Jun 21, 2011 -- posted by Achal

## Reverse-engineering the cascode

(from – **http://analogcircuits.posterous.com/reverse-engineering-the-cascode**)

Cascode (shown in Fig. 1) is well known and widely used circuit for creating large-impedances in integrated circuits. Cascode circuits also provide another advantage i.e. isolation between input and output ports and thereby reducing the Miller-effect and as a result cascodes have a good frequency performance. And when impedances achieved from cascode circuits are not good enough, we go on to use active-cascodes or regulated-cascodes. Even though cascodes are affected by the limitations of headroom (voltage swing), there are work-arounds for this problem that have been figured out by circuit designers (the smart engineers again). The question that I want to discuss here is – *how did someone (must be a genius) think of this circuit which provides an elegant area-efficient solution for achieving large impedances consequently large voltage gains?*

Fig 1. a) NMOS cascode sink b) regulated NMOS cascode sink.

I think its highly unlikely that the engineers randomly tried putting transistors in series thinking that impedances add when we put them in series and discovered that in case of a constant current bias, the impedance actually gets magnified by the internal gain of the transistor stacked on top. Even if that was the case, the other question then to consider is – *how did the smart design engineers figure out the idea of regulated-cascodes when they realized the impedance from cascode was not good enough?*

I personally think there must have been a systematic engineering process behind developing such an idea, epiphany is unlikely, because fortunately or unfortunately epiphanies are few and far in between. And an engineering process involves solving a problem by breaking it into parts or smaller problems and layering the solutions of those smaller problems to create an intelligent and elegant solution to a broader/bigger problem. And that is why, for the development of new elegant idea/solution for a problem, requires the solutions to the smaller problems to exist. Anyhow, I am digressing from the topic of discussion, the philosophical issues like this are much better discussed in this terrific radiolab podcast. All readers are strongly encouraged to listen to it.

Fig 2. a) source degeneration by a resistor b) source degeneration by active impedance of MOSFET M_{2}

Now coming back to the inception of cascode. Casode circuit is actually source degenerated circuit, just like the circuit shown in Fig 2. a). In the circuit shown in Fig 2. a) a resistor R_{S} source degenerates MOSFET M_{1}, while in cascode its the active impedance of a MOSFET (M_{2} in Fig 2. b)) that acts like a degenerating resistor. We know that source degeneration improves the output impedance by a factor of (1 + g_{m}R_{S}), so larger the R_{S}, larger the impedance that can be achieved from a cascode. However, R_{S} can only be so large in an integrated circuit without increasing the area significantly, and one of the larger impedances available on integrated circuits is an active impedance of MOSFET biased in saturation, so degeneration using active impedance of MOSFET can easily provide higher gains in the output impedance at no significant cost of area. The fourth chapter on current sources, of the landmark text on analog circuits by professors Gray, Hurst, Lewis and Meyer also implicitly suggests the same idea, considering the way section discussing cascode is presented following the section discussing source degeneration.

Still how and why source degeneration? My guess is, source degeneration circuit was devised during the bipolar days. Unlike MOSFETs bipolar devices exhibit small input impedance and source degeneration improves both input impedance and output impedance by a factor of (1 + g_{m}R_{S}). The small-signal analysis and algebra illustrating it is given below.

Fig 3. Small-signal equivalent of source degenerated bipolar transistor.

Again the small-signal analysis only tells us why source degeneration helps or works, but not how would we think of it as the solution to the problem of small input and output impedances. In order to think of source degeneration as the solution to the problem of small input and output impedances, we need to think of source degeneration as series-series feedback. And this is not something explicitly apparent. Let me try to explain, if you can’t see it already.

Fig 4. MOS Transconductance amplifier.

The NMOS transistor M_{2} in the circuits shown in Fig 1. can be looked upon as a transconductance amplifier with a current gain g_{m} (shown in Fig 4. above) with gate and source being positive and negative terminals respectively. One strange property of MOSFET transconductance amplifier is that the output current flows to ground through the negative terminal of the amplifier, it turns out to be really useful, as we will see later.

Fig 5. series-series feedback loop as source degeneration.

The series-series feedback loop (shown in Fig. 5) can be constructed as resistor connected between the negative terminal of transconductance and ground, if the output current flows through the negative terminal of the transconductance amplifier. If we do that in the resulting circuit the output current gets sampled and converted into voltage using the degenerating resistor then fed to the amplifier at the negative terminal. Hence the result is series-series negative feedback. Series-series feedback increases both input and output impedances by a factor of (1 + g_{m}?) ( ? (1 + g_{m}R_{S}), algebra given below). Note here ? has the dimensions of an impedance(? = V_{f}/I_{out}). If we understand this, then we can use the series-series feedback to fulfill the need for higher input and output impedances at the expense of current gain (g_{m}). This is the real ingenious idea behind source degeneration, consequently the cascode circuit i.e. transforming the series-series feedback loop to source degeneration circuit. Once we do that moving on to regulated-cascode for even higher impedances is somewhat straight-forward. If you have been paying attention, you would have guessed it already. If you haven’t, then let me explain.

Fig 6. series-series feedback

The feedback factor in the case of source degeneration is the degenerating impedance R_{S} (= V_{f}/I_{out}), so larger the feedback factor ? larger the gain (1 + g_{m}? ? 1 + g_{m}R_{S}) in output impedance. If we can’t increase R_{S} beyond a certain value, we can start playing with V_{f} and I_{out}. The quantity being fed back in series-series feedback scheme is a voltage, and the nice thing about voltages is that they can be easily amplified. Now, if we break the series-series feedback loop after feedback element and put an amplifier (of gain A) as shown in Fig. 7, we can improve increase the feedback factor ?by a factor of A and consequently increase output impedance by a factor A more.

Fig 7. modifying the series-series feedback loop.

However there is one minor problem if we do that in the source degeneration circuit. We can’t sample the voltage at the source node and feed the amplified voltage back at the same node, it will put the voltage amplifier in a positive feedback loop (Fig. 8), however we also have the positive terminal of the transconductance amplifier to feed the voltage and we can change the gain from A to -A to achieve the same effect. If we do that we arrive at regulated-cascode/source degenerated circuit. The idea behind regulated-cascode is often described by need to ensure a almost constant drain bias at (D_{1}) drain of M_{1} (Fig 1.) irrespective of the variation in the voltage at the drain of transistor M_{2} and the feedback using amplifier ensures that, but that doesn’t quite explain inception of regulated-cascode, it just explains why it works. And for that matter, in principle, even regular cascodes have the same basis. The idea behind the origins of regulated-cascode must have been series-series feedback.

Fig 8. constructing the regulated-cascode.

I am guessing, the person who came up with the idea of regulated-cascode must have had a shared dream about Giants winning the World *Series *(please don’t mind the bad joke).

**Originally posted at http://analogcircuits.posterous.com/reverse-engineering-the-cascode **(the text on that web-page doesn’t have any problems with greek symbols i.e. no ‘?’s in the post)

If you notice any mistake from a typo to something fundamental, please email me at achal [dot] kathuria at gmail. I will appreciate any kind of feedback.