Learn Analog Circuits with Zundamon

Episode 6

Cascodes: Raising Output Resistance

How can we get more gain from the same 1.8 V supply? Stack transistors to build a cascode, then examine the headroom tradeoff.

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  • Shikoku Metan (explains)
  • Zundamon (asks)

1Intro: more gain from a 1.8 V supply

Zundamon

My design assignment's in trouble again, nanoda! I want more gain without changing the 1.8 V supply.

Illustration: My design assignment's in trouble again, nanoda! I want more gain without changing the 1.8 V supply.
Figure 1

Shikoku Metan

Today: the cascode. Stack transistors vertically to raise the output resistance.

Illustration: Today: the cascode. Stack transistors vertically to raise the output resistance.
Figure 2

Zundamon

Let's do it, nanoda! This time we solve the assignment!

Illustration: Let's do it, nanoda! This time we solve the assignment!
Figure 3

2Review: the intrinsic-gain limit

Shikoku Metan

Recap: episode four's active-load gain was Av = −gm1(ro1∥ro2).

Shikoku Metan: Gain is gm times output resistance. It came to about −35, half the intrinsic gain gm·ro of about 70.

Illustration: Gain is gm times output resistance. It came to about −35, half the intrinsic gain gm·ro of about 70.
Figure 4

Shikoku Metan

Today we raise Rout by circuit design. The key: last time's common gate.

Illustration: Today we raise Rout by circuit design. The key: last time's common gate.
Figure 5

Shikoku Metan

Looking into the source, the resistance is about 1/gm; looking into the drain, it's large. We'll use that difference.

Illustration: Looking into the source, the resistance is about 1/gm; looking into the drain, it's large. We'll use that difference.
Figure 6

3The cascode circuit

Shikoku Metan

Stack a common-gate M2 on top of the common-source M1. That's the cascode.

Illustration: Stack a common-gate M2 on top of the common-source M1. That's the cascode.
Figure 7

Shikoku Metan

M2's gate is fixed at a constant voltage Vb. Call the node between them X.

Shikoku Metan: In series, the current is the same. M1 turns input into current; M2 passes it up.

Illustration: In series, the current is the same. M1 turns input into current; M2 passes it up.
Figure 8

Shikoku Metan

Last time I said the common-gate stage is good at receiving signals that arrive as currents. That's exactly what we use here.

Illustration: Last time I said the common-gate stage is good at receiving signals that arrive as currents. That's exactly what we use here.
Figure 9

Shikoku Metan

VDD=1.8 V. M1's gate is about 0.70 V like episode four's mirror, and Vb=1.25 V.

Illustration: VDD = 1.8 V. M1's gate is about 0.70 V like episode four's mirror, and Vb = 1.25 V.
Figure 10

Shikoku Metan

M2's body is grounded and its source is X, so we include the body effect.

4Why output resistance increases

ZundamonQuestion

Why does stacking alone make the output resistance bigger?

Illustration: Why does stacking alone make the output resistance bigger?
Figure 11

Shikoku Metan

Look at the voltage at X, VX. M1's VDS is exactly VX.

Illustration: Look at the voltage at X, VX. M1's VDS is exactly VX.
Figure 12

Shikoku Metan

Raise the output from 1.0 V to 1.5 V, and VX, about 0.45 V, moves by only about 6 mV.

Zundamon

From M1's view, VDS hardly changes, nanoda!

Shikoku Metan

So M1's current barely changes either. M2 shields M1 from changes in the output voltage.

Illustration: So M1's current barely changes either. M2 shields M1 from changes in the output voltage.
Figure 13

ZundamonQuestion

But M2's VDS changes a lot, nanoda. Doesn't M2's current increase?

Illustration: But M2's VDS changes a lot, nanoda. Doesn't M2's current increase?
Figure 14

Shikoku Metan

VX only has to rise slightly: that lowers M2's VGS=Vb−VX enough to cancel the extra current.

Illustration: VX only has to rise slightly: that lowers M2's VGS = Vb − VX enough to cancel the extra current.
Figure 15

Shikoku Metan

For that same 1.0 V to 1.5 V step, the current rises about 4.5% for a single MOSFET alone (M1 by itself, gate at 0.70 V), but only about 0.07% for the cascode.

Illustration: For that same 1.0 V to 1.5 V step, the current rises about 4.5% for a single MOSFET alone (M1 by itself, gate at 0.70 V), but only about 0.07% for the cascode.
Figure 16

5Deriving output resistance with a small-signal model

Shikoku Metan

Let's find Rout with equations. Attach a test source vt to the output.

Illustration: Let's find Rout with equations. Attach a test source vt to the output.
Figure 17

Shikoku Metan

Call the incoming current it. The input is zero, and Vb is ground for small signals.

Illustration: Call the incoming current it. The input is zero, and Vb is ground for small signals.
Figure 18

Shikoku Metan

M1 has vgs1=0, so its current source vanishes, leaving only ro1.

Illustration: M1 has vgs1 = 0, so its current source vanishes, leaving only ro1.
Figure 19

Shikoku Metan

All of that current flows through M2 into ro1, so vx=it·ro1.

Illustration: All of that current flows through M2 into ro1, so vx = it·ro1.
Figure 20

Shikoku Metan

M2's gate and body are grounded, so vgs2 and vbs2 both equal −vx.

Illustration: M2's gate and body are grounded, so vgs2 and vbs2 both equal −vx.
Figure 21

Shikoku Metan

Then M2's current source, gm2·vgs2+gmb2·vbs2, equals −(gm2+gmb2)vx: it tries to push (gm2+gmb2)vx upward.

Illustration: Then M2's current source, gm2·vgs2 + gmb2·vbs2, equals −(gm2 + gmb2)vx: it tries to push (gm2 + gmb2)vx upward.
Figure 22

Zundamon

Upward, nanoda? The test current flows downward, nanoda.

Shikoku Metan

Right. So ro2 carries the test current plus the current source's share, both flowing downward.

Shikoku Metan: Put vx=it·ro1 into vt=vx+ro2[it + (gm2+gmb2)vx], and Rout=vt/it=ro1+ro2 + (gm2+gmb2)ro2·ro1.

Illustration: Put vx = it·ro1 into vt = vx + ro2[it + (gm2 + gmb2)vx], and Rout = vt/it = ro1 + ro2 + (gm2 + gmb2)ro2·ro1.
Figure 23

Shikoku Metan

The last term is far larger, so Rout ≈ (gm2+gmb2)ro2·ro1.

Shikoku Metan: Ignoring the body effect, this is ro1 multiplied by M2's intrinsic gain gm2·ro2, about 70.

Illustration: Ignoring the body effect, this is ro1 multiplied by M2's intrinsic gain gm2·ro2, about 70.
Figure 24

Zundamon

What the source connects to changes so much, nanoda!

Illustration: What the source connects to changes so much, nanoda!
Figure 25

Shikoku Metan

Yes. Last time the source was grounded; now M2's source connects to ro1.

Shikoku Metan: At VOUT=1.2 V, Rout is about 7.1 MΩ: 66 times the single device's 107 kΩ.

Illustration: At VOUT = 1.2 V, Rout is about 7.1 MΩ: 66 times the single device's 107 kΩ.
Figure 26

6Comparing output current–voltage curves

Shikoku Metan

Plot VOUT on the horizontal axis and compare the single device with the cascode.

Illustration: Plot VOUT on the horizontal axis and compare the single device with the cascode.
Figure 27

Zundamon

The cascode is much flatter, nanoda!

Shikoku Metan

Rout is the inverse of the slope. Read off the graph, it's also about 7.1 MΩ, matching the formula.

ZundamonQuestion

But the cascode's current is a bit smaller, nanoda?

Shikoku Metan

Because M1's VDS stops at VX, about 0.45 V. It's the single device's current at VDS=0.45 V on its curve.

Shikoku Metan: Also, the curve flattens out only from about 0.75 V, to the right of the single device's 0.3 V. That will matter later.

Illustration: Also, the curve flattens out only from about 0.75 V, to the right of the single device's 0.3 V. That will matter later.
Figure 28

7The cascode current mirror

ZundamonQuestion

Can this shrink episode four's current-mirror error too, nanoda?

Illustration: Can this shrink episode four's current-mirror error too, nanoda?
Figure 29

Shikoku Metan

Yes. Episode four's mirror carried about 7% extra current at VDS=1.5 V.

Shikoku Metan: Stack two diode-connected devices on the input side too, and use the upper gate as Vb. Then VX on the output side nearly equals the lower VGS on the input side, so the two lower devices have the same VDS.

Illustration: Stack two diode-connected devices on the input side too, and use the upper gate as Vb. Then VX on the output side nearly equals the lower VGS on the input side, so the two lower devices have the same VDS.
Figure 30

Shikoku Metan

In this matched model, even at 1.5 V output the error is under 0.01%.

Shikoku Metan: This is called a cascode current mirror.

Illustration: This is called a cascode current mirror.
Figure 31

Shikoku Metan

But the upper gate voltage Vb is two VGS's stacked, about 1.55 V, and the output must stay above about 1.0 V for the curve to be flat.

8The cascode amplifier

Shikoku Metan

Let's use it in an amplifier too: make both the bottom and the top of episode four's active-load stage into cascodes.

Shikoku Metan: NMOS M1 and M2 below, PMOS M3 and M4 above. The input is M1's gate.

Illustration: NMOS M1 and M2 below, PMOS M3 and M4 above. The input is M1's gate.
Figure 32

Shikoku Metan

Doubling the PMOS W/L matches its Vov to the NMOS, about 0.3 V.

Illustration: Doubling the PMOS W/L matches its Vov to the NMOS, about 0.3 V.
Figure 33

Shikoku Metan

PMOS bodies connect to their own sources, so there's no body effect.

Shikoku Metan: Gain is −gm1 times output resistance: the lower and upper cascodes in parallel.

Illustration: Gain is −gm1 times output resistance: the lower and upper cascodes in parallel.
Figure 34

Shikoku Metan

If gm and ro match and we ignore body effect, each is about gm·ro·ro; in parallel, half that.

Shikoku Metan: So Av ≈ −(gm·ro)²/2.

Illustration: So Av ≈ −(gm·ro)²/2.
Figure 35

ZundamonQuestion

The intrinsic gain squared, nanoda! If gm·ro≈70, that's about 2500, nanoda?

Illustration: The intrinsic gain squared, nanoda! If gm·ro ≈ 70, that's about 2500, nanoda?
Figure 36

Shikoku Metan

Computed right near the operating point, about −1400. It matches the graph's slope.

Illustration: Computed right near the operating point, about −1400. It matches the graph's slope.
Figure 37

Shikoku Metan

It falls short of the estimate because each VDS is small, which lowers ro. We'll come back to that with headroom.

Illustration: It falls short of the estimate because each VDS is small, which lowers ro. We'll come back to that with headroom.
Figure 38

Zundamon

About 40 times episode four's −35, nanoda! Just two more transistors, nanoda.

Illustration: About 40 times episode four's −35, nanoda! Just two more transistors, nanoda.
Figure 39

9The cost: voltage headroom

Shikoku Metan

But it's not all good. The price is voltage headroom.

Shikoku Metan: Saturation needs NMOS VDS and PMOS VSD each at least Vov. Four of them: about 1.2 V.

Illustration: Saturation needs NMOS VDS and PMOS VSD each at least Vov. Four of them: about 1.2 V.
Figure 40

Zundamon

The supply is 1.8 V, so the output can only move about 0.6 V, nanoda!

Shikoku Metan

With margin left for M1 and M4 in this bias, every device stays saturated only for outputs of about 0.76 to 1.05 V (M2 sets the lower limit, M3 the upper).

Illustration: With margin left for M1 and M4 in this bias, every device stays saturated only for outputs of about 0.76 to 1.05 V (M2 sets the lower limit, M3 the upper).
Figure 41

Shikoku Metan

Episode four's circuit ran about 0.32 to 1.37 V, a span of 1 V. Now it's about 0.3 V.

Shikoku Metan: With a gain of about 1400, that span is only about 0.34 mV at the input. The input DC level must be set precisely too.

Illustration: With a gain of about 1400, that span is only about 0.34 mV at the input. The input DC level must be set precisely too.
Figure 42

Shikoku Metan

A small VDS also lowers ro. That's why we fell short of the estimate of about 2500.

Illustration: A small VDS also lowers ro. That's why we fell short of the estimate of about 2500.
Figure 43

Shikoku Metan

Advanced processes, with their low intrinsic gain, also run on low supply voltages, so stacking is hard. That's the dilemma.

Illustration: Advanced processes, with their low intrinsic gain, also run on low supply voltages, so stacking is hard. That's the dilemma.
Figure 44

10Summary

Shikoku Metan

To sum up: stacking a common gate on a common source is the cascode.

Shikoku Metan: M2 shields M1, raising the output resistance to about gm·ro·ro.

Illustration: M2 shields M1, raising the output resistance to about gm·ro·ro.
Figure 45

Zundamon

Mirrors and amplifiers both improve, but we lose voltage headroom, nanoda!

Zundamon: In the assignment, I'll check headroom as well as gain, nanoda!

Illustration: In the assignment, I'll check headroom as well as gain, nanoda!
Figure 46

Shikoku Metan

Next time: the differential pair, which amplifies only the difference of two inputs.

Illustration: Next time: the differential pair, which amplifies only the difference of two inputs.
Figure 47

Zundamon

Thanks for reading, nanoda!