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Breaking Down the Input Impedance Formula

In this video, we derive the formula for the Input Impedance (Z_in) of a transmission line. We start from the fundamental voltage V(x) and current I(x) equations and work step-by-step to find the ratio V(-l)/I(-l) at the input. This is a key derivation for Electromagnetics and RF Engineering courses. Understanding how Z_in changes with line length is crucial for impedance matching and Smith Chart applications. Key Formulas Derived: • Input Definition: Z_in = V(-l) / I(-l) • Voltage Wave: V(x) = V₀⁺e^(-jβx) + V₀⁻e^(+jβx) • Current Wave: I(x) = (V₀⁺/Z₀)e^(-jβx) - (V₀⁻/Z₀)e^(+jβx) • Reflection Coefficient: Γ = V₀⁻ / V₀⁺ Final Input Impedance Formula: Z_in = Z₀ · [ (Z_L + j·Z₀·tan(βl)) / (Z₀ + j·Z_L·tan(βl)) ] Where: • Z₀ = Characteristic Impedance • Z_L = Load Impedance • β = Phase Constant (Beta) • l = Length of the line • j = Imaginary unit #TransmissionLines #Electromagnetics #ElectricalEngineering #InputImpedance #Zin #EngineeringTutorial #RFEngineering