ECSE 354 -Laboratory 3 - Solved

 

In this laboratory, we will develop several simple functions relating the load impedance ZL, reflection coefficient Γ and input impedance Zin for a lossless transmission line with characteristic impedance Z0, phase constant β and length l.

 

 

1.  Reflection coefficient and input impedance
 

Write a function that calculates the reflection coefficient Γ from the normalized load impedance zL =  ZL / Z0,

 

[ Gamma ]= refcoeff(zL)

 

a function that calculates a rotated (phase shifted) reflection coefficient Γ’ from the reflection coefficient Γ and round-trip phase  = 2βl,

 

[ Gammarot ]= rotrefcoeff(Gamma,theta)

 

and a function that calculates the normalized input impedance zin =  Zin / Z0 from the reflection coefficient Γ and round-trip phase  = 2βl,

 

[ zin ]= inputZ(Gamma,theta)

 

as defined by the following equations,

 

                         𝑧 − 1                                                                  1 + Γ'      1 + Γexp(−𝑗 )

             Γ =                   Γ' = Γexp(−𝑗 )             𝑧    =  =            = 2𝛽𝑙

                         𝑧 + 1                                                                  1 − Γ'      1 − Γexp(−𝑗 )

 

Alternatively, you might wish to calculate input impedance zin directly from the rotated coefficient Γ’. In the following, you may find it useful to recall,  

2𝜋𝑓

𝛽=  

𝑣

 

For all of the calculations that follow, consider the case of a lossless transmission line with a characteristic impedance Z0 = 75 Ω, phase velocity vp = 2 x 108 m/s, and length l = 0.25 m.

 

Take a frequency range f = 1 MHz to 300 MHz in steps of Δf = 1 MHz.

 

 

 

 

             

2.  Reflection coefficient of RC, RL and RLC circuits on a Smith Chart
 

Consider three load impedances:

a)       series RC circuit, ZL = R + 1/jωC, with R = 75 Ω and C = 100 pF

b)      series RL circuit, ZL = R + jωL, with R = 75 Ω and L = 200 nH

c)       series RLC circuit, ZL = R + 1/jωC  + jωL, with R = 75 Ω, L = 200 nH and C = 100 pF

d)      parallel RLC circuit, ZL = 1/( 1/R + 1/ jωL  + jωC ), with R = 75 Ω, L = 200 nH and C = 100 pF

 

Calculate the reflection coefficient Γ for the frequency range specified above, and plot Γ in the complex plane on a Smith chart for each of the load impedances. This can be conveniently done using a MATLAB command from the RFToolbox,  s = smithplot(f,Gamma) where s is the handle for your Smith chart. You can add a title with the commend s.TitleTop = 'RC circuit' for example. The values of Γ can be read explicitly by floating your cursor over the data point of interest.

 

Inspect your Smith charts. Notice that |Γ|, and the VSWR s, varies with frequency for these load impedances.

 

Why does the reflection coefficient Γ for the series RC circuit follow the r = 1 locus in the lower half plane? Why does the reflection coefficient Γ for the series RL circuit follow the r = 1 locus in the upper half plane?

Why does the reflection coefficient Γ for the series RLC circuit follow the r = 1 locus?

At what frequency f is the series RLC circuit impedance matched, Γ = 0, with the transmission line? What is the reactive component of the load impedance, xL = Re{zL}, when the impedance matching condition is satisfied?  

 

№ 1: Show your results to the teaching assistant.

 

Optional: Why does the reflection coefficient Γ for the parallel RLC circuit follow the locus of constant normalized conductance,  g = Re{Z0/ZL} = 1.

 

3.  Reflection and input impedance of transmission line with a resistive load
 

Consider a resistive load impedance ZL = 15 Ω. Calculate the reflection coefficient Γ, the rotated reflection coefficient Γ’ and the normalized input impedance zin for the frequency range specified above.

 

Plot the rotated reflection coefficient Γ’ in the complex plane on a Smith chart, using for example the MATLAB command s = smithplot(f,Gammarot). Plot the real and imaginary parts of the normalized input impedance zin versus frequency. This can be done simply using h = plot(f,real(zin),f,imag(zin)) for example.

 

Inspect your Smith chart and your impedance versus frequency plots.

 

Where on your Smith chart are the points corresponding to low frequency operation ( f = 1 MHz ) and high frequency operation ( f = 300 MHz ) ?

At what frequency is the transmission line acting as a ¼ wave transformer?

Use your Smith chart to determine the length of the transmission line in terms of wavelengths at f = 300 MHz, by determining the rotation angle  = 2βl at f = 300 MHz, and then calculating l / λ(300 MHz) = βl/2π = /4π. Does this agree with direct calculation of l / λ(300 MHz) = l x 300 MHz / vp ? Use your Smith chart to determine the VSWR.

 

№ 2: Show your results to the teaching assistant.

4.  Reflection and input impedance of transmission line with an inductive load
 

Consider an inductive load impedance ZL = jωL with L = 20 nH. Calculate the reflection coefficient Γ, the rotated reflection coefficient Γ’ and the normalized input impedance zin for the frequency range specified above.

 

Plot the rotated reflection coefficient Γ’ in the complex plane on a Smith chart.  Plot the real and imaginary parts of the normalized input impedance zin versus frequency. Restrict your impedance figure axes to normalized impedance components within the range ±15.

 

What is the minimum frequency f required to achieve a capacitive ( xin < 0 ) input impedance zin ?

Will increasing the inductance L increase, or decrease, the minimum frequency f required to achieve a capacitive

( xin < 0 ) input impedance zin ?

 

 

№ 3: Show your results to the teaching assistant.

 

5.  Reflection and input impedance of transmission line with a series RLC circuit
 

Consider a load impedance ZL = R + jωL + 1/ jωC  of a series combination of resistance R = 75 Ω, inductance L = 200 nH and capacitance C = 100 pF (as in exercise 2 c). Calculate the reflection coefficient Γ, the rotated reflection coefficient Γ’ and the normalized input impedance zin for the frequency range specified above.

 

Plot the rotated reflection coefficient Γ’ in the complex plane on a Smith chart. The result is non-trivial. Plot the real and imaginary parts of the normalized input impedance zin versus frequency. Restrict your figure axes to impedance components within the range ±15.

 

Inspect your Smith chart and your impedance versus frequency plots, and consider the following questions.

At what frequency f is the impedance matching condition Γ = 0 satisfied?  

Does the length l of the transmission line affect the frequency at which Γ = 0 is achieved?

Why does |Γ| ≈ 1 in the low-frequency ( f = 1 MHz ) and high-frequency ( f = 300 MHz ) limits?