ELEC 353 – Assignment #8 2nd part Input line quarter-wave transformer line #3 1.An antenna operating at 1900 MHz has input impedance 4090 jZL −= ohms. The matching circuit shown above has an input line of length 5 cm. The quarter wave transformer has length tL and line #3 has length L . The characteristic resistance of the input line and of line #3 is cR and the characteristic resistance of the transformer is ctR . The speed of travel on all three lines is 20 cm/ns. i)Design the matching circuit by choosing L , tL , and ctR . Choose L so that the input impedance of line #3 is real, 033 jRZ += . Then choose tL and ctR to match 3R . Note that there are two solutions, for L and ctR . ii)Use TRLINE to model the circuit and find the bandwidth of each solution for a return loss of 20 dB or better. Which solution has a wider bandwidth? Solution The load reflection coefficient is

φjL

L

LL e

ZZZZ

Γ=+−

=Γ0

0

The reflection coefficient along the transmission line is (see class notes set 14):

( ) ( )zLjLzj

zjLjL

zj

zj

eeV

eeVeVeVz −−

−+

−+

−+

−

Γ=Γ

==Γ ββ

ββ

β

β2

2

where L is the length of line #3. At the input of line #3 we have z=0 so

)2(223

LjL

LjjL

LjL eeee βφβφβ −−− Γ=Γ=Γ=Γ

Using Euler’s identity, )2sin()2cos(3 LjL LL βφβφ −Γ+−Γ=Γ

To make the input impedance of line #3 real with zero imaginary part, it is sufficient to make the reflection coefficient real, so choose

0)2sin( =− Lβφ hence

πβφ nL ±=− 2 and the length of line #3 is

βπφ

2nL

=

The Input impedance of line #3 is

LjZZLjZZZZ

L

L

ββ

tantan

0

003 +

+=

We can simplify the calculation substantially using the choice of L to satisfy 0)2sin( =− Lβφ

Starting with the input impedance in terms of the reflection coefficient (lecture notes set 10)

zjLjL

zj

zjLjL

zj

c eeVeVeeVeVRZ βββ

βββ

2

2

−+−+

−+−+

Γ−Γ+

=

Simplify to

zjLjL

zj

zjLjL

zj

c eeeeeeRZ βββ

βββ

2

2

−−

−−

Γ−Γ+

=

Factor an exponential to get

zjLjL

zjLjL

c eeeeRZ ββ

ββ

22

22

11

−

−

Γ−Γ+

=

Then at z=0 the input impedance is

LjL

LjL

c eeRZ β

β

2

2

3 11

−

−

Γ−Γ+

=

Use φjLL eΓ=Γ to get

LjjL

LjjL

c eeee

RZ βφ

βφ

2

2

3 11

−

−

Γ−Γ+

=

Collect terms

)2(

)2(

3 11

LjL

LjL

c ee

RZ βφ

βφ

−

−

Γ−Γ+

=

Use Euler’s identity to get

))2sin()2(cos(1))2sin(()2(cos(1

3 LjLLjL

RZL

Lc βφβφ

βφβφ−+−Γ−−+−Γ+

=

Recall tha L is chosen to satisfy 0)2sin( =− Lβφ

By chooing So the sine terms will disappear

πβφ nL ±=− 2 So the sine terms will disappear and the cosine terms become

1)cos()2cos( =±=− πβφ nL So

011

)2cos(1)2cos(1

33 jRRLL

RZL

Lc

L

Lc +=

Γ±Γ

=−Γ−−Γ+

=

βφβφ

L

LcRR

Γ±Γ

=11

3

Numerical solution: ZL=90-j40 ohms so

401404040

504090504090

0

0

jj

jj

ZZZZ

L

LL −

−=

+−−−

=+−

=Γ 1.293885.09.1560.145

45568.56−∠=

−∠−∠

=

The wavelength is

526.109.1

20===

fuλ cm

And the phase constant is

2.34526.10

360360===

λβ degree/cm

Using

βπφ

2nL

=

We calculate two values of L as

206.22.3421801.29

=+−

=x

L cm

And

838.42.3423601.29

=+−

=x

L cm

The corresponding input resistance values for line #3 are for L=2.206 cm

02.223885.013885.0150

11

3 =+−

=Γ+Γ−

=L

LcRR ohms

And for L=4.838 cm

5.1133885.013885.0150

111

3 =−+

=ΓΓ+

=L

LcRR ohms

The transformer characteristic impedance values are 18.3302.22503 === xRRR ct ohms

And 34.755,113503 === xRRR ct

Using TRLINE to verify the solution and evaluate the bandwidth.

Solution #1: L=2.207 cm, R3=22.2 ohms, Transformer: Rct=33.18057 We get an almost perfect match at 1.9 GHz.

The bandwidth for a return loss of -20 dB is 0.205 GHz.

Solution #2 uses L=4.838 cm, R3=113.5 ohms and Rct=75.34 ohms. We get an almost perfect match at 1.9 GHz.

The bandwidth is 0.116 GHz so this is the solution with the narrower bandwidth.

2. The load LZ on a transmission-line circuit operating at 2.00 GHz consists of an RLC circuit with component values L =1 nH, C =2 pF, and =LR 14 ohms.

The load LZ terminates the transmission line circuit shown in the figure. The characteristic impedance of the line is =0Z 73 ohms, and the speed of travel is =u 12 cm/ns. The length of the line is 14=L cm. The generator has amplitude 10=sV volts and internal resistance =sR 50 ohms. 2.1) What is the value of the load impedance LZ ? 14.5+j3.55ohms 12.5+j8.2 68.2+j11.7 9.56+j17.9 none of these

2.2) What is the reflection coefficient Γ𝐿𝐿 at the load? -0.689+j0.367 -0.027+j0.085 -0.693+j0.162 -0.666+j0.068 none of these

2.3) What is the input impedance of the transmission line terminated with load LZ ? 44.8-j99.1 ohms 32.9-j90.9 64.2-j8.53 18.4-j73.2 none of these

2.4) What is the amplitude of the voltage 𝑉𝑉𝑖𝑖𝑖𝑖 at the input of the transmission line? 7.53 volts 5.65 7.90 4.62 none of these

2.5) What is the power delivered to the load? 109 mW 91.6 119 245 none of these

Solutions to problem 2: (numbered as problem 4, incorrectly)

3.Two transmission lines are connected in series as shown in the circuit above. Line #1 has characteristic resistance 501 =CR ohms and length =1L 9 cm. Line #2 has 752 =CR ohms and length

=2L 5 cm. The speed of wave travel on both lines is 200=u meters per microsecond. There is a shunt capacitor at the junction between the lines, of value =C 0.42 pF, and the load terminating line #2 is =LZ 120+j60 ohms. The generator has open-circuit voltage ( )ttVs ωcos10)( = volts, and internal resistance 50=sR ohms. The operating frequency is 2500 MHz.

3.1)Find the impedance AZ at the input to line #2. a) 158.8+j110.3 ohms

b) 27.2-j14.7 ohms

c) 92.3-j63.5 ohms

d) 67.5-j90 ohms

e) none of these

3.2)What is the impedance of the capacitor? a) -j151.6 ohms b) j22.3 ohms c) –j97.9 ohms d) –j205.4 ohms e) none of these

3.3)What is the value of the load impedance terminating transmission line #1? a) 19.4-j17.5 ohms

b) 67.5-j90.0 ohms

c) 195.5-j88.3 ohms

d) 38.7-j61.4 ohms

e) none of these

3.4)Find the input impedance to line #1. a) 13.9-j9.94 ohms

b) 17.0-j37.9 ohms

c) 19.7+j18.4 ohms

d) 23.1+j9.58 ohms

e) none of these

3.5)Find the amplitude of the voltage 1V at the input to line #1. a) 5.40 volts b) 7.96 volts c) 2.65 volts d) 3.74 volts e) none of these

3.6)Find the power delivered to load impedance LZ . a) 87 mW b) 166 mW c) 190 mW d) 143 mW e) none of these

Solution to problem 3: (numbered as problem 4, incorrectly)

4. An engineer measures the standing-wave pattern on a transmission line which has a characteristic resistance of 50 ohms and a speed of travel of 30 cm/ns. There is a standing-wave maximum at z=12.433 cm of value 8.699 volts and a standing-wave minimum at z=16.278 cm of value 1.301 volts. The load is located at z=20 cm. The operating frequency is 1.95 GHz. 4.1) What is the standing-wave ratio? 10.0 2.00 27.8 6.68 none of these

4.2) What is the magnitude of the reflection coefficient? 0.333 0.818 0.740 0.930 none of these

4.3) What is the angle of the reflection coefficient? -0.231 degrees 90.0 13.9 -5.86 none of these

4.4) What is the value of the load impedance? 100-j100 ohms 30-j10 100+j37 4-j3 none of these

Solution to problem 4: (numbered as problem 5, incorrectly)