# T Network Impedance Matching for Sub-50 Impedances

### Key Takeaways

• T network impedance matching uses three reactive components to build a filter with reasonable bandwidth and filter response.

• T filters are excellent options for low-impedance filters.

• The added component (compared to an L filter) allows designers to control the Q factor of the T filter.

T network impedance matching builds from a simplified model of two L filters.

Filters are important in separating targeted signals for amplification and further signal processing and are an indispensable building block of system design. Filter response is a tradeoff between the coverage bandwidth about the carrier frequency, and three- or four-component filters offer a narrow bandwidth but sharper response than their constituent L filters. The three-component filter forms a middle ground between the broad response of the L filter and the sharp response of the four-component filter networks. T network impedance matching (alongside its dual pi network) offers excellent filter characteristics for low-impedance applications.

Comparing Three-Component Networks

 Composition Impedance Suitability T Network 2 series, 1 parallel component <50 Ω Pi Network 2 parallel, 1 series component >50 Ω

## Calculations for Impedance Matching

Like the pi network filter, the T network filter is a three-component impedance-matching network. Implementation of the network requires an assessment of the source and load impedance from one of two possibilities:

1. The source and load impedance are purely real-valued, i.e., the current and voltage waveforms are in-phase.
2. Either (or both of) the source and load impedance are complex-valued due to the presence of inductive or capacitive reactance, which shifts the angular position of the voltage and current waveforms out-of-phase.

An entirely real-valued impedance is ideal, as even circuits built out of “only” resistive components contain some amount of inductive or capacitive reactance due to parasitics. However, this contribution may be negligible enough that the error generated by the reactance component relative to the ideal value is acceptable – this will depend on the accuracy desired by the circuitry. Symbolically, circuit design can instead represent the matching between the load and source impedance using the voltage standing wave ratio equation:

where Γ is the reflection coefficient, ZL and Z0 are the load impedance and characteristic impedance (respectively), and Vr and Vf are the reflected voltage and forward voltage (respectively). Circuit design wishes to minimize the value of VSWR to maximize power transfer from source to load, which occurs when Γ approaches 0 (or equivalently, when ZL - Z0 = 0 or when Vf >> Vr). Because mismatch loss is a logarithmic function, exact impedance matching can be unnecessary due to the diminishing returns of S11 (reflection coefficient) reductions: for example, a -10 dB S11 results in a mismatch of 0.45 dB, a -15 dB S11 results in a mismatch of 0.13 dB, a -20 dB S11 results in a mismatch of 0.04 dB, etc.

It’s important to understand that impedance matching is frequency dependent, meaning matching will only occur around some center frequency where the response is maximum (i.e., the -3 dB bandwidth). This value directly correlates to another constraint of the filter network: the Q-factor. In physical terms, the Q-factor measures the reactive energy stored by a filter over the resistive energy dissipated; Q also represents the damping of the oscillator (most often underdamped in these cases). Generally, a three-component filter is more beneficial for narrowband antennas as the additional component (compared to a two-component filter) affords an additional degree of freedom for the frequency, bandwidth, and Q-factor.

## T Network Impedance Matching and the Quality Factor

The T network gets its name from the arrangement of the network elements: two components in series with the load line with a parallel component tied to ground between them. Like the pi network, the T network has opposing reactance values on the series and parallel component: this doesn’t force the series or parallel component to adopt a specific reactance absolutely, only relatively.  Designers can consider a T network a series of two L networks with the virtual resistance formed between the parallel combination of the two ground-side components. This comparison is useful, as the Q factor of the T network comes from the constituent L network with the higher Q. Specifically, the equation for Q is:

where R< is the smaller of the two terminating resistances. Notably, the Q-factor uses the same calculation as the pi network. After determining the Q-factor, designers can calculate the virtual resistance formed between the parallel components of the cascading L networks (this will also determine whether the resulting component is a capacitor or inductor).

Given the similarities between the two network types, designers may wonder when to apply the different topologies. Circuits can handle harmonics with a high- or low-impedance low-pass filter that either reflects the associated energy toward the source or shunts it to the ground plane (respectively). Typically, 50 ohms will be the cutoff point between the two models, with T networks preferred at lower impedances and pi networks desired at higher impedances.

## Cadence Solutions Are a Match for Filter Design

T network impedance matching is necessary to ensure signal integrity and safe power transfer without the possibility of damaging load or source components through standing wave reflections. Like the pi filter, the T filter offers a sharper response than a two-component L network but with a broader bandwidth than more expansive four-component filter networks. Filter design requires extensive simulation tools to assess responses around the tuning frequency correctly. Cadence’s  PCB Design and Analysis Software suite provides electronics teams with a robust DFM simulation environment. Simulation results are then readily transferable to OrCAD PCB Designer for rapid ECAD production.

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