Subthreshold CMOS Circuits

While the implementation of the sum-product algorithm as an analog circuit may seem difficult, recall from section 4 that the marginalization steps consist entirely of multiplication and addition, hence the name of algorithm:

$$\sum_{\sim\{x\}} \left ( f(x)\prod_{y\in n(f)\setminus\{x\}}\mu_{h\rightarrow x}(y) \right )$$

This type of calculation maps naturally into analog circuitry if we are provided the correct building blocks.

Addition of currents in analog circuits is particularly elegant. Kirchhoff's current law states the sum of currents into a node is zero, and so a current adder is simply a short of one wire to another. On the other hand, building analog CMOS circuits that perform multiplication is difficult. In the normal operating regions of a MOS transistor, the saturation current increases as the square of its gate voltage, and quadratic functions do not lend themselves to multiplication.

Traditionally, multipliers are built by combining elements that respond exponentially and logarithmically to their inputs. In his multiplier circuit, Gilbert showed that bipolar transistors could be used as the exponential elements and logarithmic circuits could be formed from diode-connected transistors. This project uses MOS transistors in their subthreshold operating region to achieve the same type of exponential response.

Subthreshold MOS Transistors

In an MOS transistor, the amount of current flowing from source to drain is controlled by an electric field from an applied voltage at the transistor's gate. The electric field attracts charge carriers from either side of the channel at the source and drain, forming a thin conductive layer between them. A higher voltage and stronger electric field translates to more current flowing through the transistor.

In a semiconductor, there are two modes by which current can flow: diffusion and drift. Diffusion is the natural flow of particles from higher to lower concentration, and drift is the flow of particles subject to an applied force. In the subthreshold region of a MOS transistor, the flow of current from source to drain is due to diffusion [14]. Diffusion current in a MOS transistor is given by:

$$I = -WqD\frac{\delta N}{\delta z}$$ (5)

where W is the channel width, D is the diffusion constant of the carriers, N is their density, and z is the distance between source and drain. The density of the carriers decreases linearly along the channel, so $\delta N/\delta z$ can be simplified to $(N_{d} - N_{s})/L$. $N_{s}$ is the density of carriers at the source, and $N_{d}$ is their density at the drain:

$$N_{s} = N_{1}e^{\frac{-q\psi_{s}}{kT}}e^{\frac{qV_{s}}{kT}}$$ (6)

$$N_{d} = N_{1}e^{\frac{-q\psi_{s}}{kT}}e^{\frac{qV_{d}}{kT}}.$$ (7)

We first replace $N_{d}$ and $N_{s}$ with their respective formulae in the following equation:

$$I = -WqD\frac{N_{d} - N_{s}}{L}$$

to get:

$$I = \frac{qW}{L} N_{1} D e^{\frac{-q\psi_{s}}{kT}} \left(e^{\frac{qV_{s}}{kT}} - e^{\frac{qV_{d}}{kT}}\right),$$ (8)

where $\psi_{s}$ is the surface potential at the source and along the channel. If we then assume that excursions are small around the operating point, we can replace $\psi_{s}$ with $\kappa V_{g}$. Collecting all of the pre-exponential constants into one term, $I_{0}$, we have:

$$I = I_{0}e^{\frac{-q\kappa V_{g}}{kT}} \left(e^{\frac{qV_{s}}{kT}} - e^{\frac{qV_{d}}{kT}}\right),$$ (9)

which correctly describes subthreshold PMOS transistor operation. In an NMOS transistor, an increase in gate voltage attracts rather than repels charge carriers, so the voltages are reversed:

$$I = I_{0}e^{\frac{q\kappa V_{g}}{kT}} \left(e^{\frac{-qV_{s}}{kT}} - e^{\frac{-qV_{d}}{kT}}\right).$$ (10)

To simplify the equation, we can assume the substrate is intrinsic and so no charge from ionized donors or acceptors in the substrate reduces the effectiveness of an applied electric field at the gate. $\kappa$ is the term that accounts for the differences in surface potential along the channel from $V_{g}$, so we eliminate it. Because $kT/q$ is a voltage that only changes with temperature, we replace it with a single variable, $V_{T}$:

$$I = I_{0}e^{\frac{V_{g}}{V_{T}}} \left(e^{\frac{V_{s}}{V_{T}}} - e^{\frac{V_{d}}{V_{T}}}\right).$$ (11)

Another form of this equation, which is less accurate, but makes further algebraic manipulations possible, assumes that the transistor is in the saturation mode where changes in $V_{d}$ do not have a strong effect on the current flowing through the transistor. This region starts when $\vert V_{d} - V_{s}\vert \geq 4kT/q \approx 100mV$. In this mode, the $V_{d}$ term drops out and we have:

$$I = I_{0}e^{\frac{V_{g}}{V_{T}}}e^{\frac{-V_{s}}{V_{T}}} = I_{0}e^{\frac{V_{g}-V_{s}}{V_{T}}},$$ (12)

which shows that we can model the transistor as having a simple exponential response when saturated and in the subthreshold region. This response is illustrated by figure 3.

Figure 3: Drain current vs. gate voltage for a saturated subthreshold CMOS transistor.

Diode-Connected Transistors

As previously mentioned, a Gilbert multiplier is typically built of exponential and logarithmic components, and the logarithmic component is typically a diode-connected bipolar transistor. By solving equation 12 for $V_{g}$, we see that a saturated diode-connected subthreshold CMOS transistor does takes the logarithm of its input current:

$$V_{g} = V_{T} ln\left(\frac{I}{I_{0}}\right) + V_{s}.$$ (13)

For typical processes, $I_{0}$ is high enough that even for very small drain currents, the voltage at the gate of the transistor will be about 0.4V [14]. This ensures that a diode-connected transistor is always saturated and the equation above will be valid for all useful subthreshold values. A plot of this logarithmic behavior is given in figure 4.

Figure 4: Gate voltage vs. drain current for a diode-connected transistor.

Current Mirrors

A natural extension to the diode-connected transistor configuration is a current mirror where the gate of another transistor is connected to the diode-connected transistor's input. In this configuration, the attached transistor creates a mirror copy of the current flowing through the diode-connected transistor, provided the geometries of the two transistors are the same. The derivation is simple, we need only plug the diode-connected transistor equation, 13, into the transistor equation, 12:

$$V_{d1} = V_{g1} = V_{g2}$$

$$V_{s1} = V_{s2}$$

$$I_{2} = I_{0} e^{ \frac{ V_{T} ln \left( \frac{I_{1}}{I_{0}} \right) + V_{s1} - V_{s2}}{V_{T}}} = I_{1}$$ (14)

A chain of transistors can be connected to the diode-connected transistor in the same way as the first. This is possible because only a negligible amount of current leaks through the electrically-isolated gate of an MOS device. Each added transistor takes very little from the input signal, and so little accuracy is lost as additional transistors are connected. This technique is valuable for the summation part of the marginalization step of the sum-product algorithm. The marginalization requires multiplying the same value from a single message by several other values before summing. It is done by taking a message, which will be transmitted as a current on a wire, and making several copies of it with a current mirror as described above.

Another important feature of current mirrors is their ability to scale the currents flowing through them. Equation 14 assumes that the two transistors that form the current mirror have the same geometry. If the second transistor has a larger width or length than the diode-connected transistor, more or less current will flow through it at the same gate voltage. Recall that the $I_{0}$ in the transistor equations we have used so far includes the $W/L$ geometry term from equation 8. If we temporarily remove this term from the $I_{0}$ in equation 12, we have:

$$I = \frac{W}{L}\tilde{I}_{0}e^{\frac{V_{g}-V_{s}}{V_{T}}}.$$ (15)

Plugging this into equation 12 as we did in equation 14, we get:

$$I_{2} = I_{1}\frac{L_{1}W_{2}}{L_{2}W_{1}}.$$ (16)

While this gives us a fairly accurate way to multiply a current by a fixed value, it has limitations. This method relies on making the transistors physically smaller or larger, which can only be extended to a point. For example, to scale a current to 10 percent of its value, the width of the output transistor would have to be 10 percent of the input transistor's width, assuming the lengths remain constant. This would indicate that the width of the input transistor would have to be at least 10 times the minimum resolution of the process, and no scaling less than 10 percent could be achieved in a single stage.

Differential Pairs

With a logarithm circuit in hand, all that is needed to build a multiplier is a circuit that has exponential response, and we have already shown that the response of a single subthreshold MOS transistor has that. This presents the question of how to take advantage of the transistor's exponential response. Our messages are sent in the form of currents, but a diode-connected transistor converts a current to a voltage during its computation. We need a circuit that takes the exponential of a voltage and outputs a current, and that circuit can be built from a simple differential pair, shown in figure 5.

Figure 5: Schematic of a differential pair circuit. The bias current $I_{b}$ is set by $V_{b}$ and divided between $I_{1}$ and $I_{2}$ depending on $V_{1}$ and $V_{2}$.

Again, we will assume that the transistors Q1 and Q2 are saturated and operating in the subthreshold mode. From Kirchhoff's current law, we know that $I_{1} + I_{2} = I_{b}$. Substituting equation 12 for $I_{1}$ and $I_{2}$, we get:

$$I_{b} = I_{0}e^{-V}\left( e^{V_{1}} + e^{V_{2}}\right).$$ (17)

To extract the value of $I_{1}$ that does not include $V$, we first solve for V:

$$e^{-V} = \frac{I_{b}}{I_{0}\left( e^{V_{1}} + e^{V_{2}}\right)},$$

and then substitute the solution back into equation 17:

$$I_{1} = I_{b}\frac{e^{V_{1}}}{e^{V_{1}} + e^{V_{2}}}$$ (18)

$$I_{2} = I_{b}\frac{e^{V_{2}}}{e^{V_{1}} + e^{V_{2}}}.$$ (19)

If we then replace $I_{b}$ in equations 18 and 19 with equation 12, we have:

$$I_{1} = \frac{e^{V_{g}}e^{V_{1}}}{e^{V_{1}} + e^{V_{2}}}$$ (20)

$$I_{2} = \frac{e^{V_{b}}e^{V_{2}}}{e^{V_{1}} + e^{V_{2}}}.$$ (21)

While the circuit clearly has some type of exponential response, it is not immediately clear how to take advantage of it. The trick is to eliminate the $e^{V_{1}} + e^{V_{2}}$ term in the divisor of equations 18 and 19. This will happen by combining the differential pair with three diode-connected transistors.

The Gilbert Multiplier

The classic Gilbert transconductance multiplier is named for Barrie Gilbert who designed the circuit in 1968 with bipolar transistors [6]. The circuit combines diode-connected transistors, current mirrors, summing junctions, and differential pairs to multiply two differential signals. Rather than examine the entire circuit, we will examine half of the circuit, as shown in figure 6. The circuit shown is exactly the differential pair circuit, but with three diode-connected transistors added.

Figure 6: Schematic of half of a Gilbert multiplier. This circuit multiplies $I_{in1}$ and $I_{in2}$ by $I_{inb}$ if $I_{in1}$ and $I_{in2}$ sum to 1.

Each added diode-connected transistor, $Q_{4}$, $Q_{5}$, and $Q_{6}$, forms a current mirror with the transistor it is attached to, $Q_{1}$, $Q_{2}$, and $Q_{3}$. Each diode-connected transistor takes the logarithm of its input as a voltage, and each connected transistor takes the exponential of that value, so the current running through both transistors is the same. Because these are just current mirrors, we can rewrite equations (20) and (21) as:

$$I_{1} = \frac{I_{inb}I_{in1}}{I_{in1} + I_{in2}}$$ (22)

and

$$I_{2} = \frac{I_{inb}I_{in2}}{I_{in1} + I_{in2}}.$$ (23)

If we can then assure ourselves that $I_{in1} + I_{in2} = 1$, we can simplify the equations to read:

$$I_{1} = I_{inb}I_{in1}$$ (24)

and

$$I_{2} = I_{inb}I_{in2}.$$ (25)

Which is the multiplication we are interested in, as long as we can satisfy the requirement that $I_{in1}$ and $I_{in2}$ always sum to some constant value. In the context of the sum-product algorithm, we are always multiplying the probabilities that events happened or did not happen. By treating $I_{in1}$ as the probability that an event happened and $I_{in2}$ as the probability that it did not happen, we are assured that their values always sum to one, which in reality is some reference current. It is because of this that we must always compute both products even if we do not need one of them for our calculation.

Figure 7: Plot showing the linear multiplicative nature of the partial Gilbert multiplier. $I_{in1}$ sweeps from 1nA to 10nA, and $I_{in2}$ sweeps from 10nA to 1nA while $I_{inb}$ is held constant. The plot is of $I_{1}$ vs. $I_{in1}$, which shows that the circuit multiplies $I_{inb}$ and $I_{in1}$.