Transmission Line Input Impedance

Transcription

1 1/26/25 Transmission e Input Impedance.doc 1/9 Transmission e Input Impedance Consider a lossless le, length, termated with a load. I(z) I + V (z) -, β + V - z z et s determe the put impedance of this le! Q: Just what do you mean by put impedance? A: The put impedance is simply the le impedance seen at the begng (z ) of the transmission le, i.e.: V ( z ) ( z ) I ( z ) Note equal to neither the load impedance nor the characteristic impedance! and

2 1/26/25 Transmission e Input Impedance.doc 2/9 To determe exactly what is, we first must determe the voltage and current at the begng of the transmission le (z ). + + jβ jβ V (z ) V e + Γ e Therefore: V + + jβ jβ I(z ) e e Γ ( ) ( ) V z e +Γ e + jβ jβ + jβ jβ I z e Γe We can explicitly write terms of load usg the previously determed relationship: Γ + Combg these two expressions, we get: + j β j β ( + ) e + ( ) e + β β ( + ) ( ) + jβ jβ + jβ jβ ( e + e ) + ( e e ) + jβ jβ + jβ jβ ( + ) ( ) e e j j e e e e Now, recall Euler s equations: e e + j β j β + j s β j s β

3 1/26/25 Transmission e Input Impedance.doc 3/9 Usg Euler s relationships, we can likewise write the put impedance without the complex exponentials: + j j + s β + j tan β + tan j β s β Note that dependg on the values of β, and, the put impedance can be radically different from the load impedance! Special Cases Now let s look at the for some important load impedances and le lengths. You should commit these results to memory! 1. λ 2 If the length of the transmission le is exactly one-half wavelength ( λ 2), we fd that: meang that: β 2π λ π λ 2 cos π 1 and s β s π

4 1/26/25 Transmission e Input Impedance.doc 4/9 and therefore: + j j s β + s β ( 1) + j () ( 1) + () j In other words, if the transmission le is precisely one-half wavelength long, the put impedance is equal to the load impedance, regardless of or β., β λ 2 2. λ 4 If the length of the transmission le is exactly one-quarter wavelength ( λ 4 ), we fd that: meang that: β 2π λ π λ 4 2 cos β cos π 2 and s β s π 2 1

5 1/26/25 Transmission e Input Impedance.doc 5/9 and therefore: + j j + s β () + (1) j ( ) () + j (1) 2 s β In other words, if the transmission le is precisely one-quarter wavelength long, the put impedance is versely proportional to the load impedance. Thk about what this means! Say the load impedance is a short circuit, such that. The put impedance at begng of the λ 4 transmission le is therefore: ( ) ( ) 2 2! This is an open circuit! The quarter-wave transmission le transforms a short-circuit to an open-circuit and vice versa!, β λ 4

6 1/26/25 Transmission e Input Impedance.doc 6/9 3. If the load is numerically equal to the characteristic impedance of the transmission le (a real value), we fd that the put impedance becomes: + j j + s β + s j β + j s β s β In other words, if the load impedance is equal to the transmission le characteristic impedance, the put impedance will be likewise be equal to regardless of the transmission le length., β 4. j X If the load is purely reactive (i.e., the resistive component is zero), the put impedance is:

7 1/26/25 Transmission e Input Impedance.doc 7/9 + j j s β + s β jx + j s β 2 + j X s β X + s β j s X β In other words, if the load is purely reactive, then the put impedance will likewise be purely reactive, regardless of the le length. j X, β jx Note that the opposite is not true: even if the load is purely resistive ( R), the put impedance will be complex (both resistive and reactive components). Q: Why is this? A:

8 1/26/25 Transmission e Input Impedance.doc 8/9 5. λ If the transmission le is electrically small its length is small with respect to signal wavelength λ --we fd that: and thus: 2π β 2π λ λ cos 1 and s β s so that the put impedance is: + j s β + j s β (1) + j () (1) + j () In other words, if the transmission le length is much smaller than a wavelength, the put impedance will always be equal to the load impedance. This is the assumption we used all previous circuits courses (e.g., EECS 211, 212, 312, 412)! In those courses, we assumed that the signal frequency ω is relatively low, such that the signal wavelength λ is very large (λ ).

9 1/26/25 Transmission e Input Impedance.doc 9/9 Note also for this case ( the electrically short transmission le), the voltage and current at each end of the transmission le are approximately the same! V ( z ) V ( z ) and I( z ) I( z ) if λ If λ, our wire behaves exactly as it did EECS 211!