rectangular waveguides x z y 0 a b 0 TE or H mode

rectangular waveguides

z22

z2

t HH xz

y

0 a

b

0

TE or H mode

2z

2

2z

2

z2

t y

H

x

HH

z22 H

222y

2x kk

ykcosDyk sin C

xkcosBxksinAH

yy

xxz


xz

y

0 a

b

0

TE or H mode

lowest order mode

TE10 (or TE01) mode

H10 (or H01) mode

y

H

x

E

k

jE zz

22x

x

H

y

E

k

jE zz

22y

y

E

x

H

k

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22x

x

E

y

H

k

jH zz

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xksinBxkcosAk

jkE xx22

xy

x

H

k

jE z

22y


xz

y

0 a

b

0

TE or H mode

a

mk x

A = 0 m = 1

xa

sinBk

jkE

22x

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@ x = 0 @ x = a lowest order mode

a

xsin B

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xz

y

0 a

b

0

TE or H mode

a

xcosBHz

a

xsinB

k

jkE

22x

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xz

y

0 a

b

0

TE or H mode

0a

x

Ey

z

Movie to illustrate the propagation of the lowest order

mode in a waveguide.


xz

y

0 a

b

0

TE or H mode

2

2 2ma

222y

2x kk

22 m

a

2

2c

ma

c


TE10 mode2

c a

1

2

1f

2

c b

1

2

1f

y

xz 0 a

b

0

y

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xz 0 a0

yb

xz 0 a0

a = 2b

a2

cfc

cf2

ca

2


xz

y

0 a

b

0

TE or H mode

2

c a

m1

2

1f

2

c a

m1

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1f

28

a

m

2

103

28

03.0

1

2

103

2

1010 9 GHz 5

a = 3 cm


xz

y

0 a

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0

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xk2

c1c

phasev2

c1c

2

c1

c


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0 a

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groupv

2

c1c

1

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c1c


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vg

v

v c

1

c


ztje

z22

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z22

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y

0 a

b

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TM or E mode


z22

z2

t EE xz

y

0 a

b

0

TM or E mode

2z

2

2z

2

z2

t y

E

x

EE

z22 E

222y

2x kk

ykcosDyk sin C

xkcosBxksinAE

yy

xxz

a

mk x

b

nk y


xz

y

0 a

b

0

TM or E mode

2222

b

n

a

m

b

ynsin

a

xmsinEE zoz

y

H

x

E

k

jE zz

22x

x

H

y

E

k

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y

E

x

H

k

jH zz

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x

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y

H

k

jH zz

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xz

y

0 a

b

0

TM or E mode

2222

b

n

a

m

b

ynsin

a

xmsinEE zoz

Note ==> m = 1 and n = 1 is the lowest mode


xz

y

0 a

b

0

lowest order modeTE10 mode

22

c b

n

a

m12

1

a

b


xz

y

0 a

b

0

lowest order modeTE10 mode

2

1

a

b

10TE11

11

TM

TE

mode TE c

c

10

1 2 3

01TE20TE

Transmission Lines Demonstration

High Frequency Electronics Course EE527

Andrew Rusek Oakland University

Winter 2007

Demonstration is based on the materials collectedfrom measurement set up to show sinusoidal and step responses of a transmission line with various terminations. Results of selected simulations are included.

Fig.1a Test circuits

Fig. 1b Low frequency sine-wave (1MHz), TL matched (50 ohms), observe small delays and almost identical amplitudes

Fig. 1c Low frequency sine-wave (1MHz), TL matched (50 ohms) Channel 4 (output) shows the voltage for grounded center conductor and a probe input connected to the outer conductor (shield), observe the phase inversion of the last wave (180 degrees)

Fig. 2a Sine-wave of 17 MHz, matched load The waves have the same amplitudes, the phases are different.

Fig. 2b Sine-wave of 17 MHz, matched load Channel 4 (output) shows the voltage for grounded center conductor and a probe input connected to the outer conductor (shield).

Fig. 3 Open ended TL, sine-wave of 1 MHz applied, observe 2X larger amplitude in comparison with previous tests, amplitudes are almost the same for all waves.

Fig. 4a Open ended TL, 3.5 MHz, observe minimum (input)One quarter wave pattern is shown

Fig. 4b Open ended TL, 3.5 MHz, observe minimum (input)One quarter wave pattern is shown

Fig. 4c Open ended TL, 3.5 MHz, observe minimum (input)One quarter wave pattern is shown

Fig. 5 Open ended TL, 5.5 MHz, observe shift of the minimum

The minimum is located quarter wave from the end.

Fig. 6 Open ended TL, 11 MHz, observe two minima

Fig. 7 Shorted TL, low frequency,1MHz applied, observe zero output voltage

Fig. 8 Shorted TL, 5 MHz applied

Fig. 9 Shorted TL, 7 MHz, observe two minima (half wave). If the length of the line is known, the dielectric constant can be calculated (Lambda_cable/2 = 12m, open space Lambda = 42.8m).

Fig. 10 Shorted TL, 7 MHz, increased vertical sensitivity; observe two minima as before and effects of stray inductance of the source and probe leads (half wave),

Fig.11 Shorted TL, 11 MHz, two minima, first shifted towards the load, ¼ wavelength + ½ wavelength

Fig. 12 Pulse response of open ended TL, slow pulse (0.3us rise time), no reflections observed, Channel 2 – Input, Channel 4 – Output, observe the delay.

Fig. 13a Open ended TL, Input Pulse rise time = 240 ns, Output = 120 ns, Long pulse applied, measurement circuit

Fig. 13b Open ended TL, Input Pulse rise time = 240 ns, Output = 120 ns,Why Output is faster than Input ? End of TL reflection adds to incident (Real rise time of the input wave is120 ns), and this effect doubles Input signal rise time. Long pulse applied, simulations.

Fig. 13c Open ended TL, Input Pulse rise time = 240 ns, Output = 120 ns,Why Output is faster than Input ? End of TL reflection adds to incident (Real rise time of the input wave is120 ns), and this effect doubles Input signal rise time. Long pulse applied, measurements.

Fig. 14a Open ended TL, long pulse applied, source matched, measurement circuit.

Fig. 14b Open ended TL, long pulse applied, source matched, simulations.

Fig. 14c Open ended TL, Input – Channel 2 shows incident step and reflected step (doubled TL delay), source matched, Output – Channel 4 shows doubled incident wave level, delayed (about 60 ns), long pulse applied. Distance between steps of Channel 2 – 2X TL delay time, measurements.

Fig. 15c Open ended TL, short pulses applied to show “radar effect”, circuit.

Fig. 15c Open ended TL, short pulses applied to show “radar effect”. Echo is observed (Upper Channel – Input), doubled amplitude – Lower Channel, simulations.

Fig. 15c Open ended TL, short pulses applied to show “radar effect”. Echo is observed (Channel 2 – Input), doubled amplitude – Channel 4 – Output, observe effects of the losses of TL – echo is slower and smaller. Distance between pulses of Channel 2 – 2X TL delay time. Measured unit delay yields 20cm/ns.

Fig 16a Shorted TL, narrow pulses, circuit.

Fig 16b Shorted TL, narrow pulses, observe change of polarity of a reflected pulse (Upper Channel – Input).

Fig 16c Shorted TL, narrow pulses, “short” is not really short at HF (Channel 4), observe change of polarity of reflected pulse (Channel 2 – Input).

Fig. 17a Transmission line and the inductive load, the source resistance is matched (50 ohms), circuit.

Fig. 17b Transmission line and the inductive load, the source resistance is matched (50 ohms), simulated waves.

Fig. 17c Transmission line and the inductive load, the source resistance is matched (50 ohms), measurements.

Fig. 17d Transmission line and the inductive load, the source resistance is matched (50 ohms), larger time scale

Fig. 17e Transmission line and the inductive load, the source resistance is matched (50 ohms), display adjusted to calculate the time constant and inductance (L = 100 uH).

Fig. 17f Transmission line and the capacitive load, the source resistance is matched (50 ohms), circuit.

Fig. 17g Transmission line and the capacitive load, the source resistance is matched (50 ohms), display adjusted to calculate the time constant and capacitance (C = 10nF), simulated waves.

Fig. 17h Transmission line and the capacitive load, the source resistance is matched (50 ohms), display adjusted to calculate the time constant and capacitance (C = 10nF), measured waves.

Fig. 18 a. Matched TL, reversed connections of Output Probe (center conductor is grounded} – the waves show that outer conductor of TL also participates in signal delay, circuit.

Fig. 18 b. Matched TL, reversed connections of Output Probe (center conductor is grounded} – the waves show that outer conductor of TL also participates in signal delay, simulated waves .

Fig. 18 c. Matched TL, Input – Channel 2, reversed connections of Output Probe (center conductor is grounded} – Channel 4, shows that outer conductor of TL also participates in signal delay

Fig. 19a Reflection from the unmatched load of the TL (Rload =27 ohms), source is matched, circuit.

Fig. 19b Reflection from the unmatched load of the TL (Rload =27 ohms), source is matched, simulated waves.

Fig. 19c Reflection from the unmatched load of the TL (Rload =27 ohms), source is matched, measured waves.

Fig. 20a Reflection from the unmatched load of the TL (Rload =100 ohms), source is matched

Fig. 20c Reflection from the unmatched load of the TL (Rload =100 ohms), source is matched

Fig. 21a Reflection from the unmatched load and the source of the TL (Rsource = 25 ohms Rload =open circuit), circuit.

Fig. 21b Reflection from the unmatched load and the source of the TL (Rsource = 25 ohms Rload =open circuit), simulated waves.

Fig. 21c Reflection from the unmatched load and the source of the TL (Rsource = 25 ohms Rload =open circuit)

Fig. 22a Reflection from the unmatched load and the source of the TL (Rsource = 100 ohms Rload =open circuit), circuit.

Fig. 22b Reflection from the unmatched load and the source of the TL (Rsource = 100 ohms Rload =open circuit), simulated waves.

Fig. 22c Reflection from the unmatched load and the source of the TL (Rsource = 100 ohms Rload =open circuit), measured waves.

Fig. 23a Reflection from the shorted load of the TL, source is matched, circuit.

Fig. 23b Reflection from the shorted load of the TL, source is matched, simulation results

Fig. 23c Reflection from the shorted load of the TL, source is matched (one inch length wire = Rload), measurements.

Fig. 24a Reflection from the shorted load of the TL, source is matched (one inch length wire = Rload), observe effects of TL losses (elevated “tail” that follows the pulse, and the step of the output signal)

Fig. 24b Reflection from the shorted load of the TL, source is matched (less than 0.5 inch length wire = Rload)

Fig. 25a Reflection from the shorted load of the TL, source resistance is not matched (25 ohms), circuit.

Fig. 25b Reflection from the shorted load of the TL, source is not matched (source resistance is 25 ohms, simulated waves.

Fig. 25c Reflection from the shorted load of the TL, source is not matched (25 ohms), measurements.

Fig. 26a Reflection from the shorted load of the TL, source is matched (100 ohms), circuit.

Fig. 26b Reflection from the shorted load of the TL, source is not matched (100 ohms), simulations.

Fig. 26c Reflection from the shorted load of the TL, source is not matched (100 ohms), measurements.

Rsource = 100 ohms, Rload = 6 ohms




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rectangular waveguides x z y 0 a b 0 TE or H mode