Text Section: 2.4

Portfolio Question:

1) Describe and Demonstrate how to use a Smith Chart to find impedance, Vmin, Vmax, SWR, reflection coefficient, etc.

See Chapter 2, Problems 7-12

A Smith chart is a graphical representation of the complex reflection coefficient, G

Reflection Coefficient and Load (Z_L) are directly related:

G = (Z_L / Zo - 1) / (Z_L/Zo+ 1) = (z_L - 1) / (z_L + 1)

OR

Z_L / Zo = z_L = (1 + G ) / (1 - G ) ß This is NORMALIZED load impedance

z_L = r_L + j x_L

The real and imaginary parts of z_L are functions of G , and these functions can be plotted on the same chart. Remember |G | £ 1.

Example: Given Z_L , find G using Smith Chart

See transparencies (Copies to be made available in copy room)

How to find G :

1) Find Normalized load Impedance, z_L = Z_L / Zo = r_L + j x_L

1. Find intercept of semicircles for r_L and x_L and PLOT z_L
1. Draw line from center of smith chart to (or through) z_L
2. Read angle of G from outside of Smith chart
3. Measure | G | with a protractor and compare to line on bottom of smith chart labeled "Ref. Coeff. E or I"

Zo = 100 ohms

Z_L = open circuit

1. z_L = ¥ = ¥ + j 0
2. PLOT (far right)
3. Draw Line through z_L . Read Ð 0
4. Measure using a protractor (or this one is obviously =1) | G | = 1

G =1 Ð 0 (which is what we expect for an open circuit)

Z_L = open circuit

1. z_L = 0 = 0+ j 0
2. PLOT (far left)
3. Draw Line through z_L . Read Ð 180°
4. Measure using a protractor (or this one is obviously =1) | G | = 1

G =1 Ð 180° = -1 (which is what we expect for an short circuit)

Z_L = 100 + j 0 ohms

1. z_L = Z_L / Zo = 1 + j 0
2. PLOT (center of smith chart)
3. Draw Line through z_L . Not so easy … Ð ?
4. Measure using a protractor (or this one is obviously =0) | G | = 0

G =0 Ð ? (which is what we expect for a matched load)

Z_L = 100 + j 100 ohms

1. z_L = Z_L / Zo = 1 + j 1
2. PLOT (top right quadrant)
3. Draw Line through z_L . about Ð 63°
4. Measure using a protractor | G | = 0.45

G =0.45 Ð 63°

G = (z_L - 1) / (z_L + 1) = (0+j1) / (2+j1) =1Ð 90° / 2.236 Ð 26.56 ° = 0.45 Ð 63.43°

How do you find load impedance if given G ?

1. Plot G
2. Read z_{L =} z_L +j z_L
3. Unnormalize: Z_L = z_L * Zo

Admittance y_L = 1 / z_L

G = (z_L - 1) / (z_L + 1) = (1/y_L - 1) / (1/y_L + 1) = - (y_L - 1) / (y_L + 1) = 180 out of phase

Steps to find G from y_L:

1. Find normalized y_L = Zo / Z_L = g_L +jb_L
2. Plot it (Using same curves g=r and b=x)
3. "Transform it through the origin" … Rotate 180 degrees = draw a line of equal length through the origin. Now you have found z_L
4. Read G as before

EXAMPLE (See transparencies)

Z_in = Zo [1 + G e ^{-jb l} ] / [1 - G e ^{-jb l} ]

z_in = Z_in / Zo = [1 + G e ^{-j2b l} ] / [1 - G e ^{-j2b l} ]

Define reflection coefficient at the input (NOT G g ) as the reflection coefficient looking into the load frm the input location. G _l = G _L Ð -2b l

This represents moving 2b l radians towards the generator.

You can convert this distance to degrees, and read it off the outer circles on the Smith Chart (notice DIRECTION to the generator is marked)

OR 2b l = 2(2p / l ) l = 4p (l / l ) This has been normalized for you on the outside circle around the Smith Chart. Observe that if l=l , this represents 2 complete rotations around the Smith Chart. L = l /2 represents one complete rotation.

Does this make sense? For a Transmission line of length L = l /2, traveling from generator to the load and back would represent a phase shift of 360 degrees … one complete rotation.

Then z_in = [1 + G _l] / [1 - G _l ]

How to find Zin :

1. Normalize z_L = Z_L / Zo
2. Plot z_L. This also gives you G _L.
3. Rotate G distance l (given in wavelengths) TOWARDS the generator.
4. Read z_in and G _l
5. Zin = z_in * Zo

EXAMPLE (see transparencies)

To read SWR from the Smith Chart:

1. PLOT z_L
2. Draw a circle through it.
3. Read SWR from real axis to right (SWR ³ 1)

EXAMPLE (See transparencies)

To read Voltage maxima off Smith Chart:

1. PLOT z_L
2. First Voltage maximum occurs on right side of real axis. First Voltage minimum occurs on left side of real axis.

EXAMPLE (See transparencies)