Notes 16 ECE Microwave Engineering Fall 2015 Impedance Matching Prof. David R. Jackson Dept. of ECE 1

Impedance Matching Impedance matching is used to:  Maximize power from source to load  Minimize reflections  Set terminating condition Considerations: Complexity Implementation Bandwidth A djustability Matching circuit typically requires at least 2 degrees of freedom. 2 Two constraints

We will consider: 1)Lumped element matching circuits 2)Transmission line matching circuits 3)Quarter-wave Impedance transformers 3 Impedance Matching (cont.)

Lumped-Element Matching Circuits Examples 4

5 Lumped-Element Matching Circuits (cont.) m m+2

Smith Charts Review Short-hand version 6  plane

7 Smith Charts Review (cont.)  plane

ZY - Chart 8 Smith Charts Review (cont.)  plane

Series and Shunt Elements 9  plane Note: The Smith chart is not actually being used as a transmission-line calculator but an impedance/admittance calculator. Hence, the normalizing impedance is arbitrary. Center of Smith chart: Z in (real)

High Impedance to Low Impedance The shunt element decreases the impedance; the series element is used to “tune out” unwanted reactance. 10 s hunt-series “ell” Two possibilities Use when G L < Y in (We are outside of the red G = 1 circle.)

The series element increases the impedance; the shunt element is used to “tune out” the unwanted reactance. Low Impedance to High Impedance 11 Series-shunt “ell” Two possibilities Use when R L < R in (We are outside of the black R = 1 circle.)

Example 12 Use low impedance to high impedance matching. Here L means inductor. Choose normalizing impedance 100 [  ]

13 Example (cont.) Here, the design example was repeated using a 50 [  ] normalizing impedance. Note that the final normalized input impedance is 2.0.

14 Matching with a Pi Network Note that this solution is not unique. Different values for B c2 could have been chosen. Note: We could have also used parallel inductors and a series capacitor, or other combinations. This works for low-high or high-low. The same example (1000   100  ) is used to illustrate matching with a pi network.

15 Here, the design example was done using a 50 [  ] normalizing impedance. Note that the final normalized input impedance is 2.0. Pi Network Example

Transmission Line Matching 16  Single-stub matching  Double-stub matching  This has already been discussed (will be reviewed briefly).  This is an alternative matching method.

Transmission-Line Matching 17 Single-Stub Tuning  plane Note: Only one of two possible solutions is shown.

18 Transmission-Line Matching (cont.)  plane

Short ckt. (Can also use open ckt.) Series stub 19 Transmission-Line Matching (cont.)  plane Note: Only one of two possible solutions is shown.

Double-Stub Tuner 20 The advantage of the double-stub tuner is that we can use a fixed distance d between the stubs (hence we can re-tune easily if the load changes). Note: d is arbitrary (it simply changes the load). The distance d is arbitrary but fixed.

21 Double-Stub Tuner (cont.) 1)Add shunt stub 1 in order to intersect the rotated (green) 1 + jB circle. 2)Rotate on the Smith chart a distance d (black dashed curve) to intersect the 1+jB circle. 3)Add shunt stub 2 to go to the center of the Smith chart The green circle is the 1+jB circle that has been rotated counter-clockwise a distance d. Note: Both stubs are capacitive here.

22 Double-Stub Tuner (cont.) Alternative solution (The stubs are both inductive.) 1)Add shunt stub 1 in order to intersect the rotated (green) 1 + jB circle. 2)Rotate on the Smith chart a distance d (black dashed curve) to intersect the 1+jB circle. 3)Add shunt stub 2 to go to the center of the Smith chart

Quarter-Wave Transformer 23 Note: If Z L is not real, we can always add a reactive load in series or parallel to make it real (or add a length of transmission line between the load and the transformer to get a real impedance).

At a general frequency: 24 Quarter-Wave Transformer (cont.) After some algebra (omitted):

After some more algebra (omitted): 25 Quarter-Wave Transformer (cont.)

26 Quarter-Wave Transformer (cont.) The bandwidth is defined by the limit  m. (For example, using  m = 1/3 corresponds to SWR = 2.0. We could also s ay  m = dB. ) Bandwidth of transformer: Solving for  m : We set

For TEM lines: 27 Quarter-Wave Transformer (cont.) Hence, using  m from the previous slide, Note: Multiply by 100 to get BW in percent.

28Example Given: We’ll illustrate with two choices: