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title: 'Lecture 8: Enhancement type MOSFET'
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  ## DI02011011: Electronics Circuit and Application (ECA)
  Field Effect Transistors(FETs)
---

# DI02011011: Electronics Circuit and Application (ECA)
## Lecture 8: Enhancement type MOSFET
### Field Effect Transistors(FETs)

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Welcome to Lecture 8. Today we build upon the structural physics of Lecture 7 to derive the mathematical V-I characteristics of E-MOSFETs. We will analyze the triode and saturation region equations, channel length modulation, and small-signal transconductance.
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# Lecture Outline: E-MOSFET V-I Characteristics & Modeling

- Operating Modes Overview: Cutoff, Triode, and Saturation region conditions.
- Triode / Ohmic Region Derivation: Low $V_{DS}$ channel resistance modeling.
- Pinch-off Mechanics & Onset of Saturation ($V_{DS(sat)} = V_{GS} - V_{Th}$).
- Saturation Region Derivation: Square-law drain current equation.
- Channel Length Modulation ($\lambda$) and Early Voltage ($V_A$).
- Small-Signal Parameters: Transconductance $g_m$ and Output Resistance $r_o$ formulas.
- Numerical Circuit Problem Walkthrough & Output Curve Plotting.

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Here is our roadmap. We will define operating boundaries, derive the fundamental current equations, introduce second-order channel length modulation, compute transconductance, and solve a comprehensive numerical problem.
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# Cutoff Region Physics ($V_{GS} < V_{Th}$ for NMOS)

- Sub-Threshold Condition: Gate-to-source voltage is less than threshold voltage ($V_{GS} < V_{Th}$).
- Absence of Inversion Channel: Insufficient surface potential prevents formation of mobile electron inversion layer; channel region remains depleted.
- Ideal Current: Ideal drain current is strictly zero ($I_D = 0\text{ A}$); device acts as an open switch.
- Subthreshold Conduction (Real-World): Small diffusion-dominated leakage current flows: $I_D(sub) = I_{S0} \exp\left(\frac{V_{GS} - V_{Th}}{n V_T}\right)$.
- Off-State Power Dissipation: In VLSI CMOS chips with billions of transistors, static subthreshold leakage power ($P_{static} = V_{DD} \times I_{sub}$) is a major power constraint.

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The working principle of an enhancement type MOSFET is governed by the threshold voltage. Below this critical voltage, no channel forms in the body region, and thus no current flows between the source and drain.
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# Triode / Ohmic Region Mechanics ($V_{GS} \ge V_{Th}$ & $V_{DS} < V_{GS} - V_{Th}$)

- Conduction Conditions: Gate voltage exceeds threshold ($V_{GS} \ge V_{Th}$) AND drain voltage is low ($V_{DS} < V_{GS} - V_{Th}$).
- Continuous Channel Profile: Inversion layer remains continuous from Source to Drain along entire channel length $L$.
- Drain Current Derivation: $I_D = k'_n \frac{W}{L} \left[ (V_{GS} - V_{Th}) V_{DS} - \frac{1}{2} V_{DS}^2 \right]$
- Process Transconductance Parameter ($k'_n$): Defined as $k'_n = \mu_n C_{ox} = \mu_n \frac{\epsilon_{ox}}{t_{ox}}$ (measured in $\text{A/V}^2$ or $\mu\text{A/V}^2$).
- Device Transconductance Parameter ($K_n$): Defined as $K_n = k'_n \frac{W}{L} = \mu_n C_{ox} \frac{W}{L}$.

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In the triode region, VGS > VTh forms the channel, and VDS is small enough that the channel remains un-pinched. The drain current depends on both VGS and VDS, exhibiting a parabolic dependence on VDS.
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# Deep Triode Region: Voltage-Controlled Resistance (VCR)

- Very Small $V_{DS}$ Limit ($V_{DS} \ll V_{GS} - V_{Th}$): The quadratic term $\frac{1}{2} V_{DS}^2$ becomes negligible compared to $(V_{GS} - V_{Th}) V_{DS}$.
- Linearized Drain Current Equation: $I_D \approx k'_n \frac{W}{L} (V_{GS} - V_{Th}) V_{DS} = K_n (V_{GS} - V_{Th}) V_{DS}$
- Ohmic Channel Resistance Formula ($R_{DS(on)}$): $R_{DS(on)} = \frac{V_{DS}}{I_D} = \frac{1}{k'_n \frac{W}{L} (V_{GS} - V_{Th})} = \frac{1}{K_n V_{OV}}$
- Gate-Voltage Modulated Resistance: Channel resistance $R_{DS(on)}$ is inversely proportional to overdrive voltage $V_{OV} = V_{GS} - V_{Th}$.
- Practical Applications: Used in analog voltage-controlled attenuators, electronic potentiometers, and analog switches.

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When VDS is very small, the I-V curve is purely linear! The MOSFET acts as a pure resistor whose value RDS(on) is electrically tuned by changing VGS. Higher gate voltage yields lower ON-resistance.
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# Pinch-Off Physics and Saturation Boundary Condition

- Channel Tapering: As $V_{DS}$ increases, potential along channel $V(x)$ rises from $0\text{V}$ at source to $V_{DS}$ at drain, reducing local gate-to-channel voltage $V_{GS} - V(x)$.
- Pinch-Off Onset Voltage ($V_{DS(sat)}$): At the drain end ($x=L$), local gate-to-channel voltage drops to exactly threshold voltage: $V_{GS} - V(L) = V_{Th} \implies V_{DS(sat)} = V_{GS} - V_{Th} = V_{OV}$.
- Physical Pinch-Off Point: Inversion charge at drain drops to zero ($Q_n(L) = 0$), pinching off the channel tip near the drain contact.
- Electron Drift Across Depletion Region: Electrons reaching the pinch-off point are swept across the narrow high-electric-field depletion region into the drain, sustaining steady current flow.
- Current Saturation Independence: Further increases in $V_{DS} > V_{DS(sat)}$ extend the pinch-off region length toward the source, but do not increase the voltage drop across the active channel.

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Pinch-off does NOT stop current flow! It simply caps the voltage drop across the active channel at VDS(sat) = VGS - VTh. Any additional VDS beyond VDS(sat) drops across the pinched-off depletion zone, keeping current saturated.
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# Saturation Region Mathematics: Ideal Square-Law Model

- Saturation Conditions: Gate voltage above threshold ($V_{GS} \ge V_{Th}$) AND drain voltage above saturation onset ($V_{DS} \ge V_{GS} - V_{Th}$).
- Square-Law Equation Derivation: Substituting $V_{DS} = V_{GS} - V_{Th}$ into triode equation yields: $I_D = \frac{1}{2} k'_n \frac{W}{L} (V_{GS} - V_{Th})^2 = \frac{1}{2} K_n (V_{GS} - V_{Th})^2$
- Drain Voltage Independence: Ideal saturation current $I_D$ depends quadratically on gate overdrive voltage $(V_{GS} - V_{Th})$ and is completely independent of $V_{DS}$.
- Amplifier Region of Operation: Analog voltage amplifiers bias the MOSFET strictly in Saturation, where high incremental transconductance $g_m$ yields maximum voltage gain.
- PMOS Saturation Equation: $I_D = \frac{1}{2} k'_p \frac{W}{L} (V_{GS} - V_{Th,p})^2 = \frac{1}{2} k'_p \frac{W}{L} (V_{SG} - |V_{Th,p}|)^2$.

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The square-law formula ID = 1/2 Kn (VGS - VTh)^2 is the core equation of analog IC design! Notice that ID depends quadratically on gate voltage VGS, but is independent of VDS in an ideal transistor.
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# Channel Length Modulation ($\lambda$) and Output Resistance

- Physical Origin: As $V_{DS}$ increases above $V_{DS(sat)}$, the pinch-off point moves towards the source, reducing effective channel length from $L$ to $L' = L - \Delta L$.
- Aspect Ratio Increase ($W / L'$): Since effective length $L'$ shrinks, factor $W / L'$ increases, causing drain current to rise slightly with increasing $V_{DS}$.
- Channel Length Modulation Parameter ($\lambda$): Quantifies finite output resistance, measured in $\text{V}^{-1}$; inverse of Early Voltage ($V_A = 1/\lambda$).
- Modified Saturation Current Equation: $I_D = \frac{1}{2} k'_n \frac{W}{L} (V_{GS} - V_{Th})^2 (1 + \lambda V_{DS})$
- Length Dependency: $\lambda$ is inversely proportional to nominal channel length $L$ ($\lambda \propto 1/L$); longer channels exhibit flatter saturation curves.

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In real transistors, increasing VDS shortens the effective channel length L' (Channel Length Modulation). This introduces a small positive slope in saturation characterized by lambda or Early Voltage VA = 1/lambda.
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# Derivation of Small-Signal Transconductance ($g_m$)

- Definition of Transconductance: Rate of change of drain current with respect to gate-to-source voltage at a fixed Q-point: $g_m = \left. \frac{\partial I_D}{\partial V_{GS}} \right|_{V_{DS}=\text{const}}$.
- Formulation 1 (Overdrive Voltage Form): $g_m = k'_n \frac{W}{L} (V_{GS} - V_{Th}) = K_n V_{OV}$
- Formulation 2 (Current & Aspect Ratio Form): $g_m = \sqrt{2 k'_n \frac{W}{L} I_D} = \sqrt{2 K_n I_D}$
- Formulation 3 (Current & Overdrive Form): $g_m = \frac{2 I_D}{V_{GS} - V_{Th}} = \frac{2 I_D}{V_{OV}}$
- Design Implications: To maximize $g_m$ at a fixed DC bias current $I_D$, designers must minimize overdrive voltage $V_{OV}$ by increasing channel width $W$.

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Transconductance is a measure of how much the drain current changes with respect to changes in the gate-to-source voltage. Learn all three forms of gm: Form 1 shows gm increases with overdrive VOV; Form 2 shows gm scales with sqrt(ID); Form 3 shows gm = 2 ID / VOV.
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# Derivation of Small-Signal Output Resistance ($r_o$)

- Definition of Small-Signal Output Resistance: Incremental internal resistance looking into the drain terminal: $r_o = \left[ \frac{\partial I_D}{\partial V_{DS}} \right]^{-1}_{V_{GS}=\text{const}}$.
- Mathematical Derivation: Differentiating modified saturation equation yields $\frac{\partial I_D}{\partial V_{DS}} = \frac{1}{2} K_n (V_{GS} - V_{Th})^2 \lambda = \frac{\lambda I_D}{1 + \lambda V_{DS}} \approx \lambda I_D$.
- Output Resistance Formulas: $r_o = \frac{1}{\lambda I_{D0}} = \frac{V_A}{I_D}$
- Early Voltage ($V_A$): Extrapolated negative voltage intercept on the $V_{DS}$ axis where saturation curves converge ($V_A = 1/\lambda$ or $V_A = V'_{A} L$).
- Impact on Gain: Finite output resistance $r_o$ limits intrinsic voltage gain of single-stage amplifier to $A_{v0} = -g_m r_o = -\frac{2 V_A}{V_{GS} - V_{Th}}$.

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Output resistance ro represents the finite slope of the saturation curve caused by channel length modulation. Higher Early Voltage VA (achieved with longer channel length L) gives higher ro and higher amplifier gain.
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# Drain Output Characteristics Family Curves Analysis

- Three Operating Regions Mapping: Cutoff region along horizontal axis ($V_{GS} < V_{Th}$); Triode region to the left of parabolic boundary ($V_{DS} < V_{GS} - V_{Th}$); Saturation region to the right ($V_{DS} \ge V_{GS} - V_{Th}$).
- Pinch-Off Parabola Boundary: Locus of pinch-off points defined by curve $I_{D(sat)} = \frac{1}{2} K_n V_{DS}^2$.
- Gate Voltage Parameter Steps: Family of curves plotted for increasing $V_{GS}$ steps ($V_{Th} + 0.5\text{V}$, $V_{Th} + 1.0\text{V}$, $V_{Th} + 1.5\text{V}$, etc.).
- Saturation Slope Visualization: Curves slope slightly upward in saturation due to finite $\lambda$; extrapolating backward converges at $-V_A$ on horizontal axis.
- Breakdown Boundary: At extreme $V_{DS} > V_{DS(max)}$, drain-substrate junction breakdown causes sharp vertical current rise.

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Study the drain characteristics graph: the parabolic boundary separates the Triode region (left) from Saturation (right). In Saturation, curves are nearly horizontal, lifting slightly due to channel length modulation.
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# Numerical Analysis of E-MOSFET Bias Point & Region Identification

- Circuit Parameters: Given NMOS with $V_{Th} = 1.0\text{V}$, $k'_n = 100\text{ }\mu\text{A/V}^2$, $W/L = 10$, $\lambda = 0.02\text{ V}^{-1}$. Circuit biased at $V_{GS} = 3.0\text{V}$ and $V_{DS} = 5.0\text{V}$.
- Step 1: Determine Overdrive Voltage ($V_{OV}$): $V_{OV} = V_{GS} - V_{Th} = 3.0\text{V} - 1.0\text{V} = 2.0\text{V}$.
- Step 2: Check Saturation Condition: $V_{DS} = 5.0\text{V} \ge V_{OV} = 2.0\text{V} \implies$ Transistor is in SATURATION.
- Step 3: Calculate $K_n$: $K_n = k'_n \frac{W}{L} = 100\text{ }\mu\text{A/V}^2 \times 10 = 1.0\text{ mA/V}^2$.
- Step 4: Compute Drain Current $I_D$: $I_D = \frac{1}{2} (1.0\text{ mA/V}^2) (2.0\text{V})^2 \times [1 + (0.02)(5.0)] = 0.5 \times 4.0 \times (1.10) = 2.20\text{ mA}$.
- Step 5: Compute Transconductance $g_m$ and Output Resistance $r_o$: $g_m = K_n V_{OV} = (1.0\text{ mA/V}^2)(2.0\text{V}) = 2.0\text{ mS}$; $r_o = \frac{1}{\lambda I_{D0}} = \frac{1}{(0.02\text{ V}^{-1})(2.0\text{ mA})} = 25\text{ k}\Omega$.

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Follow this systematic 5-step problem solving approach: First compute overdrive voltage VOV = VGS - VTh; second test if VDS >= VOV to verify saturation; third apply the modified square law; fourth calculate ID; fifth compute small-signal gm and ro.
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# Parasitic Capacitances and High-Frequency Model

- Gate Oxide Capacitances: Gate-to-Source capacitance $C_{gs}$ and Gate-to-Drain capacitance $C_{gd}$.
- Triode Region Capacitances: $C_{gs} = C_{gd} = \frac{1}{2} W L C_{ox} + W L_{ov} C_{ox}$.
- Saturation Region Capacitances: Channel pinches off at drain; $C_{gs} = \frac{2}{3} W L C_{ox} + W L_{ov} C_{ox}$, while $C_{gd} = W L_{ov} C_{ox}$ (overlap capacitance only).
- Cutoff Region Capacitances: $C_{gs} = C_{gd} = W L_{ov} C_{ox}$; Gate-to-Body capacitance $C_{gb} = W L C_{ox}$.
- Transition Frequency ($f_T$): Frequency at which short-circuit current gain drops to unity: $f_T = \frac{g_m}{2\pi (C_{gs} + C_{gd})} \approx \frac{\mu_n (V_{GS} - V_{Th})}{2\pi L^2}$.

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At high frequencies, parasitic capacitances Cgs and Cgd limit speed. In saturation, Cgs equals 2/3 of gate oxide capacitance plus overlap, while Cgd drops to overlap capacitance only. Unity gain bandwidth fT scales inversely with channel length squared L^2!
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# Summary of E-MOSFET Operational Equations

- Cutoff ($V_{GS} < V_{Th}$): $I_D = 0\text{ A}$.
- Triode ($V_{GS} \ge V_{Th}, V_{DS} < V_{GS} - V_{Th}$): $I_D = k'_n \frac{W}{L} \left[ (V_{GS} - V_{Th}) V_{DS} - \frac{1}{2} V_{DS}^2 \right]$.
- Deep Triode ($V_{DS} \ll V_{GS} - V_{Th}$): $R_{DS(on)} = \frac{1}{k'_n (W/L) (V_{GS} - V_{Th})}$.
- Saturation ($V_{GS} \ge V_{Th}, V_{DS} \ge V_{GS} - V_{Th}$): $I_D = \frac{1}{2} k'_n \frac{W}{L} (V_{GS} - V_{Th})^2 (1 + \lambda V_{DS})$.
- Small-Signal Formulas: $g_m = \sqrt{2 K_n I_D} = K_n V_{OV}$; $r_o = \frac{V_A}{I_D} = \frac{1}{\lambda I_D}$.

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In summary, enhancement type MOSFETs have a p-substrate with n+ source and drain regions. Channel forms above threshold voltage, controlling current flow according to triode or square-law saturation equations.
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