ELEC 412 Lecture 5 1

ELEC 412 RF

& Microwave Engineering

Fall 2004

Lecture 5

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Smith Chart Use

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Smith Chart: Reflection

• If we plot on the polar plot, and overlay the circles of constant r and x, this yields the Smith Chart, on which we can convert from to Z (or the reverse) by inspection.

• To see how the Smith Chart works, first consider a matched load, Z = Zo and = 0. This point is at the origin of the plot, since = 0 +j0. This is plotted below left.

• Next, consider a transmission line terminated with an open circuit at d=0.

ELEC 412 Lecture 5 9

Smith Chart - Reflection

Zo Z(d)

d=0d

Z(0)

At d=0, the plane of the open, the current is constrained to be zero, so the reflected wave current must equal the incident wave current and be out of phase (i.e., I- = - I+, so that V- = V+). The impedance Z(0) at this point is ∞, and the reflection coefficient is = = 1/0.

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Smith Chart – Matched & Open Circuit Loads

= 0 r = 1

Matched load (=0)

= 1 r =

Open circuit load (=+1)

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Smith Chart – Short Circuit Load

Zo Z(d)

d=0d

Z(0)

= -1 r = 0

Short circuit load (=-1)

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= -1 d = 0

= 1 d = /4

Moving away from short toward generator

d = /8

(d)

Short through arbitrary line length

Smith Chart – Short Circuit Load Through Arbitrary Line Length

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= -1 d = /4

= 1 d = 0

Moving away from open toward generator

d = /8

(d)

Open through arbitrary line length

Toward generator

Toward load

Only the angle of changes

Smith Chart –Arbitrary Line Length

Any arbitrary impedance z or reflection coefficient will have the same behavior if we move along the transmission from the point it is measured toward the generator. And if the impedance is measured at a point on the transmission line other than at the termination, we can move toward the load as well. It is this variation only of the phase angle, and not the magnitude, of .

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x=1

x=-1

x=0.5

x=-0.5

x=0

r=0 r=0.5 r=1 r=2

x=1

x=-1

x=0.5

x=-0.5

x=0

r=0 r=0.5 r=1 r=2

SWR=2

Smith Chart –Resistance, Reactance, & SWR

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Smith Chart Benefits

The Smith Chart has at least four benefits:1. All possible values of , hence all possible values of Z, lie

within the unit circle.2. For a given termination, the variation of with transmission

line position is simply a rotation on the chart with no change in magnitude ||, and hence, no change in SWR.

3. Lines of constant R and X are uniquely defined circles on the chart, so we can input data in format and read the result in Z format by inspection.

4. Data from a slotted line can be entered directly in terms of SWR and distance between minima.

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The Smith Chart is a mapping onto the complex plane from the complex z plane. We let

= u + jv and z = r + jx

= u + jv =

Smith Chart Z Plane Mapping