Module 5.3 Input impedance

Introduction:

    In this module, we discuss the characteristics of input impedance along a lossless transmission line terminated with a load. By Observe the dynamic behavior and distribution of input impedance along the transmission line by adjusting the range sliders.
    Ex.1：When the load impedance is matched ( \(\left|\Gamma_L\right|\) = 0 ), the input impedance along the transmission line is equal to the characteristic impedance ( \(Z_0\) ) of the line; otherwise, the input impedance varies along the transmission line if the load impedance is mismatched ( \(\left|\Gamma_L\right|\) ≠ 0 )
    Ex.2：For any fixed and mismatched load impedance ( \(\left|\Gamma_L\right|\) ≠ 0 ), the reflection coefficient has the same magnitude but different phases along the transmission line.
    Ex.3：When observation distance is prolonged or the frequency ( \(f\) ) increases, it can be found from the distribution of impedance that the period of the input impedance is a half of the wavelength.

Formula:

1. \(Z_{in}(z) (\Omega)\) : Input impedance looking into the transmission line at z.
    \( Z_{in} \left(z\right) = Z_{0} \cdot \frac { Z_{L} + j Z_{0} { tan\beta }\left|z\right|}{Z_{0} + j Z_{L} { tan\beta }\left|z\right|} (\Omega )\)
2. \(\Gamma(z)\) : Voltage reflection coefficient at z, defined by
    \(\Gamma (z)=\left|\Gamma (0)\right|\cdot e^{j\theta } \cdot e^{-j2\beta \left|z\right|} \)
3. \(SWR\) : Voltage standing wave ratio, defined by
    \(SWR=\frac{V_{\max } }{V_{\min } } =\frac{1+\left|\Gamma (0)\right|}{1-\left|\Gamma (0)\right|} \)

Parameters:

1. \(\left|\Gamma (0)\right|\) : Voltage reflection coefficient at z = 0, defined by
      \(\Gamma (0)=\left|\Gamma (0)\right|\cdot e^{j\theta } =\frac{Z_{L} -Z_{0} }{Z_{L} +Z_{0} } \)
2. \(\theta\) : Phase angle of the voltage reflection coefficient, (rad)
3. \(Z_{L}\) : Load impedance, defined by
      \(Z_{L} =R_{L} +jX_{L} =Z_{0} \cdot \frac{1+\Gamma (0)}{1-\Gamma (0)}   (\Omega) \)
     a. \(R_{L}\) : Real part of \(Z_{L}\),
     b. \(X_{L}\) : Imaginary part of \(Z_{L}\).
4. \(\beta\) : Phase constant, (rad/m)
5. \(f\) : Operating frequency, (GHz)
6. \(d\) : Length of the transmission line, value equal to 10 cm
7. \(Z_{0}\) : Characteristic impedance of the transmission line, \((\Omega)\)