MOSFET Transistor Equations: Derivations, Worked Examples, and Part Selection

The primary MOSFET transistor equations governing drain current depend entirely on the operating region. For the saturation (active) region, the foundational square-law equation is I_D = ½ μ_nC_ox(W/L)(V_GS - V_th)². However, for power switching applications, the triode (ohmic) region equation I_D = V_DS / R_DS(on) dictates real-world thermal performance. Below, we derive the saturation formula, track units through solved problems, and map these textbook math concepts directly to discrete power MOSFET datasheets.

The Core MOSFET Transistor Equations: Saturation vs. Triode

The Shichman-Hodges model describes MOSFET behavior across two primary operating states. The formula applies strictly under the assumption of a long-channel device, utilizing the gradual channel approximation, and assuming constant carrier mobility. It ignores short-channel effects like velocity saturation and channel-length modulation (setting the lambda parameter to zero for the base derivation).

Symbol Definition and Spec Sheet Table

When V_DS ≥ (V_GS - V_th), the channel pinches off at the drain end, and the device enters saturation. This is where the square-law equation applies. When V_DS < (V_GS - V_th), the device operates in the triode (linear) region, acting as a voltage-controlled resistor.

Rearranged Forms: Solving for Every Variable

In bench design and reverse-engineering, you rarely solve for I_D directly. Here are the algebraically rearranged forms of the saturation equation, solving for each critical variable:

• Solving for Gate Drive (V_GS):
  V_GS = √[ 2 · I_D / (μ_nC_ox · W/L) ] + V_th
• Solving for Threshold Voltage (V_th):
  V_th = V_GS - √[ 2 · I_D / (μ_nC_ox · W/L) ]
• Solving for Geometry Ratio (W/L):
  W/L = 2 · I_D / [ μ_nC_ox · (V_GS - V_th)² ]
• Solving for Process Parameter (μ_nC_ox):
  μ_nC_ox = 2 · I_D / [ (W/L) · (V_GS - V_th)² ]

Bridging Textbook Math to Datasheet Specs

A common point of failure for hobbyists is trying to plug textbook variables into a power MOSFET datasheet. Manufacturers like Infineon or Vishay rarely publish μ_nC_ox or the exact W/L ratio for discrete power FETs like the IRLZ44N. Instead, they group the physical constants into a single device-specific transconductance parameter, often denoted as K or derived from the I_D(on) specification at a fixed V_GS (usually 10V or 4.5V).

For power switching, you are intentionally driving the MOSFET deep into the triode region to minimize V_DS. Therefore, the saturation equation is primarily useful for constant-current loads, LED drivers, and analog amplification. For switching, your governing equation becomes Ohm's law applied to the channel resistance: P_dissipated = I_D² × R_DS(on).

Worked Examples with Unit Tracking

Problem 1: Calculating Saturation Current for a Small-Signal FET

Given: A 2N7000-style small-signal MOSFET with a process transconductance μ_nC_ox = 150 μA/V², a geometry ratio W/L = 40, a threshold voltage V_th = 2.0V, and an applied gate voltage V_GS = 4.5V.

Find: The drain current I_D in the saturation region.

1. Convert μ_nC_ox to base SI units: 150 μA/V² = 150 × 10^-6 A/V² = 0.00015 A/V².
2. Calculate the overdrive voltage: (V_GS - V_th) = 4.5V - 2.0V = 2.5V.
3. Square the overdrive voltage: (2.5)² = 6.25 V².
4. Apply the formula: I_D = ½ × (0.00015 A/V²) × 40 × 6.25 V².
5. Multiply the constants: ½ × 0.00015 × 40 = 0.003 A/V².
6. Final calculation: I_D = 0.003 × 6.25 = 0.01875 A (or 18.75 mA).

Problem 2: Finding Required Gate Drive for a Target Current

Given: A custom integrated MOSFET designed to deliver exactly 50 mA in saturation. The device has μ_nC_ox = 100 μA/V², W/L = 100, and V_th = 1.5V.

Find: The required V_GS.

1. Convert target current and transconductance: I_D = 0.050 A; μ_nC_ox = 0.0001 A/V².
2. Use the rearranged formula: V_GS = √[ 2 · I_D / (μ_nC_ox · W/L) ] + V_th.
3. Calculate the denominator: 0.0001 A/V² × 100 = 0.01 A/V².
4. Calculate the numerator: 2 × 0.050 A = 0.10 A.
5. Divide and take the square root: √(0.10 / 0.01) = √(10) ≈ 3.162 V.
6. Add threshold voltage: V_GS = 3.162V + 1.5V = 4.662 V.

Common Unit Mistakes and Magnitude Sanity Checks

Magnitude Sanity Checks

• Small-Signal FETs (e.g., 2N7000, BS170): Realistic I_D magnitudes are between 10 mA and 250 mA. If your saturation equation yields 5A for a TO-92 package, you have missed a decimal point.
• Power FETs (e.g., IRF520, IRLZ44N): Realistic continuous I_D magnitudes are 10A to 100A. If your math yields 0.005A for a power switching application, your V_GS is barely above V_th, placing the device in the subthreshold (weak inversion) region where the square-law model breaks down entirely.
• W/L Ratio Reality: In IC design, W/L is typically between 1 and 100. In discrete power MOSFETs, the die consists of thousands of parallel hexagonal or trench cells. The effective W/L ratio is massive (often >10,000), which is why power FETs can pass massive current even with relatively low μ_nC_ox values.

Decision Path: Picking the Right Power MOSFET for Your Load

Theoretical equations dictate silicon physics, but practical circuit design requires selecting a physical part that won't melt under load. Use this decision tree to terminate your design phase with a concrete component selection.

Default Recommendation: If you are building a 12V or 24V DIY load switch (motor, solenoid, high-power LED array) driven by a 5V microcontroller, bypass standard MOSFETs entirely. Buy the IRLZ44N. Its datasheet explicitly guarantees a low R_DS(on) at V_GS = 5V, ensuring the device operates fully in the low-loss triode region without requiring a secondary 12V gate driver circuit.