Let's solve **part (e)** step by step: --- ## **Given Data** - \( V_{DD} = 2 \, \text{V} \) - \( R_D = 15\, \text{k}\Omega \) - \( V_t = .5\,\text{V} \) - \( k_n' = .4\,\text{mA/V}^2 \) - \( \frac{W}{L} = 12.5 \) - \( V_{GS} = .7\,\text{V} \) - \( \lambda = \) - \( v_{gs} = .015 \sin \omega t \) - Eq. (7.28): \( i_D = \frac{1}{2} k_n (V_{GS} + v_{gs} - V_t)^2 \) Where \( k_n = k_n' \frac{W}{L} = .4 \times 12.5 = 5\,\text{mA/V}^2 \). --- ## **1. Substitute \( v_{gs} \) into the Equation** \[ i_D = \frac{1}{2} k_n (V_{GS} + v_{gs} - V_t)^2 \] Let \( V_{ov} = V_{GS} - V_t = .7 - .5 = .2\,\text{V} \): \[ i_D = \frac{1}{2} k_n (V_{ov} + v_{gs})^2 \] Expand: \[ i_D = \frac{1}{2} k_n [V_{ov}^2 + 2V_{ov}v_{gs} + v_{gs}^2] \] \[ i_D = \frac{1}{2} k_n V_{ov}^2 + k_n V_{ov} v_{gs} + \frac{1}{2} k_n v_{gs}^2 \] --- ## **2. Substitute \( v_{gs} = .015 \sin \omega t \)** Let \( A = .015 \): \[ i_D = \frac{1}{2} k_n V_{ov}^2 + k_n V_{ov} (A \sin \omega t) + \frac{1}{2} k_n (A \sin \omega t)^2 \] Expand the last term: \[ (A \sin \omega t)^2 = A^2 \sin^2 \omega t \] Recall the identity: \[ \sin^2 \omega t = \frac{1}{2} (1 - \cos 2\omega t) \] So, \[ i_D = \frac{1}{2} k_n V_{ov}^2 + k_n V_{ov} A \sin \omega t + \frac{1}{2} k_n A^2 \frac{1}{2} (1 - \cos 2\omega t) \] \[ i_D = \frac{1}{2} k_n V_{ov}^2 + k_n V_{ov} A \sin \omega t + \frac{1}{4} k_n A^2 (1 - \cos 2\omega t) \] --- ## **3. Identify Harmonic Components** - **DC Component:** \( \frac{1}{2} k_n V_{ov}^2 + \frac{1}{4} k_n A^2 \) - **Fundamental (\(\omega\)) Component:** \( k_n V_{ov} A \sin \omega t \) - **Second Harmonic (\(2\omega\)) Component:** \( -\frac{1}{4} k_n A^2 \cos 2\omega t \) --- ## **4. Find the Second-Harmonic Distortion** Second-harmonic distortion (SHD) is defined as: \[ \text{SHD} = \frac{\text{amplitude of } 2\omega \text{ component}}{\text{amplitude of } \omega \text{ component}} \times 100 \] - Amplitude at \(\omega\): \( k_n V_{ov} A \) - Amplitude at \(2\omega\): \( \frac{1}{4} k_n A^2 \) \[ \text{SHD} = \frac{\frac{1}{4} k_n A^2}{k_n V_{ov} A} \times 100 = \frac{A}{4V_{ov}} \times 100 \] Substitute \( A = .015 \), \( V_{ov} = .2 \): \[ \text{SHD} = \frac{.015}{4 \times .2} \times 100 = \frac{.015}{.8} \times 100 = .01875 \times 100 = 1.875\% \] --- ## **Summary Table** | Component | Value | |---|---| | **DC Component** | \( \frac{1}{2} k_n V_{ov}^2 + \frac{1}{4} k_n A^2 \) | | **\(\omega\) Component** | \( k_n V_{ov} A \sin \omega t \) | | **\(2\omega\) Component** | \( -\frac{1}{4} k_n A^2 \cos 2\omega t \) | | **Second-Harmonic Distortion** | \( 1.88\% \) | --- ## **Final Answers** ### **Various Components of \( i_D \):** \[ i_D = \underbrace{\frac{1}{2} k_n V_{ov}^2 + \frac{1}{4} k_n A^2}_{\text{DC}} + \underbrace{k_n V_{ov} A \sin \omega t}_{\omega} \underbrace{-\frac{1}{4} k_n A^2 \cos 2\omega t}_{2\omega} \] ### **Second-Harmonic Distortion:** \[ \boxed{1.88\%} \] --- **Let me know if you need the values plugged in for the DC and AC components!**

Sure! Let's plug in the values for the DC and AC components based on the data provided. --- ## **Values Used** - \( k_n = 5 \, \text{mA/V}^2 \) - \( V_{ov} = 0.2 \, \text{V} \) - \( A = 0.015 \) --- ## **1. Calculate the DC Component** The DC component is given by: \[ \text{DC Component} = \frac{1}{2} k_n V_{ov}^2 + \frac{1}{4} k_n A^2 \] Substituting the values: \[ \text{DC Component} = \frac{1}{2} \cdot 5 \cdot (0.2)^2 + \frac{1}{4} \cdot 5 \cdot (0.015)^2 \] Calculating: \[ = \frac{1}{2} \cdot 5 \cdot 0.04 + \frac{1}{4} \cdot 5 \cdot 0.000225 \] \[ = 0.1 + 0.00028125 \] \[ = 0.10028125 \, \text{mA} \] --- ## **2. Calculate the Amplitude of the Fundamental (\(\omega\)) Component** The amplitude at \(\omega\) is given by: \[ \text{Amplitude}_{\omega} = k_n V_{ov} A \] Substituting the values: \[ \text{Amplitude}_{\omega} = 5 \cdot 0.2 \cdot 0.015 \] Calculating: \[ = 5 \cdot 0.003 = 0.015 \, \text{mA} \] --- ## **3. Calculate the Amplitude of the Second Harmonic (\(2\omega\)) Component** The amplitude at \(2\omega\) is given by: \[ \text{Amplitude}_{2\omega} = -\frac{1}{4} k_n A^2 \] Substituting the values: \[ \text{Amplitude}_{2\omega} = -\frac{1}{4} \cdot 5 \cdot (0.015)^2 \] Calculating: \[ = -\frac{1}{4} \cdot 5 \cdot 0.000225 \] \[ = -\frac{1}{4} \cdot 0.001125 = -0.00028125 \, \text{mA} \] --- ## **Final Summary of Components** | Component | Value (mA) | |---|---| | **DC Component** | \( 0.10028125 \) | | **\(\omega\) Component Amplitude** | \( 0.015 \) | | **\(2\omega\) Component Amplitude** | \( -0.00028125 \) | --- ## **Final Second-Harmonic Distortion Calculation** Using the previously calculated values: \[ \text{SHD} = \frac{0.00028125}{0.015} \times 100 \] Calculating: \[ = \frac{0.00028125 \, \text{mA}}{0.015 \, \text{mA}} \times 100 = 1.875\% \] --- ## **Final Answers** ### **Various Components of \( i_D \):** \[ i_D = 0.10028125 \, \text{mA} + 0.015 \sin \omega t - 0.00028125 \cos 2\omega t \] ### **Second-Harmonic Distortion:** \[ \boxed{1.875\%} \] --- This concludes the calculations for the DC and AC components, as well as the second-harmonic distortion. Let me know if you need any further assistance!