Let's solve both **2.12** and **2.13** step-by-step using Smith Chart concepts for a **50 Ω transmission line**. --- # **2.12: Short-circuited 50 Ω Stub** Given: - Characteristic Impedance \( Z_0 = 50\,\Omega \) - The stub is **short-circuited** at the end. We need the **shortest length** of stub that presents the following input impedances: --- ## **Step 1: Normalize the Impedance** Normalized impedance \( z_{in} = \frac{Z_{in}}{Z_0} \): - (a) \( z_{in} = 0 \) - (b) \( z_{in} = \infty \) - (c) \( z_{in} = j \) - (d) \( z_{in} = -j \) --- ## **Step 2: Locate on Smith Chart** For a **short-circuited stub**: - At the short (\( z = 0 \)), and as you move away, you travel **clockwise** on the Smith Chart. Lengths are expressed as a **fraction of a wavelength (\( \lambda \))**. --- ## **Step 3: Use the Smith Chart Principles** ### **(a) \( Z_{in} = 0 \) (\( z_{in} = 0 \))** - At the short-circuit (stub length \( l = 0 \)), the normalized input impedance is 0. - **Shortest length:** \(\boxed{0}\) --- ### **(b) \( Z_{in} = \infty \) (\( z_{in} = \infty \))** - At a distance \( \lambda/4 \) from the short, impedance transforms from short to open. - **Shortest length:** \(\boxed{\frac{\lambda}{4}}\) --- ### **(c) \( Z_{in} = j50\,\Omega \) (\( z_{in} = +j \))** - This is a normalized **+j** (inductive reactance). - On the Smith Chart, from the origin (\( z=0 \)), move **clockwise** to \( +j \) (the top of the chart). - This is at \( \lambda/8 \) from the short. **Shortest length:** \(\boxed{\frac{\lambda}{8}}\) --- ### **(d) \( Z_{in} = -j50\,\Omega \) (\( z_{in} = -j \))** - This is a normalized **-j** (capacitive reactance). - On the Smith Chart, from the origin (\( z=0 \)), move **clockwise** to \( -j \) (the bottom of the chart). - This is at \( 3\lambda/8 \) from the short (or equivalently, \( -\lambda/8 \) going counterclockwise, but shortest positive length is \( 3\lambda/8 \)). **Shortest length:** \(\boxed{\frac{3\lambda}{8}}\) --- ## **Summary Table (Short-circuited stub)** | Input Impedance | Normalized | Shortest Length (\( l \)) | |---------------------|------------|-----------------------------| | \( Z_{in} = 0 \) | 0 | \( 0 \) | | \( Z_{in} = \infty \) | \( \infty \) | \( \frac{\lambda}{4} \) | | \( Z_{in} = j50\,\Omega \) | \( +j \) | \( \frac{\lambda}{8} \) | | \( Z_{in} = -j50\,\Omega \) | \( -j \) | \( \frac{3\lambda}{8} \) | --- # **2.13: Open-circuited Stub** Now, **repeat for open-circuited stub** (i.e., the end is open, \( Z_{L} = \infty \)). - At the open, \( z = \infty \). You **also move clockwise** from the open. --- ## **(a) \( Z_{in} = 0 \) (\( z_{in} = 0 \))** - At \( \lambda/4 \) from the open, the impedance transforms from open to short. **Shortest length:** \(\boxed{\frac{\lambda}{4}}\) --- ## **(b) \( Z_{in} = \infty \) (\( z_{in} = \infty \))** - At the open itself (\( l = 0 \)), the input impedance is infinite. **Shortest length:** \(\boxed{0}\) --- ## **(c) \( Z_{in} = j50\,\Omega \) (\( z_{in} = +j \))** - From \( z = \infty \), move **clockwise** to \( +j \) (top). - This is at \( 3\lambda/8 \) from the open. **Shortest length:** \(\boxed{\frac{3\lambda}{8}}\) --- ## **(d) \( Z_{in} = -j50\,\Omega \) (\( z_{in} = -j \))** - From \( z = \infty \), move **clockwise** to \( -j \) (bottom). - This is at \( \lambda/8 \) from the open. **Shortest length:** \(\boxed{\frac{\lambda}{8}}\) --- ## **Summary Table (Open-circuited stub)** | Input Impedance | Normalized | Shortest Length (\( l \)) | |---------------------|------------|-----------------------------| | \( Z_{in} = 0 \) | 0 | \( \frac{\lambda}{4} \) | | \( Z_{in} = \infty \) | \( \infty \) | \( 0 \) | | \( Z_{in} = j50\,\Omega \) | \( +j \) | \( \frac{3\lambda}{8} \) | | \( Z_{in} = -j50\,\Omega \) | \( -j \) | \( \frac{\lambda}{8} \) | --- ## **Summary** **2.12 (Short-circuited):** - (a) \( 0 \) - (b) \( \lambda/4 \) - (c) \( \lambda/8 \) - (d) \( 3\lambda/8 \) **2.13 (Open-circuited):** - (a) \( \lambda/4 \) - (b) \( 0 \) - (c) \( 3\lambda/8 \) - (d) \( \lambda/8 \) --- ### **Smith Chart Reference Diagram** The above results are based on the standard traversal of the Smith Chart for lossless lines. If you need a visual, let me know!