### Supporting Information

- Discrete Semiconductor Circuits: Differential Amplifier
- Discrete Semiconductor Circuits: Simple Op-Amp
- Insulated-Gate Field-Effect Transistors (MOSFET)

### Differential or Single-Ended?

Introductory studies of active circuits generally devote a significant amount of time to standard single-ended amplifier configurations—e.g., common-source, common-gate, emitter-follower. This is certainly a worthy endeavor in the context of becoming familiar with transistor operation, small-signal analysis, and amplifier characteristics. But the *practical* value of single-ended amplifier configurations is a different story—the fact is, differential amplifiers dominate modern analog ICs. There are a few reasons for this:

- Differential amplifiers apply gain not to one input signal but to the
*difference*between two input signals. This means that a differential amplifier naturally eliminates noise or interference that is present in both input signals. - Differential amplification also suppresses common-mode signals—in other words, a DC offset that is present in both input signals will be removed, and the gain will be applied only to the signal of interest (assuming, of course, that the signal of interest is not present in both inputs). This is particularly advantageous in the context of IC design because it eliminates the need for bulky DC-blocking capacitors.
- The subtraction that occurs in a differential pair makes it easy to incorporate the circuit into a negative-feedback amplifier, and if you’ve read the Negative Feedback series, you know that negative feedback is about the best thing that could ever happen to an amplifier circuit.

It is only sensible to expect these benefits to be accompanied by significant disadvantages, but the nature of IC fabrication has rendered the differential configuration almost wholly beneficial. Two concerns are 1) higher component count and 2) the importance of symmetrical component characteristics. You can forget about number 1 because the cost of adding a few more transistors to an IC is negligible. As for number 2, IC technology happens to be very good at achieving consistency among the components within a chip (this consistency is referred to as “matching”).

In this article, we will explore the basic MOSFET differential-amplifier configuration by means of conceptual discussion and simulations (i.e., not too much math or complicated circuit analysis). Because this topic is relevant primarily to IC implementation, we will use an NMOS model that is specific to 0.35 µm CMOS technology; the various LTspice files and some related information can be found here.

### A Pair of MOSFETs

This is the circuit:

Note the following:

- In real life, the current-source symbol would be replaced by a circuit that generates a constant current. (Refer to The Basic MOSFET Constant-Current Source for more information.) However, we want to keep things nice and simple for this introductory analysis, so in our simulations we’ll use an ideal current source instead of a constant-current circuit.
- An actual IC implementation of this circuit would replace the resistors with a current mirror functioning as an “active load.” However, if our goal is to understand the functionality of the differential pair, I think we should start with the resistor version.
- The differential pair is all about balance. Thus, for optimal performance the resistors and MOSFETs must be matched. This means that the channel dimensions of both FETs must be the same and that R
_{1}must equal R_{2}. The resistance value chosen for the two resistors will be referred to as R_{D}(for__d__rain resistance).

### DC Analysis

Let’s determine the biasing conditions of this circuit when both inputs are grounded.

The sum of the two drain currents I_{D1} and I_{D2} must equal I_{BIAS}. We also know that the two drain currents are equal because, in this idealized analysis, both halves of the circuit are identical. Thus,

\[I_{D1}=I_{D2}=\frac{I_{BIAS}}{2}\]

Let’s assume for the moment that the transistors are in saturation. The equation for saturation-mode drain current is the following:

\[I_D=\frac{1}{2}\mu_nC_{ox}\frac{W}{L}(V_{GS}-V_{TH})^2\]

(We will ignore channel-length modulation throughout this article.) The drain current is already established (by the current source) and the gates are tied to the ground node; this means that the source voltage will settle on whatever value creates a gate-to-source voltage (V_{GS}) corresponding to a drain current of I_{BIAS}/2. We’ll let the simulation figure this out for us. The output voltages are easier: Calculate the voltage drop across the resistor as (I_{BIAS}/2) × R_{D}, then subtract this voltage drop from the positive supply. Here is an example:

The output voltages are as expected. The source voltage seems reasonable considering that the threshold voltage (V_{TH}) for this SPICE model is about 0.5 V; the simulation is telling us that the V_{GS} corresponding to a drain current of 250 µA is about 0 V – (–725 mV) = 725 mV, which is about 225 mV above V_{TH}.

Let’s go back to our assumption about the transistors being in saturation (aka “active mode”). A MOSFET amplifier needs to remain in the saturation portion of its transfer characteristic, because the gain is higher and more stable in the saturation region compared to the triode region. To ensure saturation, the drain voltage must always be higher than the gate voltage minus the threshold voltage:

\[V_{DS}\geq V_{GS}-V_{TH}\ \ \Rightarrow \ \ V_{GD}\leq V_{TH}\]

In this example, the drain voltage (aka V_{OUT}) is fixed at 2.05 V. This means that we have a restriction on V_{IN}: the common-mode input voltage cannot exceed 2.05 V + 0.5 V = 2.55 V, because when the input voltage reaches V_{TH} volts above the drain voltage, the MOSFET enters the triode region.

### Common-Mode Rejection

Let’s do a quick simulation to prove to ourselves that the differential pair will not amplify common-mode voltages. Here is the circuit:

As you can see, even with a 1 V common-mode input, the output voltage is still at the bias voltage—2.05 V. A simple, conceptual explanation for this common-mode rejection behavior is the following: The magnitude of the output voltage is governed by the drain current, not the input voltage. As long as the two input voltages are the same, the fixed bias current is evenly divided between the two transistors, and thus V_{OUT1} and V_{OUT2} do not change.

Note also that the gate-to-source voltage is about the same (because the drain current hasn’t changed), though the source voltage has increased to compensate for the fact that the gate is now at 1 V instead of ground.

### Differential Gain

You can understand this circuit’s differential action if you remember the following:

- I
_{D1}+ I_{D2}= I_{BIAS} - V
_{S1}= V_{S2}

If the voltage at the gate of Q_{1} is higher than the voltage at the gate of Q_{2}, V_{GS1} must also be higher than V_{GS2}, because both transistors have the same potential at the source terminal. A higher gate-to-source voltage means more drain current, but the sum of the drain currents remains the same—thus, I_{D1} increases and I_{D2} decreases, and this causes a corresponding decrease in V_{OUT1} and a corresponding increase in V_{OUT2}. For example:

We’ll finish up this introductory analysis by simulating the differential small-signal response and comparing the simulated gain to the theoretical gain. Let’s take the common-mode voltage back to 0 V and then apply a 1 mV sine wave to the gate of Q_{1}:

We will define the output voltage as V_{OUT1} – V_{OUT2}; this doubles the gain relative to using only V_{OUT1} or V_{OUT2}, and it also eliminates the DC offset associated with the bias voltages. Here is a plot with V_{IN1} and the differential output voltage:

Here we have an output amplitude of 10 mV and an input amplitude of 1 mV; hence, our simulated differential gain is 10. The formula for theoretical differential gain is

\[A_{diff}=g_m\times R_D\]

where g_{m} can be calculated as follows:

\[g_m=\sqrt{2\mu_nC_{ox}\left(\frac{W}{L}\right)I_D}=\sqrt{\mu_nC_{ox}\left(\frac{W}{L}\right)I_{BIAS}}\]

The SPICE model has µ_{n} = 148.2 cm^{2}/V·s and t_{ox} = 7.754 × 10^{–9}. We can calculate C_{ox} as

\[C_{ox}=\frac{\epsilon_{ox}}{t_{ox}}=\frac{3.45\times10^{-11}\ \frac{F}{m}}{7.754\times 10^{-9}\ m}=4.449\times10^{-3}\ \frac{F}{m^2}\]

Thus,

\[\mu_nC_{ox}=148.2\ \frac{cm^2}{V\cdot s}\times \left(4.449\times10^{-3}\ \frac{F}{m^2}\right)=65.9\ \frac{\mu A}{V^2}\]

Almost done:

\[g_m=\sqrt{65.9\ \frac{\mu A}{V^2}\times\left(\frac{35\ \mu m}{0.35\ \mu m}\right)\times500\ \mu A}=0.00182\ \frac{A}{V}\]

Done:

\[A_{diff}=0.00182\ \frac{A}{V}\times5\ k\Omega=9.1\]

Calculated = 9.1, simulated = 10: I’d say that’s pretty close.

### Conclusion

The basic MOSFET differential pair is an important circuit for anyone who wants to delve into analog IC design. There is a lot more we could say about this circuit, but we’ll leave it here for now. In the next article, we’ll look at the improved performance that can be achieved by using an active load instead of drain resistors.

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