Impedance Matching and the Smith Chart

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Audio systems are designed such that the output from the amplifier is matched to the impedance presented by the speaker. This maximizes power output, and mimizes reflection between the two.

The same principle applies to microwave systems; we want the output of a source to be matched to the input of the next stage. If two components are connected (e.g. a transmission line to an amplifier), there will be a reflection if their impedances are unequal, reducing power transfer.

We will look at methods of designing networks that match one impedance to another.

Motivation

Power reflected from the input of a device, like an LNA, reduces the useable power.

• When two components are connected (e.g. a transmission line to an amplifier), there will be a reflection if their impedances are not matched
• This is important for several reasons:
  • Reflection reduces power transfer (e.g. carrying power from transmitter to an antenna).
  • The signal to noise ratio is related to reflection (e.g. when carrying input signals from an antenna to a receiver).
  • Minimizing reflection reduces amplitude and phase errors in power distribution networks (e.g. when designing a distribution network for an antenna array).
  • When transmitting high-power over cables, reflection reduces the power-handling capability of the cable.
• We want to be able to design a network between two items ("matching network") to reduce the reflection between them.

The use of a matching network to reduce power loss at the input of a transistor. E.g. if the input to a transistor at a particular frequency is taken to be ZL = 50 − j60 and source applied to the input of the transistor has an output impedance of Zs = 50, we need to find a matching network such that , or equivalently at the frequency of interest. This will maximize the amount of power transferred from the signal source to the transistor. We often match two devices over a narrow range of frequencies, as it is usually more difficult to match devices over a wide bandwidth.

Selecting items to have conjugate impedances (differing in phase by 180 degrees) is analogous to the requirement that angled ends of water pipes have complementary angles (differing by 180 degrees). It is important to remember that when we want maximum power transfer between two items (e.g. a filter connected to an amplifier), we need to ensure that their impedances are complex conjugates of one another, i.e. the real parts are equal, and the imaginary parts have the opposite signs. Or another way, they have equal magnitudes, but their angles have the opposite signs. This is similar to how pipes with angled ends need to have complementary angles to maximize water flow.

We design matching networks at the input and output of an amplifier.

We will be designing matching networks for amplifiers, to provide the desired source and load impedances to the transistor. Before we do that, we need to review Smith chart use, since we will be using them for matching network design.

What is in a matching network?

Lumped element matching networks (top), distributed matching networks (bottom)

Matching networks consist of lumped and/or distributed elements; we will work exclusively with passive elements. Circuits can be designed to match a particular impedance using active elements (e.g. a common gate input stage has an input impedance of 1/gm, and so can be designed to have a largely real impedance.).

Lumped elements, like resistors, inductors, and capacitors, are usually more compact. However at high frequencies parasitics reduce the effectiveness of the components. We are limited by practicality of components (e.g. can't have 1 mH inductor on monolithic circuit).

Distributed matching networks use series transmission lines and stubs. These may take up much more space (which means larger cost for MMIC), but the reactive parasitics associated with these usually much lower. Ohmic losses may be significant due to their large size.

Matching networks are often designed (or at least "tweaked") using Smith charts, so we need to diverge to review Smith charts before examining these matching networks.

Smith Charts

Smith chart with the reflection coefficient lines of magnitude and phase shown. Impedance lines are shown in black, magnitude of reflection coefficient are shown in blue, and angle of reflection coefficient are shown in green,

• Designed by Philip Smith over period from 1932 to 1937 with help of others
• Two separate "graphs" on one page
  • Impedance lines are always drawn
  • Reflection coefficient (Γ) lines are usually assumed, though shown in the plot on the right.
• Smith Chart uses include:
  • Calculation of Γ and Z_in from Z_L and visa versa
  • 
  • VSWR, and position of maxima and minima
  • Matching networks, stability, gain analysis

Smith Chart Derivation

• Lines of constant resistance and reactance derived from expression for Γ

Let Γ = Γ_r + jΓ_i then

• Equate real and imaginary parts, rearrange

• These are both equations of circles, and produce the lines of constant resistance (r), and constant reactance (x) on the Smith chart

Smith chart examples

Pair activity: Plot on a Smith chart the items below:

• The impedance of a short circuit termination
• The impedance of an open circuit termination
• The impedance of a length of transmission line that is λ / 8 long at a frequency ω_o, terminated with an open or short circuit, from DC to ω_o.
• Same, but for a λ / 4 length of transmission line
• Same, but for a λ / 2 length of transmission line

Impedance/admittance Smith chart

The Smith chart derivation above shows how to derive the curves on the impedance Smith by relating impedance to reflection coefficient. A similar analysis could be used to relate admittance and reflection coefficient, yielding the admittance form of the Smith chart (shown in red)

• The admittance chart is rotated relative to the impedance chart. E.g. a load with impedance Z = j1 at the top of the Smith chart corresponds to the admittance Y = 1 / Z = − j1, at the same location.
• Black lines indicate constant impedance, and red lines constant admittance

Caption

Motion on the Smith Chart

We would like to be able to calculate, using the Smith chart, the input impedance resulting from adding a series or shunt element to a load. We will see that this is how we can design a matching network.

Example: If we were to take a particular load ZL = RL + jXL, and add an inductor X = jωL in series with it its impedance changes to Z' = ZL + X = RL + j(XL + X). The real part of the input impedance remains constant, and the imaginary part increases by jX. On the Smith chart this corresponds to moving along a line of constant resistance, and increasing imaginary impedance. If ZL = 50 − j75 and the system impedance were 50 Ω, the normalized load impedance would be zL = ZL / Zo = 1 − j1.5. Adding an inductor with a reactance of X = jωL = j50 (which normalizes to x = j50 / 50 = j1) at the frequency of interest yields a normalized input impedance of z'L = 1 − j1.5 + j1 = 1 − j0.5. The shift on the Smith chart is along the line of constant resistance (r = 1), from j = − j1.5 to j = − j0.5.

Adding a series inductor shifts the load impedance along a line of constant resistance, and to a larger positive reactance.

It is easy enough to calculate the effect of adding a single series resistor mathematically, but it becomes more complex if we were to add multiple items in series and parallel. However, it is fairly simple to calculate the change in impedance with the Smith chart.

The change in impedance resulting from adding R, L, C elements to a load.

• The addition of various lumped and distributed elements can be seen as "movement" around the Smith chart
• For a particular normalized load with impedance z, or admittance y = 1 / z, the effect of adding various elements is shown to the right
• Series inductor: Z_L = + jωL adds a positive reactance, move clockwise on resistance circle
• Series capacitor: Z_C = − j / ωC adds a negative reactance, move counter-clockwise on resistance circle
• Parallel inductor: Y_L = − j / ωL adds a negative admittance, move counter-clockwise on conductance circle
• Parallel capacitor: Y_C = + jωC adds a positive reactance, move clockwise on conductance circle

The general rule is: adding positive reactance moves clockwise on a constant resistance circle; adding positive susceptance moves clockwise on a constant conductance circle. Either way, adding positive moves clockwise.

Matching Using the Smith Chart

• If we know how a L, C elements shift the load impedance, we can add appropriate elements to move a load impedance to any point on the Smith chart. We will do that in L-section Matching Networks.