1. 16.360 Lecture 8 Last lecture short circuit line open circuit line
quarter-wave transformer
matched transmission line

2. 16.360 Lecture 8 e - + e short circuit line Ii A B Zg Vg(t) Zin Z0 VLsc Z0 VL
ZL = 0 l
z = - l
z = 0
ZL= 0,  = -1, S =  (e
-jz e
jz +
V(z) = V ) -
= -2jV0sin(z) (e
-jz + V0 Z0 e
jz +
i(z) = + )
= 2V0cos(z)/Z0
V(-l)
Zin =
= jZ0tan(l)
i(-l)

3. 16.360 Lecture 8 short circuit line V(-l) Zin = = jZ0tan(l) i(-l)
If tan(l) >= 0, the line appears inductive,
jLeq = jZ0tan(l),
If tan(l) <= 0, the line appears capacitive,
1/jCeq = jZ0tan(l),
The minimum length results in transmission line as a capacitor:
l = 1/[- tan (1/CeqZ0)], -1

4. 16.360 Lecture 8 e + - e open circuit line Ii A B Zg Vg(t) Zin Z0 VLoc Z0 VL
ZL =  l
z = - l
z = 0
ZL= 0,  = 1, S =  (e
-jz e
jz
= 2V0cos(z) +
V(z) = V ) + (e
-jz + V0 Z0 e
jz +
i(z) = - )
= 2jV0sin(z)/Z0
V(-l)
Zin oc =
= -jZ0cot(l)
i(-l)

5. 16.360 Lecture 8 Application for short-circuit and open-circuit
Network analyzer
Measure S paremeters
Measure
Zin oc sc
and
Calculate Z0
Zin
= jZ0tan(l) sc
Zin
= -jZ0cot(l) oc = Z0
Zin sc oc
Calculate l
= -j Z0
Zin sc oc

6. 16.360 Lecture 8 + e - Transmission line of length l = n/2
tan(l) = tan((2/)(n/2)) = 0, + (1  e
-j2l ) - Z0
Zin(-l) =
= ZL
Any multiple of half-wavelength line doesn’t modify the load impedance.

7. 16.360 Lecture 8 + e - + e - e Quarter-wave transformer l = /4 + n/2(1  e
-j2l ) - Z0
-j  (1 +  e )
(1 + ) Z0
(1 - )
Zin(-l) = = Z0
-j  = (1 -  e )
= Z0²/ZL

8. 16.360 Lecture 8 Matched transmission line: ZL = Z0  = 0
 = 0
All incident power is delivered to the load.

9. Lecture 8
Today: Power flow on a lossless transmission line
Instantaneous power
Time-average power + e
-jz
jz
V(z) = V0 ( ) +  e (e
-jz + V0 Z0 e
jz
i(z) = -  )
At load z = 0, the incident and reflected voltages and currents: + V0 Z0 i + i
V = V0
i = - V0 Z0 r - r
V = V0
i =

10. 16.360 Lecture 8 Instantaneous power
P(t) = v(t) i(t) = Re[V exp(jt)] Re[ i exp(jt)] + + + +
= Re[|V0|exp(j )exp(jt)] Re[|V0|/Z0 exp(j )exp(jt)] + +
= (|V0|²/Z0) cos²(t +  ) r r r
P(t) = v(t) i(t) = Re[V exp(jt)] Re[ i exp(jt)] - + - +
= Re[|V0|exp(j )exp(jt)] Re[|V0|/Z0 exp(j )exp(jt)] + +
= - ||²(|V0|²/Z0) cos²(t +  + r)

11. 16.360 Lecture 8 Time-average Time-domain approach: T 1  Pav =
(|V0|²/Z0) cos²(t +  )dt +
P (t)dt = 2 +
= (|V0|²/2Z0) r +
Pav
= -||² (|V0|²/2Z0)
Net average power: i r
Pav
= Pav
+ Pav +
= (1-||²) (|V0|²/2Z0)

12. 16.360 Lecture 8 Time-average Phasor-domain approach Pav = (½)Re[V i*]+ +
Pav = (1/2) Re[V0 V0* /Z0]
= (|V0|²/2Z0) r +
Pav
= -||² (|V0|²/2Z0) +
Pav
= (1-||²) (|V0|²/2Z0)

13. Lecture 8
Next lecture
The Smith Chart