1/20/2005 The Complex Propagation Constant.doc 1/4 The Complex Propagation Constant γ Recall that the current and voltage along a transmission line have the form: + −−+γ zzγ V (z)= V00 e+ V e VV+− I(z)=−00 e−+γ zz e γ ZZ00 where Z0 and γ are complex constants that describe the properties of a transmission line. Since γ is complex, we can consider both its real and imaginary components. γ = ( RjL)(GjC)++ωω αβ+ j where α ==Re{γ } and β Im {γ }. Therefore, we can write: ee−−+−−γαβαzjzzjBz= ()= ee Since e − j βz =1, then e −αz alone determines the magnitude of e −γ z . Jim Stiles The Univ. of Kansas Dept. of EECS 1/20/2005 The Complex Propagation Constant.doc 2/4 I.E., ee−−γ zz= α . e −αz z Therefore, α expresses the attenuation of the signal due to the loss in the transmission line. Since e −αz is a real function, it expresses the magnitude of e −γ z only. The relative phase φ()z of e −γ z is therefore determined by ee−−j βzjz= φ () only (recall e − jzβ = 1). From Euler’s equation: eej φβ( zjz) ==cos(ββ zjz ) + sin( ) Therefore, βz represents the relative phase φ()z of the oscillating signal, as a function of transmission line position z. Since phase φ(z ) is expressed in radians, and z is distance (in meters), the value β must have units of : φ radians β = z meter Jim Stiles The Univ. of Kansas Dept. of EECS 1/20/2005 The Complex Propagation Constant.doc 3/4 The wavelength λ of the signal is the distance ∆z2π over which the relative phase changes by 2π radians. So: 2(π =+∆φφββλzz22ππ )-() z == ∆ z or, rearranging: 2π β = λ Since the signal is oscillating in time at rate ω rad sec, the propagation velocity of the wave is: ωωλ ⎛⎞m rad m v p == =f λ ⎜⎟ = βπ2secsec⎝⎠rad where f is frequency in cycles/sec. Recall we originally considered the transmission line current and voltage as a function of time and position (i.e., vzt(),, and izt()). We assumed the time function was sinusoidal, oscillating with frequency ω : v (,)zt= Re(){ V z ejωt } izt(,)= Re(){} Izejωt Jim Stiles The Univ. of Kansas Dept. of EECS 1/20/2005 The Complex Propagation Constant.doc 4/4 Now that we know V(z) and I(z), we can write the original functions as: +−ααzz−−jzt()βω − jzt () βω + v (,)zt=+ Re{ Ve00 e Ve e } ++ ⎧⎫VV−−βω βω + izt(,)=− Re⎨⎬00 e−ααzz ejzt( )() ee jzt ⎩⎭ZZ00 The first term in each equation describes a wave propagating in the +z direction, while the second describes a wave propagating in the opposite (-z) direction. +−αβωzjzt −( − ) V0 ee − αz j ( βωzt+ ) Z0 ,γ V0 ee z Each wave has wavelength: 2π λ = β And velocity: ω v = p β Jim Stiles The Univ. of Kansas Dept. of EECS .