1/20/2005 The Complex .doc 1/4

The Complex Propagation Constant γ

Recall that the current and along a have the form:

V (z)= V+ e−−+γ zz+ V e γ 00 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 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 , 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 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