Incident Reflected and Absorbed Power - I2S
1/27/2009
Incident Reflected and Absorbed Power.doc
1/8
Incident, Reflected, and Absorbed Power
We have discovered that two waves propagate along a
transmission line, one in each direction (V + (z ) and V - (z ) ).
I (z ) = I +(z ) + I -(z )
z = -A
+
V (z ) =V +(z ) +V -(z )
-
A
IL
+
VL
ZL
-
z =0
The result is that electromagnetic energy flows along the transmission line at a given rate (i.e., power).
Q: How much power flows along a transmission line, and where does that power go?
A: We can answer that question by determining the power absorbed by the load!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/27/2009
Incident Reflected and Absorbed Power.doc
2/8
You of course recall that the time-averaged power (a real value!) absorbed by a complex impedance ZL is:
{ } Pabs
=
1 2
Re
VL
IL
Of course, the load voltage and current is simply the voltage an current at the end of the transmission line (at z = 0 ). A happy result is that we can then use our transmission line theory to determine this absorbed power:
{ } Pabs
=
1 2
Re
VL
I
L
{ } = 1 Re V (z = 0)I (z = 0) 2
{( ) ( ) } = 1 Re 2 Z0
V0+ e - j 0 + 0 e + j 0
V0+ e - j 0 - 0 e + j 0
{ ( ) } = V0+ 2 Re 1 - 2 Z0
0 - 0
- 0 2
( ) = V0+ 2 2 Z0
1 - 0 2
The significance of this result can be seen by rewriting the expression as:
( ) Pabs
= V0+ 2 2 Z0
1 - 0 2
= V0+ 2 - V0+0 2 = V0+ 2 - V0- 2 2 Z0 2 Z0 2 Z0 2 Z0
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/27/2009
Incident Reflected and Absorbed Power.doc
3/8
The two terms in above expression have a very definite physical meaning. The first term is the time-averaged power of the wave propagating along the transmission line toward the load.
We say that this wave is incident on the load:
Pinc
= P+
= V0+ 2 2Z 0
Likewise, the second term of the Pabs equation describes the power of the wave moving in the other direction (away from the load). We refer to this as the wave reflected from the load:
Pref
= P-
= V0- 2 2Z0
=
LV0+ 2 2Z0
=
L
2
V+2 0
2Z0
=
L 2 Pinc
Thus, the power absorbed by the load (i.e., the power delivered to the load) is simply:
Pabs
=
V+2 0
2 Z0
- V0- 2 2 Z0
= Pinc
- Pref
or, rearranging, we find: Pinc = Pabs + Pref
This equation is simply an expression of the conservation of energy !
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/27/2009
Incident Reflected and Absorbed Power.doc
4/8
It says that power flowing toward the load (Pinc) is either
absorbed by the load (Pabs) or reflected back from the load
(Pref).
Pabs
Pinc
Pref
ZL
Now let's consider some special cases: 1. L 2 = 1 For this case, we find that the load absorbs no power!
( ) ( ) Pabs = Pinc 1 - 0 2 = Pinc 1 - 1 = 0
Likewise, we find that the reflected power is equal to the
incident:
( ) Pref
=
L
P2 inc
=
1
Pinc
= Pinc
Note these two results are completely consistent--by conservation of energy, if one is true the other must also be:
Pinc = Pabs + Pref = 0 + Pref = Pref
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/27/2009
Incident Reflected and Absorbed Power.doc
5/8
In this case, no power is absorbed by the load. All of the
incident power is reflected, so that the reflected power is equal
to that of the incident.
Pabs = 0
Pinc
Pref = Pinc
L = 1
2. L = 0
For this case, we find that there is no reflected power!
( ) Pref
=
L
P2 inc
=
0
Pinc = 0
Likewise, we find that the absorbed power is equal to the
incident:
( ) ( ) Pabs = Pinc 1 - 0 2 = Pinc 1 - 0 = Pinc
Note these two results are completely consistent--by conservation of energy, if one is true the other must also be:
Pinc = Pabs + Pref = Pabs + 0 = Pabs
In this case, all the incident power is absorbed by the load. None of the incident power is reflected, so that the absorbed power is equal to that of the incident.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
................
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