amplifier input and this input capacitance is (1+A) times larger than C1 alone.

Also, notice that the Miller theorem applies not only to capacitors but the resistors also.



And for inverting input (negative feedback) we have:

\(R_{IN} = \frac{R_F}{1-( - Ao)} = \frac{R_F}{1+|Ao|}\)

But we can use a positive feedback also (voltage follower example/Bootstrapping)