MOSFET I-V Characteristics: Saturation & Triode

Recommended resources

The freely available resources this lesson will direct you to.

Welcome back. In our previous lesson, we established a crucial strategy for analyzing MOSFET circuits: assume an operating region (typically saturation), analyze the circuit, and then verify your assumption. Today, we will put the "meat on the bones" of that strategy by mastering the specific current-voltage (I-V) equations for each region.

Your goal for this lesson is to become proficient in calculating the drain current, $I_{D}$ , using the correct I-V formula for both the saturation and triode regions. This skill is the computational engine for all DC analysis of MOSFETs and is fundamental to solving the exam-style problems you'll be facing.

The MOSFET I-V Characteristic Equations

Let's begin by recalling the MOSFET's characteristic curves. These curves visually represent the mathematical equations we'll be using today.

This graph plots drain current ($i_D$) versus drain-source voltage ($v_{DS}$) for an N-channel MOSFET at several gate-source voltages ($v_{GS}$). The equations we will study describe these curves mathematically.

For an N-channel enhancement MOSFET, there are two primary "ON" states, each with a distinct equation governing the drain current. In our analysis, we will assume the channel-length modulation effect is negligible ( $λ = 0$ ). We will address this effect in a later lesson.

1. The Saturation Region Equation

This is the most important region for amplifier design. When a MOSFET is in saturation, it behaves like a voltage-controlled current source, where the drain current is ideally independent of $V_{D S}$ .

Condition: $V_{GS} \geq V_{t}$ and $V_{D S} \geq V_{GS} - V_{t}$
Equation: $I_{D} = \frac{1}{2} k_{n}^{'} \frac{W}{L} (V_{GS} - V_{t})^{2} = \frac{1}{2} k_{n} (V_{GS} - V_{t})^{2}$

Here:

$k_{n}^{'} = μ_{n} C_{o x}$ is the process transconductance parameter, determined by the manufacturing technology.
$W / L$ is the aspect ratio of the transistor, a key design parameter.
$k_{n} = k_{n}^{'} (W / L)$ is the transistor transconductance parameter.
$V_{GS} - V_{t} = V_{O V}$ is the overdrive voltage.

Notice that the drain current has a square-law relationship with the gate-source voltage and is (ideally) independent of the drain-source voltage.

2. The Triode (or Linear) Region Equation

In the triode region, the MOSFET acts more like a voltage-controlled resistor. The drain current depends on both $V_{GS}$ and $V_{D S}$ .

Condition: $V_{GS} \geq V_{t}$ and $V_{D S} < V_{GS} - V_{t}$
Equation: $I_{D} = k_{n}^{'} \frac{W}{L} [(V_{GS} - V_{t}) V_{D S} - \frac{1}{2} V_{D S}^{2}] = k_{n} [(V_{GS} - V_{t}) V_{D S} - \frac{1}{2} V_{D S}^{2}]$

This equation describes the rising portion of the curves in the graph above.

Applying the Equations: A Systematic Approach

Now let's apply these equations within the "assume and verify" framework. We will walk through a detailed example that mirrors a typical exam problem.

The following video demonstrates the analysis of a common four-resistor biasing circuit. The key learning point here is observing the process when the initial assumption of saturation turns out to be incorrect.

MOSFET Circuits in DC, Part 3

This video, "MOSFET Circuits in DC, Part 3," from the channel unwired, provides a comprehensive walkthrough of a DC analysis problem where the initial assumption fails, forcing a re-analysis in the triode region.

Please watch the following segments carefully: Circuit Setup: The instructor calculates the gate voltage V_G using a voltage divider and sets up the expression for the source voltage V_S. This is the standard starting point. Assume Saturation: Watch how the saturation equation is used. The instructor substitutes the expressions for V_{GS} into the equation, resulting in a quadratic equation for I_D. Verification Step: This is critical. After solving for I_D, the node voltages are calculated. The instructor checks the saturation condition (V_{DS} \ge V_{GS} - V_t) and finds that it is violated, proving the assumption was wrong. Re-analyze in Triode: Now, the correct triode equation is used. Notice that this leads to a more complex quadratic equation involving I_D. Final Solution and Check: The quadratic equation yields two possible values for I_D. The instructor calculates the corresponding voltages for both and uses the triode condition (V_{DS} < V_{OV}) to identify the single physically correct solution.

The process shown in the video is the complete algorithm. First, you make a calculated guess (saturation), then you test it. If it fails, you switch to the triode model and solve again, being careful to select the physically meaningful solution.

Worked Examples from Text

Let's reinforce this with examples from your textbook resource, Microelectronic Circuits. These examples provide clear, step-by-step calculations for different scenarios.

Example 1: Calculation in the Triode Region

This example starts with a circuit where the operating region is known to be triode, providing a direct application of the triode I-V equation.

5.3 MOSFET Circuits at DC 277

Example 5.5 demonstrates a straightforward calculation for a MOSFET operating deep in the triode region.

Find and read through Example 5.5 (page 280). Note how the condition V_D < V_G - V_t is immediately apparent, confirming triode operation. The drain current I_D is then calculated using the full triode equation. The example also shows how to find the effective resistance r_{DS} in this region.

Example 2: Calculation with a Saturation Assumption

This next example is a classic voltage-divider bias circuit, very similar to the one in the video you just watched. It requires you to assume saturation and solve a quadratic equation.

5.3 MOSFET Circuits at DC 277

Example 5.6 is an excellent, comprehensive problem that illustrates the full "assume and verify" procedure.

Please study Example 5.6 (pages 281-283). Follow these key steps in the solution: Calculation of V_G via the voltage divider. Writing the expression for V_{GS} in terms of I_D. Substituting this into the saturation equation to form a quadratic equation for I_D. Solving the quadratic equation and, importantly, discarding the non-physical root. Finally, calculating all node voltages and verifying that the saturation condition V_D > V_G - V_t holds true.

A Note on PMOS Transistors

The principles for P-channel MOSFETs are identical, but you must be careful with voltage polarities. We use source-referenced voltages ( $V_{S G}$ , $V_{S D}$ ) and the absolute value of the (negative) threshold voltage, $∣ V_{tp} ∣$ .

Saturation: $V_{S G} \geq ∣ V_{tp} ∣$ and $V_{S D} \geq V_{S G} - ∣ V_{tp} ∣$ $I_{D} = \frac{1}{2} k_{p}^{'} \frac{W}{L} (V_{S G} - ∣ V_{tp} ∣)^{2} = \frac{1}{2} k_{p} (V_{S G} - ∣ V_{tp} ∣)^{2}$
Triode: $V_{S G} \geq ∣ V_{tp} ∣$ and $V_{S D} < V_{S G} - ∣ V_{tp} ∣$ $I_{D} = k_{p}^{'} \frac{W}{L} [(V_{S G} - ∣ V_{tp} ∣) V_{S D} - \frac{1}{2} V_{D S}^{2}] = k_{p} [(V_{S G} - ∣ V_{tp} ∣) V_{S D} - \frac{1}{2} V_{D S}^{2}]$

You can see a worked example for a PMOS device in Example 5.7 of the Microelectronic Circuits resource, which you also reviewed briefly in the last lesson.

Summary and Key Takeaways

In this lesson, you've learned to apply the specific mathematical models for MOSFET drain current.

The saturation region is characterized by the square-law equation where $I_{D}$ is controlled by $V_{GS}$ .
$I_{D} = \frac{1}{2} k_{n} (V_{GS} - V_{t})^{2}$
The triode region is described by a more complex equation where $I_{D}$ depends on both $V_{GS}$ and $V_{D S}$ .
$I_{D} = k_{n} [(V_{O V}) V_{D S} - \frac{1}{2} V_{D S}^{2}]$
The standard analysis procedure is to assume saturation, solve for $I_{D}$ and terminal voltages, and then verify the saturation condition. If the condition fails, you must re-analyze using the triode equation.
Solving for $I_{D}$ often requires solving a quadratic equation. Always check both mathematical solutions to identify the one that is physically valid for the circuit.

With the ability to both identify the operating region and calculate the corresponding current, you are now ready to tackle the full DC analysis of MOSFET circuits. In our next lesson, we will formalize this by defining and analyzing the DC operating point, or Q-point, for various common single-transistor amplifier circuits.