Laboratory 3 : Smith Chart

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Laboratory 3 : Smith Chart
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,
Γ =
𝑧௅ − 1
𝑧௅ + 1 Γ' = Γexp(−𝑗 ) 𝑧௜௡ =
1 + Γ'
1 − Γ' =
1 + Γexp(−𝑗 )
1 − Γexp(−𝑗 )  = 2𝛽𝑙
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,
𝛽 =
2𝜋𝑓
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.

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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.
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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?
№ 4: Show your results to the teaching assistant.
Optional: Compare the Smith charts of the series RLC circuit reflection coefficient Γ (exercise 2c) and rotated
reflection coefficient Γ’. Using the two charts, determine the length of the transmission line in terms of
wavelengths at f = 300 MHz. In other words, determine the ratio l / λ(300 MHz). Confirm that your answer is the
same as in exercise 3, as it must be, because the transmission line is unchanged.