6.9 Simple calculations with the smith chart

Introduction to physical Page 1 / 1

Instructions on how to use the Smith Chart for simple calculations such as converting from admittance to impedance.

So, what do we do for $Z_{L}$ ? A quick glance at a transmission line problem shows that at the load we have a resistor and an inductor in parallel. Thiswas done on purpose, to show you one of the powerful aspects of the Smith Chart. Based on what you know from circuit theory youwould calculate the load impedance by using the formula for two impedances in parallel $Z_{L} ω L R ω L R$ which will be somewhat messy to calculate.

Let's remember the formula for what the Smith Chart represents in terms of the phasor $r s$ .

Z_{L} Z_{0} 1 r s 1 r s

Let's invert this expression

Z_{L} Z_{0} 1 Z_{L} 1 Z_{0} Y_{L} Y_{0} 1 r s 1 r s

says that is we want to get an admittance instead of an impedance, all we have to do is substitute

r s

for

r s

on the Smith Chart plane!

Y_{0} 1 Z_{0} 150 0.02

in our case. We have two elements in parallel for the load (

Y_{L} Y B

), so we can easily add their admittances, normalize them to

Y_{0}

, put them on the Smith Chart, go

180^{°}

around (same thing as letting

r s r s

) and read off

Z_{L} Z_{0}

. For a

200 Ω

resistor,

G

, the condunctance equals

1200 0.005

Y_{0} 0.02

G Y_{0} 0.25

. The generator is operating at a frequency of

200 MHz

, so

ω 2 f 1.25 109 s -1

and the inductor has a value of

160 nH

, so

ω L 200

and

B 1 ω L -0.005

and

B Y_{0} -0.25

We plot this on the Smith Chart by first finding the real part = 0.25 circle, and then we go down onto the lower half of the chart since that iswhere all the negative reactive parts are, and we find the curve which represents $-0.25$ and where they intersect, we put a dot, and mark the location as $Y_{L} Y_{0}$ . Now to find $Z_{L} Z_{0}$ , we simply reflect half way around to the opposite side of the chart, which happens to be about $Y_{L} Y_{0} 2 2$ , and we mark that as well. Note that we can take the length of the line from the center of the Smith Chart to our $Z_{L} Z_{0}$ and move it down to the $Γ$ scale and find that the reflection coefficient has a magnitude of about 0.6. On a real Smith Chart, there is also aphase angle scale on the outside of the circle (where our distance scale is) which you can use to read off the phase angleof the reflection coefficient as well. Putting that scale on the "mini Smith Chart" would clog things up too much, but thephase angle of $Γ$ is about ${3.0}^{°}$ .

Moving down the transmission line

Now the wavelength of the signal on the line is given as

λ ν_{p} f 2.8 108 200106 1 m

The input to the line is located

21.5 cm

0.215 λ

away from the load. Thus, we start at

Z_{L} Z_{0}

, and rotate around on a circle of constant radius a distance

0.215 λ

towards the generator. To do this, we extend a line out from our

Z_{L} Z_{0}

point to the scale and read a relative distance of

0.208 λ

. We add

0.215 λ

to this, and get

0.423 λ

Thus, if we rotate around the Smith Chart, on our circle of constant radius Since, after all, all we are doing isfollowing

r s

as it rotates around from the load to the input to the line. When we get to

0.423 λ

, we stop, draw a line out from the center, and where it intercepts the circle, we read off

Z_{L} Z_{0}

from the grid lines on the Smith Chart . We find that

Z_{in} Z_{0} 0.3 -0.5

Using a smith chart to convert from admittance to impedance

Thus,

Z_{in} 15 -25

ohms . Or, the impedance at the input to the line looks like a

15 Ω

resistor in series with a capacitor whose reactance

X -25

, or, since

X_{cap} 1 ω C

, we find that,

C 1 2 200200106 31.8 pF

To find

V_{in}

, there is no avoiding doing some complex math:

V_{in} 15 -25 50 15 -25 10

Which, we write in polar notation, divide, figure the voltage and then return to rectangular notation.

V_{in} 29.1 59 69.6 -21 10

V_{in} 0.418 -38 10 4.18 -38 3.30 -2.58

Find vin

If at this point we needed to find the actual voltage phasor

V^{+}

we would have to use the equation

V_{in} V^{+} β L Γ V^{+} β L V^{+} β L Γ V^{+} θ_{r} β L

Where

β 2 λ

is the propagation constant for the line as mentioned in the last chapter , and

L

is the length of the line.

For this example, $β L 2 λ 0.215 λ 1.35 radians$ and $θ_{Γ} Γ 0.52 radians$ . Thus we have:

V_{in} V^{+} 1.35 0.52 V^{+} 0.52 1.35

Which then gives us:

V^{+} V_{in} 1.35 0.52 0.52 1.35

When you expand the exponentials, add and combine in rectangular coordinates, change to polar, and divide, you will get a phasorvalue for

V^{+}

. If you do it correctly, you will find that

V^{+} 5.04 -71.59

Many times we don't care about $V^{+}$ itself, but are more interested in how much power is being delivered to the load. Note that power delivered to the inputof the line is also the amount of power which is delivered to the load! Finding $I_{in}$ is easy, it's just $V_{in} Z_{in}$ . All we have to do is change $Z_{in}$ to polar form.

Z_{in} 15 -25 29.1 59

I_{in} V_{in} Z_{in} 4.18 38 29.1 59 0.144 21

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Introduction to physical electronics. OpenStax CNX. Sep 17, 2007 Download for free at http://cnx.org/content/col10114/1.4

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Introduction to physical electronics' conversation and receive update notifications?

Ask

	15 Dr. Amberg Pharm quiz By Brooke Delaney Start Exam
	6 Neuroanatomy 06 Head Somatic Visceral Sensory By Stephen Voron Start Quiz
	8 AP Key Terms 08 The Appendicular Skeleton By OpenStax Start Key Terms
	Worldport Outside By Rachel Carlisle Start Quiz
	22 AP Key Terms 22 The Respiratory System By OpenStax Start Key Terms
	How much do you love him? By Zarina Chocolate Start Quiz
	NCE Ch 05 Theories in Counseling Helping... By Anh Dao Start Exam
	Pre-class Online Quiz - Case Control Studies By Mucho Mizinduko Start Quiz
©flickr: Steve	C Programming Language By JavaChamp Team Start Quiz
	Clinical Issues in TB Management By Cath Yu Start Quiz