(a) Plot the wave impedances for an air-filled waveguide versus the ratio \left( f/{ f }_{ c } \right) for TM and TE modes. (b) Compare the value of { Z }_{ TM } and { Z }_{ TE } at f=1.1{ f }_{ c } and 2.2{ f }_{ c }
Question : (a) Plot the wave impedances for an air-filled waveguide ver...
We use Eqs(9-34) and (9-39) for { Z }_{ TM } and { Z }_{ TE } respectively. For air, \eta ={ \eta }_{ 0 }=120\pi \left( \Omega \right) =377\left( \Omega \right).
a) The normalized wave impedances are plotted as shown.
b) { Z }_{ TM }={ \eta }_{ 0 }\sqrt { 1-{ \left( \frac { { f }_{ c } }{ f } \right) }^{ 2 } }
{ Z }_{ TE }=\frac { { \eta }_{ 0 } }{ \sqrt { 1-{ \left( \frac { { f }_{ c } }{ f } \right) }^{ 2 } } }
At f=1.1{ f }_{ c },\quad \sqrt { 1-{ \left( \frac { 1 }{ 1.1 } \right) }^{ 2 } } =0.417
{ Z }_{ TM }=0.417{ \eta }_{ 0 }=157\left( \Omega \right)
{ Z }_{ TE }=\frac { { \eta }_{ 0 } }{ 0.417 } =904\left( \Omega \right)
At f=2.2{ f }_{ c },\quad \sqrt { 1-{ \left( \frac { { f }_{ c } }{ f } \right) }^{ 2 } } =\sqrt { 1-{ \left( \frac { 1 }{ 2.2 } \right) }^{ 2 } } =0.891
{ Z }_{ TM }=0.891{ \eta }_{ 0 }=336\left( \Omega \right) ,\quad { Z }_{ TE }=\frac { { \eta }_{ 0 } }{ 0.981 } =423\left( \Omega \right)
{ Z }_{ TM }=\eta \sqrt { 1-{ \left( \frac { { f }_{ c } }{ f } \right) }^{ 2 } } \quad \left( \Omega \right) \quad \quad \left( 9-34 \right)
{ Z }_{ TE }=\frac { { \eta }_{ 0 } }{ \sqrt { 1-{ \left( { f }_{ c }/f \right) }^{ 2 } } } \quad \quad\quad \left( \Omega \right) \quad \quad \left( 9-39 \right)
