1. Changing the reference-impedance for training... · VU 354.058 – RF Techniques Examples for Training, 29.1.2020 H.Arthaber 3. Cascade connection using T-matrices Calculate the

VU 354.058 – RF Techniques Examples for Training, 29.1.2020 H.Arthaber

1. Changing the reference-impedance The S-matrix of a network is given (referenced to 50 Ω on both ports):

𝑆𝑆 = 0.2 0.012.1 0.4

Re-normalize the S-matrix to the new reference impedances 𝑍𝑍01 = 30 Ω and 𝑍𝑍02 = 100 Ω!

Solution: The renormalized S-matrix with reference impedances 𝑍𝑍01 = 30 Ω and 𝑍𝑍02 = 100 Ω: 𝑆𝑆 = 0.435 0.010

2.102 0.071


2. Cascade connection using T-matrices

Calculate the S-matrix of the depicted circuit by using the T-matrix!

Solution:

𝑆𝑆 = 0.333 + 𝑗𝑗0. 222 0.444 − 𝑗𝑗0.2220.444 − 𝑗𝑗0.222 0.111

𝑍𝑍1 = (50 + 𝑗𝑗50) Ω 𝑍𝑍2 = (100 − 𝑗𝑗50) Ω 𝑍𝑍0 = 50 Ω


3. Cascade connection using T-matrices

Calculate the S-matrix of the depicted circuit by dividing it into four parts and use the T-matrix for cascading all parts! Use a reference impedance of 50 Ω for both ports!

𝑅𝑅 = 75Ω

Solution:

𝑆𝑆 = 0.478 0.246 − 𝑗𝑗0.246

0.246 − 𝑗𝑗0.246 𝑗𝑗0.565


4. Output matching of a power transistor (written exam of 10.3.2011) The output matching network for a pulsed 275 W power amplifier using an LDMOS transistor (Freescale MRF6V12250H, datasheet: see next page) shall be designed. The transistor shall be matched to a 50 Ω load. frequency of operation: f = 1030 MHz drain bias voltage: VDD = 50 V drain current (when idle): IDQ = 100 mA

a) Calculate an equivalent circuit of the output matching network as seen by the transistor! The equivalent one-port circuit shall be equivalent to Zload and shall consist of two elements (out of R/L/C) connected in parallel. Calculate the elements‘ values for the operation frequency!

b) Use the Smith-chart to determine the values of the capacitance C and the length of the line L of the depicted output matching network!

Solution:

a) R = 5.2 Ω , C = 16 pF

b) C = 51 pF, L = 0.2914 λ


Datasheet page of Freescale MRF6V12250H (275W, L-band, LDMOS, lateral double-diffused MOSFET) Please note:

• The datasheet does not specifiy the transistor’s S-parameters. Instead, the S-parameters of the required matching networks are stated.

• The given Smith-diagram has a reference impedance Z0 = 5 Ω (!!!)


5. Matching of an RF amplifier module (written exam of 12.4.2011) The input of an RF amplifier module operating at 𝑓𝑓𝑐𝑐 = 1 GHz shows a reflection coefficient Γ = 0.3 − 𝑗𝑗0.5. By means of two 5.5 pF capacitors, placed along a lossless 50 Ω line, the input of the entire circuit shall be matched to 50 Ω.

By using the Smith-chart determine the transmission line lengths 𝑙𝑙1 and 𝑙𝑙2 (as a ratio of the wavelength λ)!

Solution: 𝑙𝑙1 = 0.0857𝜆𝜆 and 𝑙𝑙2 = 0.1662𝜆𝜆

Alternative solution: 𝑙𝑙1 = 0.1875𝜆𝜆 and 𝑙𝑙2 = 0.3064𝜆𝜆


6. Matching range of tunable network The depicted circuit is used to match the output of a transistor to 50 Ω.

The inductor and capacitor are adjustable within the following ranges:

𝐿𝐿 = 2 … 4 nH 𝐶𝐶 = 2 … 8 pF

In a Smith-chart (with the reference impedance Zo = 50 Ω) mark the area of 𝑆𝑆22 (the output reflection coefficient of the transistor) which can be matched to 50 Ω at an operating frequency of f = 2.4 GHz!

Solution:


7. Impedance transformation with maximum VSWR The impedance Z (represented by a parallel R/L-circuit) is transformed by the following circuit to a new value. The circuit is operated at the frequency 𝑓𝑓 = 12 GHz.

Determine the range of possible lengths 𝑙𝑙2 which result in a VSWR ≤ 5 for the complete circuit (50 Ω reference impedance)!

Definition VSWR:

In telecommunications, standing wave ratio (SWR) is the ratio of the amplitude of a partial standing wave at an antinode (maximum) to the amplitude at an adjacent node (minimum), in an electrical transmission line. For a lossless line terminated with a reflection coefficient Γ, it can be calculated as follows:

VSWR =1 + |𝛤𝛤|1 − |𝛤𝛤|

Solution: 0.0947 𝜆𝜆 < 𝑙𝑙2 < 0.4528 𝜆𝜆 …to be precise: (0.0947 + 0.5𝑛𝑛)𝜆𝜆 < 𝑙𝑙2 < (0.4528 + 0.5𝑛𝑛)𝜆𝜆 ∀ 𝑛𝑛 ∈ ℕ0


8. S-Parameters of a 3-Port The depicted 3-port is operated at 𝑓𝑓 = 1.95 GHz.

a) Calculate the S-Matrix for a port impedance 𝑍𝑍0 = 50 Ω at all ports!

b) Is the given circuit active, passive, or lossless?

Solution:

a) 𝑆𝑆 = 15

1 −4 2−4 1 22 2 4

b) The given circuit is passive, not active and lossy.


9. Forced Match for Guaranteed Stability The depicted transistor amplifier is operated at f = 800 MHz and is driven by a source with an appropriate impedance (not shown in schematic). To ensure stable operation in case of a defective antenna cable an ideal (perfectly matched) attenuator is added. The transistor’s S-parameters are: S = 0.350∡ − 95° 0.055∡ + 45°

21.0∡ + 115° 0.900∡ + 62° with Z0 = 50 Ω.

a) Analyze the circuit and determine the requirements onto the reflection coefficient at reference plane (A) to ensure stable operation!

b) Determine the required attenuation to ensure stability even in case of a defective cable!

Solution: a) The requirements onto the reflection coefficient is shown below:

b) The attenuator must be greater than 2.5 dB to ensure stability even in case of a defective

cable


10. Impedance transformation The given circuit is operated at f = 1 GHz, the ports’ reference impedances are Z0 = 50 Ω. Use the Smith-chart to determine the area of possible reflection coefficients at reference planes (A), (B) and (C) for an arbitrary passive load ZL connected to port 2!

By using the Smith-chart…

a) Determine the area of possible reflection coefficients at ref. plane (A) b) Determine the area of possible reflection coefficients at ref. plane (B) c) Determine the area of possible reflection coefficients at ref. plane (C)


10. Impedance transformation Solution a)

b)

c)


11. Impedance Matching of a passive load The given circuit is operated at f = 1.73 GHz and is used to match the passive load 𝑍𝑍𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 to a 50 Ω system.

𝑍𝑍𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = (20 − j45) Ω

By using the Smith-chart determine 𝑅𝑅1, 𝑅𝑅2, and 𝐿𝐿1 in order to have a perfect match at port 1! Because this leads to an ambiguous solution…

1. Find a solution for the minimum value of 𝑅𝑅1! 2. Find another solution for the maximum value of 𝑅𝑅1 and interpret this result!

Solution: 1. 𝑅𝑅1 = 23.1𝛺𝛺, 𝑅𝑅2 = 96.2𝛺𝛺, 𝐿𝐿1 = 2.48nH 2. 𝑅𝑅1 = 50𝛺𝛺, 𝑅𝑅2 = 0𝛺𝛺, 𝐿𝐿1 = 0H

This is the trivial but in most of the cases useless solution for this matching problem. 𝑅𝑅2 and 𝐿𝐿1 are short circuited, therefore the load is also short circuited and all of the input power is dissipated in 𝑅𝑅1. So no power will be transferred into the load.

𝑍𝑍𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙

𝐿𝐿1 𝑅𝑅1

𝑅𝑅2 Port 1


12. Impedance Matching of an unknown passive load The given matching circuit is operated at f = 1.73 GHz and is used to match an unknown passive load at port 2 to the output of an amplifier which is connected at port 1. The reference impedance of both ports is Zo = 50 Ω.

By using the Smith-chart determine the maximum VSWR which will be measured into port 1 when an arbitrary passive load is connected to port 2!

Solution: 𝑉𝑉𝑆𝑆𝑉𝑉𝑅𝑅 < 2.52

𝐿𝐿1 𝑅𝑅1

𝑅𝑅2 Port 2 Port 1

𝐿𝐿1 = 1.5 nH 𝑅𝑅1 = 25 Ω 𝑅𝑅2 = 50 Ω


13. Impedance transformation The given circuit is operated at f = 1 GHz, the ports’ reference impedances are Z01 = 50 Ω and Z02 = 75 Ω.

By using the Smith-chart determine the transmission line lengths 𝑙𝑙2 and 𝑙𝑙3 (as a ratio of the wavelength λ) in order to have perfect match on both ports!

Solution: 𝑙𝑙2 = 0.2819𝜆𝜆 𝑙𝑙3 = 0.0215𝜆𝜆

Alternative solution: 𝑙𝑙2 = 0.2180𝜆𝜆, 𝑙𝑙3 = 0.4786𝜆𝜆


14. Impedance transformation The given circuit is operated at f = 1 GHz.

Use the Smith-chart to determine the values of the capacitance C and of the inductance L for perfectly matching the complex valued impedance Z = (450+j380) Ω to Port 1

Solution: 𝐶𝐶 = 3.77 pF, 𝐿𝐿 = ∞ H (no inductor needed)


15. Impedance transformation The given circuit is operated at f = 1 GHz.

Use the Smith-chart to determine the values of the capacitance C and of the inductance L for perfectly matching the complex valued impedance Z = (220-j450) Ω to Port 1

Solution: 𝐶𝐶 = 1.468 pF, 𝐿𝐿 = 43.08 nH



By using the Smith-chart determine the transmission line lengths 𝑙𝑙2 and 𝑙𝑙3 (as a ratio of the wavelength λ) in order to have perfect match on both ports!

Solution: 𝑙𝑙2 = 0.149𝜆𝜆 𝑙𝑙3 = 0.141𝜆𝜆

Alternative solution: 𝑙𝑙2 = 0.236𝜆𝜆, 𝑙𝑙3 = 0.189𝜆𝜆


17. Impedance transformation The given circuit is operated at f = 1 GHz, the ports’ reference impedances are Z0 = 50 Ω. Use the Smith-chart to determine the area of possible reflection coefficients at port 1 for an arbitrary passive load ZL connected to port 2!

Solution:


18. Matching range of tunable network The depicted circuit is used to match a load at port 2 to a 50 Ω system.

The inductor and length of the line are adjustable within the following ranges:

𝑙𝑙1 =1

12 …14 λ

𝐿𝐿 = 1 … 15 nH

In a Smith-chart (with the reference impedance Zo = 50 Ω) mark the area of impedances connected to Port 2 which can be matched to 50 Ω at an operating frequency of f = 1 GHz!

Solution:



By using the Smith-chart determine the transmission line lengths 𝑙𝑙1 and 𝑙𝑙2 (in degree) in order to have perfect match on both ports!

Solution: 𝑙𝑙1 = 109.7°, 𝑙𝑙2 = 54.4°

Alternative solution: 𝑙𝑙1 = 165.2°, 𝑙𝑙2 = 125.6°


20. S-Parameters of a 3-Port The depicted 3-port is operated at 𝑓𝑓 = 1 GHz.

𝑁𝑁 = √2 𝐿𝐿 = 10 nH 𝐶𝐶 = 8 pF

a) Calculate the S-Matrix!


Solution:

a) 𝑆𝑆 = 0.349 + 𝑗𝑗0.100 0.651 − 𝑗𝑗0.100 −0.651 + 𝑗𝑗0.1000.651 − 𝑗𝑗0.100 0.349 + 𝑗𝑗0.100 0.651 − 𝑗𝑗0.100−0.651 + 𝑗𝑗0.100 0.651 − 𝑗𝑗0.100 0.349 + 𝑗𝑗0.100

b) The given circuit is passive, not active and lossless


21. S-Parameters of a 3-Port The depicted 3-port is operated at 𝑓𝑓 = 1 GHz.

𝑆𝑆 = 0.2 0.80.8 −0.3

𝑅𝑅1 = 35 Ω 𝑅𝑅2 = 68 Ω

a) Calculate the S-Matrix!


(Hint: Use T-matrices for the solution)

Solution:

a) 𝑆𝑆 = 0.5730 0.2685 −0.26850.2685 0.5167 0.4834−0.2685 0.4834 0.5167

b) The given circuit is passive, not active and lossy


22. Transistor amplifier matching A transistor amplifier operated at f = 1 GHz is given and its input matching network is shown below. Find the range of C resulting in a gain reduction of less than 0.5 dB compared to optimum matching conditions!

The S-matrix of the transistor including the bias network is given by

𝑆𝑆 = −0.2 + 𝑗𝑗0.3 011.5 − 𝑗𝑗1.5 −0.2 + 𝑗𝑗0.1.

Solution: 0.55 pF < 𝐶𝐶 < 5.00 pF

𝐿𝐿 = 820 pH


23. S-Parameters of a 3-Port The following circuit is operated at f = 1.8 GHz:

a) Calculate the S-matrix of the given circuit! Note the different port impedances!

b) The circuit is conditionally passive/lossless. Write down two possible stimulus vectors alossless and alossy (waves entering the network): one which results in a lossless behavior and one which results in losses!

Solution:

a) 𝑆𝑆 =

−0.559 + 𝑗𝑗0.253 0.779 − 𝑗𝑗0.127 00.779 − 𝑗𝑗0.127 0.610 + 𝑗𝑗0.063 0

0 0 −0.6

b) 𝑎𝑎𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 =

100 or

010 , 𝑎𝑎𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 =

001


24. A 10 GHz Amplifier The given amplifier uses an Infineon BFP843 bipolar transistor as active element. The circuit is operated at f = 10.0 GHz and does have an output matching network only. The chosen bias point is VCE = 1.8 V, IC = 15 mA.

The datasheet of the BFP843 states the following S-Params at VCE = 1.8 V, IC = 15 mA, Z0 = 50 Ω: ! f S11 S21 S12 S22 ! GHz MAG ANG MAG ANG MAG ANG MAG ANG 9.500 0.6496 87.2 3.053 -35.6 0.0702 -3.8 0.6193 104.8 10.000 0.6881 83.0 2.737 -41.5 0.0722 -6.0 0.6663 99.6 10.500 0.7281 79.4 2.511 -47.3 0.0746 -6.9 0.7040 97.0 Please note:

• The amplifier is operated in a mixed 50 Ω / 40 Ω system • Assume the transistor to be unilateral!

a) Determine the range of capacitor values (Cmin...Cmax) to ensure an amplifier gain ≥ 10 dB!

Solution:

a) C = 0.16 pF … 1.12 pF


25. Repairing a Damaged Attenuator In the lab drawer you find the following circuit:

From the circuit structure it is quite obvious that this is an attenuator. Unfortunately one resistor burnt up and, thus, you don’t know its value. The only thing you remember is that this attenuator was intended for 50 Ω systems and it was excellently matched (the time before the resistor burnt up). Your job is to repair this attenuator!

a) What value is the damaged resistor?

b) Calculate the attenuator’s attenuation in Decibels!

c) Is it possible, by only selecting a different value for the burnt resistor, to build an attenuator with excellent matching for 20 Ω systems as well? Explain!

Solution:

a) R = 35.1 Ω

b) 10.01 dB

c) No, the minimum achievable input impedance of the attenuator is achieved if R = 0 Ω (short). Thus, the input impedance of the attenuator is ≥ 26 Ω in any case. This results in a mismatch when used in a 20 Ω system.


26. Amplifier’s Gain Variation due to Different Biasing The given amplifier uses an Infineon BFP843 bipolar transistor as active element. The circuit is operated at f = 2.0 GHz and does have an output matching network only. Relevant parts of the BFP843’s datasheet are given afterwards.

S-Parameters, VCE = 1.8 V, IC = 8 mA, Z0 = 50 Ω: ! f S11 S21 S12 S22 ! GHz MAG ANG MAG ANG MAG ANG MAG ANG 1.900 0.2201 -83.3 10.383 116.0 0.0571 -8.3 0.2708 -82.7 2.000 0.2199 -87.7 10.194 113.1 0.0573 -8.7 0.2701 -87.0 2.100 0.2177 -92.4 10.009 110.3 0.0567 -9.9 0.2679 -90.9 S-Parameters, VCE = 1.8 V, IC = 15 mA, Z0 = 50 Ω: ! f S11 S21 S12 S22 ! GHz MAG ANG MAG ANG MAG ANG MAG ANG 1.900 0.1342 -142.1 12.780 112.9 0.0464 -3.7 0.1327 -103.0 2.000 0.1412 -145.8 12.530 110.0 0.0470 -4.0 0.1350 -107.4 2.100 0.1474 -149.8 12.250 107.3 0.0459 -4.2 0.1369 -111.6

a) By using the Smith-chart find the parameters len and C to achieve maximum gain at the bias condition VCE = 1.8 V and IC = 8 mA!

b) Determine the change of the entire amplifier’s gain when the bias current is adjusted to IC = 15 mA without changing the output matching!

Solution:

a) length = 0.294 λ (105.8°), C = 1.03 pF

b) ΔG = +1.44 dB (G8mA = 20.50 dB, G15mA = 21.93 dB)


27. S-Parameters of a 2-Port The following circuit is operated at f = 1.8 GHz:

a) Calculate the S-matrix of the given circuit!

b) The circuit is driven with a +30 dBm signal at port 1. The signal source’s impedance is 50 Ω. Port 2 is terminated with 50 Ω. How much heat will be dissipated by the circuit? Please note: The sum of the power dissipation of resistors R1 and R2 is asked.

Solution:

a) 𝑆𝑆 = 0.15 0.850.85 0.15

b) 255 mW


28. Impedance transformation The given circuit is operated at f = 900 MHz, the port’s reference impedance is Z01 = 50 Ω.

By using the Smith-chart determine the transmission lines’ lengths 𝑙𝑙1 and 𝑙𝑙2 (as a ratio of the wavelength λ) in order to match the resistor R to port 1 (thus, S11 should be 0)! Please note: The transmission lines 1 and 2 have a 50 Ω wave impedance while transmission line 3 has 60 Ω!

Solution: 𝑙𝑙1 = 0.354𝜆𝜆 , 𝑙𝑙2 = 0.296𝜆𝜆

Alternative solution: 𝑙𝑙1 = 0.198 , 𝑙𝑙2 = 0.400𝜆𝜆


29. S-Parameters of a 2-Port The following circuit is operated at f = 477 MHz:

a) Calculate the S-matrix of the given circuit!

b) The circuit will be operated in a 50 Ω system. Calculate the maximum drive level from port 2 in order to limit thermal losses of the shown circuit to1 W!

Solution:

a) 𝑆𝑆 = −0.12 0.320.32 −0.52

b) 1.594 W


30. A 10 GHz Amplifier The shown amplifier circuit is operated at f = 10.0 GHz. It uses an Infineon BFP843 bipolar transistor as active element and is perfectly matched on its output. On the input side a matching network is missing, thus resulting in an imperfect match. The chosen bias point is VCE = 1.8 V, IC = 15 mA. All relevant parts of the BFP843’s datasheet are given on the next page.

S-Parameters, VCE = 1.8 V, IC = 15 mA, Z0 = 50 Ω: ! f S11 S21 S12 S22 ! GHz MAG ANG MAG ANG MAG ANG MAG ANG 9.000 0.6066 93.5 3.458 -28.0 0.0688 -0.6 0.5666 112.3 9.500 0.6496 87.2 3.053 -35.6 0.0702 -3.8 0.6193 104.8 10.000 0.6881 83.0 2.737 -41.5 0.0722 -6.0 0.6663 99.6 10.500 0.7281 79.4 2.511 -47.3 0.0746 -6.9 0.7040 97.0 11.000 0.7503 76.3 2.308 -53.2 0.0773 -8.9 0.7432 95.5

Please note: • Assume the active device to be unilateral!

a) Determine the power gain of the depicted circuit when operated with a 50 Ω source connected to port 1 and a 50 Ω load connected to port 2!

b) Determine the maximum acceptable VSWR of the load connected to port 2 to ensure the voltage gain being at least 85 % of its original value!

Solution:

a) 11.29 dB

b) 3.23


31. Impedance transformation The following circuit is to transform the load impedance ZL into the given input impedance.

Use the Smith-Chart to determine the transformer’s winding ratio and the line length (in fractions of the wavelength)! Please note:

• This example has two solutions with line lengths < λ/2!

Solution: 𝑙𝑙 = 0.2138𝜆𝜆 and 𝑛𝑛 = 2.757

Alternative solution: 𝑙𝑙 = 0.4427𝜆𝜆 and 𝑛𝑛 = 0.312


32. Power Dissipation in an Unknown Load In the following circuit, an unknown load ZL is connected to a lossless transmission line. Vector-voltmeters are used to measure the voltages V1 and V2. The signal source’s frequency is f = 1.5 GHz.

The readings from the vector-voltmeters are: 𝑉𝑉1 = 1.3 V ∡ 41° 𝑉𝑉2 = 0.3 V ∡–90° Calculate the power dissipation in the load ZL!

Solution: 𝑃𝑃 = 5.008 mW


33. Coplanar Lines Two different coplanar 50 Ω lines are connected in series. The discontinuity can be modeled by a parallel capacitor. This structure is operated at f = 26.5 GHz.

Assume a dielectric strength of 3.3 kV/mm for air and a dielectric strength of 44 kV/mm for the substrate! What is the maximum allowed peak input power (in Watts) to the adapter in order to prevent dielectric breakdown? Please note that the correct solution would require EM-field simulation. Therefore, assume the following:

- perfect terminations on both sides of the lines - edge effects are neglected (normally, sharp corners would introduce higher local field

strengths) - the voltage along the coplanar line’s lateral dimension is constant (normally, the voltage

on a microstrip or coplanar line is not constant along the line’s lateral dimension)

Solution: 𝑃𝑃 = 2.636 kW


34. Coaxial 3.5 mm to 1.85 mm Adapter A coaxial adapter (air filled) is used to interconnect measurement systems with different connector sizes. The diameter step can be modeled by a parallel capacitor.

One side of the adapter is terminated by a load |Γ| = −25 dB. What is the maximum operating frequency to ensure that the overall VSWR (adapter plus load) is < 1.4? Please note:

• When using the Smith-chart only, the solution will be an approximation. Try to find a solution which gives maximum accuracy – it can be done with ruler and compass!

Solution: 𝑓𝑓 = 10.820 GHz


35. S-Matrix of Bipolar Transistor Calculate the scattering parameters of an idealized bipolar transistor at the operating frequency f = 1.4 GHz in a Z0 = 50 Ω reference system.

gM =ICUT

IC = 100 mA UT = 28 mV

Rπ =β

gM

β = 70 Cπ = 15 pF RE = 2 Ω

Solution:

𝑆𝑆 = 0.1101 − 𝑗𝑗0.6566 0

−23.5637 + 𝑗𝑗14.9756 1

𝑅𝑅𝐸𝐸

𝐶𝐶𝜋𝜋 𝑅𝑅𝜋𝜋

𝑔𝑔𝑀𝑀 ∙ 𝑈𝑈𝜋𝜋

𝐵𝐵 𝐶𝐶

𝑈𝑈𝜋𝜋

𝐸𝐸


36. Lines with different Impedances Two lines with different wave impedances are connected in series according to the following schematic.

The S-parameters of line 2 for a 50 Ω reference impedance are 𝑆𝑆𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙2 = 0 −1

−1 0 .

a) The network is driven with 1 W from port 1. Port 2 is terminated with 50 Ω. Thus, the forwards going wave (from left to right) in line 1 has a power of 1 W. Calculate the power of the forwards going wave (from left to right) in line 2!

b) Proof that the S-parameters given for line 2 are correct!

Solution: a) 𝑃𝑃𝑓𝑓𝑓𝑓𝑙𝑙 = 2 𝑉𝑉

𝑃𝑃𝑟𝑟𝑙𝑙𝑟𝑟 = 1 𝑉𝑉


37. S-Parameters of a Circuit with a PIN-Diode A PIN diode is used as an adjustable element for a circuit operating at f= 2.4 GHz. The channel resistance of the diode is controlled by the bias current IF. Therefore, the resistance RS of the equivalent circuit is a function of IF.

a) Explain why the given RF-equivalent circuit c) represents the RF behavior of the circuit a) correctly!

b) Calculate the S-Parameters of the RF-equivalent circuit, assume a reference impedance of 50 Ω! Note that RS is a function of IF!

c) By varying IF it is possible to control RS in the range [1…100] Ω. Draw the curve which S22 describes in a 50 Ω Smith-chart when varying RS over the entire range!

Solution:

b) S =

⎝

⎛

𝑅𝑅𝑠𝑠+37.5 Ω+𝑗𝑗47.8 Ω𝑅𝑅𝑠𝑠+65.2 Ω+𝑗𝑗47.8 Ω

50 Ω𝑅𝑅𝑠𝑠+62.5 Ω+𝑗𝑗47.8 Ω

50 Ω𝑅𝑅𝑠𝑠+62.5 Ω+𝑗𝑗47.8 Ω

𝑅𝑅𝑠𝑠−37.5 Ω+𝑗𝑗47.8 Ω𝑅𝑅𝑠𝑠+65.2 Ω+𝑗𝑗47.8 Ω⎠

⎞

c)

a) circuit b) eq. circuit of the Diode c) RF equivalent circuit 𝐶𝐶1 = 10 nF; 𝐶𝐶2 = 𝐶𝐶3 = 22 nF; 𝐿𝐿𝑆𝑆 = 3.17 nH; 𝐿𝐿 = 12 µH; 𝑛𝑛 = 2; 𝑅𝑅 = 25 Ω; 𝑈𝑈𝑄𝑄 = 1 V


38. Matching with C/C-Network In a Smith-chart, mark the areas of 𝑍𝑍𝐿𝐿, which can be matched to a source impedance of 𝑍𝑍0 = 50 Ω by the given circuit!

Solution:


39. Variable phase shifter The following circuit allows controlling the phase shift between port 1 and port 2 (of the circuit enclosed by the dashed line) by changing the value of two identically large capacitances. (Note: variable capacitances are typically realized by varactor diodes).

The 90° hybrid’s scattering parameters are 𝑆𝑆90°Hybrid = 1√2

0 1 −𝑗𝑗 01 0 0 −𝑗𝑗−𝑗𝑗 0 0 10 −𝑗𝑗 1 0

.

a) Calculate the S-parameters of the resulting two-port! Simplify the result as much as

possible!

b) Draw the locus of all possible 𝑆𝑆21 values of the resulting two-port (the phase shifter) in the attached polar chart! Indicate the direction of increasing capacitances on the locus!

Please take care of the port numbering! The overall circuit (the phase shifter) is a two-port. It utilizes a 90° hybrid as a subcircuit, which is a four-port device. Thus, the overall circuit’s port 1 is connected to the 90° hybrid’s port 4, for example.

Solution:

a) 𝑆𝑆 = 0 −𝑗𝑗Γ𝐶𝐶

−𝑗𝑗Γ𝐶𝐶 0

b)

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