Breaking Down the Input Impedance Formula

This video walks through the full step-by-step derivation of the input
impedance formula for a transmission line — one of the most important results
in electromagnetic theory and microwave engineering.

Starting from the general voltage and current wave equations, we build up to
the complete expression for Z_in using the reflection coefficient and hyperbolic
functions — then simplify to the final lossless form using tan(βl).

If you're studying for an electromagnetics exam, diving into RF engineering,
or trying to understand transmission line theory before tackling the Smith Chart
— this derivation is the foundation you need.

━━━━━━━━━━━━━━━━━━━━━━━━━━
📐 FINAL FORMULA
━━━━━━━━━━━━━━━━━━━━━━━━━━
Z_in = Z0 · (ZL + j·Z0·tan(βl)) / (Z0 + j·ZL·tan(βl))

Where:
Z0 = Characteristic Impedance
ZL = Load Impedance
β = Phase Constant
l = Length of the transmission line
j = Imaginary unit

━━━━━━━━━━━━━━━━━━━━━━━━━━
📚 WHAT YOU'LL LEARN
━━━━━━━━━━━━━━━━━━━━━━━━━━
✅ How to express impedance Z(x) from incident and reflected voltage waves
✅ How to rewrite Z(x) in terms of the reflection coefficient Γ
✅ Why switching from x to x' (distance from the load) simplifies the math
✅ How to substitute the load reflection coefficient ΓL
✅ How cosh and sinh lead to the general hyperbolic impedance formula
✅ How to reduce the full formula to the lossless tan(βl) version
✅ How Z_in connects to impedance matching and Smith Chart analysis

━━━━━━━━━━━━━━━━━━━━━━━━━━
⏱️ TIMESTAMPS
━━━━━━━━━━━━━━━━━━━━━━━━━━
0:00 – Introduction: What Is Input Impedance of a Transmission Line?
0:40 – General Impedance Z(x) From Voltage and Current Waves
1:24 – Rewriting Z(x) Using the Reflection Coefficient
2:35 – Isolating the Reflection Coefficient Γ(x)
3:55 – Switching Variables: From x to x' (Distance From the Load)
5:07 – Substituting the Load Reflection Coefficient ΓL
7:16 – Introducing Hyperbolic Functions (cosh and sinh)
10:52 – Final Input Impedance Formula: Z_in at x' = L

━━━━━━━━━━━━━━━━━━━━━━━━━━
🎓 ABOUT THIS CHANNEL
━━━━━━━━━━━━━━━━━━━━━━━━━━
Byte-Sized Learning delivers short, precise technical derivations for
electronics and electrical engineering students. Straight to the point —
no filler, just the math you need.

━━━━━━━━━━━━━━━━━━━━━━━━━━
🔖 TOPICS
━━━━━━━━━━━━━━━━━━━━━━━━━━
transmission line derivation, input impedance, characteristic impedance,
electromagnetic theory, microwave engineering, reflection coefficient,
transmission line theory, electrical engineering, wave equation, smith chart,
RF engineering, impedance matching

Видео Breaking Down the Input Impedance Formula канала Byte-Sized Learning