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Laboratory 4: Impedance matching with a single stub circuit – design

In this laboratory, we will implement a code to design a single stub impedance matching circuit, and then analyze

it’s frequency dependent behaviour. The parameters of the single stub circuit are defined in the diagram below,

Y0 = 1/Z0 the characteristic admittance of the lossless transmission line, YS = 1/ZS the input admittance of the

shorted stub of length l, and YA = 1/ZA is the input admittance of the line with the load admittance YL = 1/ZL.

The impedance matching condition is: 𝑦୧୬ = 1 = 𝑦 + 𝑦ୗ

∴ 𝑦 = 1 + 𝑗𝑏 𝑦ୗ = 𝑗𝑏ୗ = -𝑗𝑏

For this problem it will be useful to use the Smith chart in admittance mode, shown below. Recall that the short

circuit ( -Γ = 1 ) is at the right hand side of this chart.

For our analysis it will be useful to make use of the following relations for admittance towards the load:

-Γ =

𝑦 − 1

𝑦 + 1 = |Γ|exp(𝑗𝜓) -Γ’ = |Γ|exp(𝑗𝜓’) 𝑦 =

1 + (-Γ’)

1 − (-Γ’)

= 𝑔 + 𝑗𝑏

Simple geometry of the triangle can be used to locate the intersection with gA = 1: cos 𝜓’ = |Γ|

The rotated reflection coefficient for the shorted stub is:

-Γ’ୗ =

(-𝑗𝑏) − 1

(-𝑗𝑏) + 1 = exp(𝑗𝜙)

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Finally, the lengths can be inferred from the phase of the reflection coefficients

-Γ

-Γ

= exp[𝑗(𝜓 − 𝜓’)] = exp(𝑗2𝛽𝑑)

1

-Γ’ୗ

= exp[𝑗(2𝜋 − 𝜙)] = exp(𝑗2𝛽𝑙)

where β is the phase constant of the lossless transmission line sections.

1. Single shorted stub impedance matching circuit design

Write a function that calculates the lengths l and d for the single shorted stub (reporting the solution with shortest

length d) given the normalized load impedance zL = ZL / Z0 and the phase constant β,

[ l, d ]= shortedstubdesign(zL,beta)

You are free to design the flow your function however you wish. An example flow is as follows:

1. Calculate the normalized load admittance yL.

2. Calculate the negative reflection coefficient for the load -ΓL. It might be useful to work directly with the

negative reflection coefficient, denoting it with a variable nGammaL.

3. Calculate the angle ψ of the reflection coefficient.

4. Calculate the angle ψ’ of the rotated reflection coefficient. Choose the value of ψ’ that minimizes ψ – ψ’.

5. Calculate the length d.

6. Calculate the rotated negative reflection coefficient -Γ’A corresponding to the selected ψ’.

7. Calculate the admittance yA.

8. Calculate the shorted stub negative rotated reflection coefficient -Γ’S.

9. Calculate the angle .

10. Calculate the length l.

You may find it useful to use the modulo function to fold phase angles into the interval [0,2π) using the MATLAB

command theta_new = mod(theta_old,2*pi).

Test your function using a normalized load impedance zL = 0.70 – j0.95 and phase constant β = 2π rad/m. You

should find l = 0.111 m and d = 0.059 m.

№ 1: Show your results to the teaching assistant.

Optional:

Can you develop a code for designing a matching circuit using a single open-circuit stub?

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Laboratory 5: Impedance matching with a single stub circuit – analysis

1. Calculating the frequency response of a single shorted stub circuit

Write a function that calculates the reflection coefficient for a single stub circuit with lengths l and d, normalized

load impedance zL, and phase constant β,

[ Gamma ]= shortedstubresponse(l,d,zL,beta)

You are free to design the flow your function however you wish. An example flow is as follows:

1. Calculate the load reflection coefficient ΓL.

2. Calculate the rotated load reflection coefficient Γ’L.

3. Calculate the admittance yA towards the load.

4. Calculate the rotated reflection coefficient Γ’S for the short circuited stub.

5. Calculate the admittance yS towards the short.

6. Calculate the total admittance yin = yA + yS.

7. Calculate the reflection coefficient Γ.

Test your function using the example of the previous laboratory, with normalized load impedance zL = 0.70 – j0.95,

phase constant β = 2π rad/m and lengths l = 0.111 m and d = 0.059 m. You should find a reflection coefficient Γ =

0, corresponding to the matched condition (with possible deviation due to numerical rounding error).

№ 1: Show your results to the teaching assistant.

2. Analyzing the frequency response of a single shorted stub circuit

Design the lengths l and d of a single shorted stub circuit for a normalized load impedance zL = 0.4, with a lossless

transmission line of phase velocity vp = 2×108

m/s and an operating frequency f0 = 1 GHz (from which you can

determine the phase constant β ).

Calculate the reflection coefficient Γ versus frequency f for your single shorted stub circuit with the lengths l and

d that you have designed. Take a frequency range f = 1 MHz to 2 GHz in steps of Δf = 1 MHz. Note that the phase

“constant” β depends upon frequency.

Plot the reflection coefficient Γ on a Smith chart, using the MATLAB command s = smithplot(f,Gamma).

Inspect your Smith chart, which is presented in the impedance mode.

Plot the magnitude of the reflection coefficient |Γ| versus frequency f.

Is impedance matching achieved at the design frequency f = 1 GHz ?

In the low frequency limit 𝑓 → 0 , what is the input impedance? Does this agree with what you expect at dc?

The reflection coefficient Γ = -1, equivalent to that of a short circuit, at a frequency f > 1 GHz. Why?

№ 2: Show your results to the teaching assistant.

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Repeat the exercise, and design a single shorted stub circuit for matching a load zL = 0.04. Calculate the reflection

coefficient Γ versus frequency f. Take a frequency range f = 1 MHz to 2 GHz in steps of Δf = 1 MHz. Plot the

reflection coefficient Γ on a Smith chart, and the magnitude of the reflection coefficient |Γ| versus frequency f.

What do you notice about the frequency dependence of |Γ| near the design frequency as compared to the single

stub circuit with zL = 0.4 ?

№ 3: Show your results to the teaching assistant.