ECE 3300 LECTURE 10: Power Flow on Lossless Transmission Lines
Text Section: 2-8
Phasor Voltage and Current on Lossless TEM Transmission Line:
V(z) = Vo+ ( e -jβz + Γ e jβz )
I(z) = (Vo+ / Zo) ( e -jβz - Γ e jβz )
Incident Voltage and Current at the LOAD (z=0) :
Plug z=0 into above equations, and extract incident (+-traveling) and reflected (traveling) waves
V I (z) = Vo+
I I (z) = ( Vo+/ Zo)
Reflected Voltage and Current at the LOAD (z=0):
V r (z) = Γ Vo+
I r (z) = -Γ ( Vo+/ Zo)
Instantaneous Incident Power (power as a function of time):
P I (t) = v I (t) • i I (t)
= Real [V I (z) ejωt] • Real [I I (z) ejωt]
= Real [ |Vo+| ejφ ejωt] • Real [ ( |Vo+| / Zo) ejφ ejωt]
= |Vo+| cos( ωt + φ+) • ( |Vo+| / Zo) cos( ωt + φ+)
= ( |Vo+| 2 / Zo) cos 2 ( ωt + φ+) watts
Instantaneous Reflected Power (power as a function of time):
Γ = |Γ| ejθ
P r (t) = v r (t) • i r (t)
= -|Γ|2 ( |Vo+| 2 / Zo) cos 2 ( ωt + φ+ + θr) watts
Negative sign? Power is flowing in the –z direction
How is instantaneous power measured? It isn’t, unless you calculate it from v(t) seen on
a scope.
What kind of power can we measure?
Incident Time – Averaged Power:
Power incident on load
(assuming no power is lost along TL, this is also power delivered by generator to TL)
Integrate over one period T = 1/f = 2π / ω
PIav = (1/T) integral from 0 to T of P I (t) dt
= (ω / 2π) integral from 0 to (2π / ω) of P I (t) dt
= ( |Vo+| 2 / 2 Zo) watts
Power reflected by load
Prav = -|Γ|2 ( |Vo+| 2 / 2 Zo) watts
Net averaged power delivered to load
Pav = PIav + Prav = ( |Vo+| 2 / 2 Zo) [ 1 – |Γ|2 ] watts Å THIS can be measured with a
power meter
Ugh.
Phasor Domain Approach: (easier)
PIav = ½ Real [ V i • I I * ] = ½ Real [Vo+ • ( Vo+/ Zo) ] = = ( |Vo+| 2 / 2 Zo) watts
Similarly: Prav = -|Γ|2 ( |Vo+| 2 / 2 Zo) watts
Pav = ½ Real [ V • I ] = ( |Vo+| 2 / 2 Zo) [ 1 – |Γ|2 ] watts
Another Way to Look at This:
From circuits, recall the concept of RMS voltage and Current?
RMS = root – mean – square
For a function v(t)
Square = v2(t)
Mean = (1/T) integral from 0 to T of v2(t) dt
Root = [(1/T) integral from 0 to T of v2(t) dt ] ½
For a sine wave Vrms = Vpeak / sqrt(2)
For the voltage incident on the load:
v I (t) = Vo+ cos(ωt + φ)
Vrms = Vo+ / sqrt(2)
i I (t) =(Vo+ / Zo) cos(ωt + φ)
Irms = (Vo+ / Zo) / sqrt(2)
P i av = Vrms • Irms = ( |Vo+| 2 / 2 Zo) watts
Review Definitions and Answer Questions