The ABCD parameters of a series impedance are given by: For instance, the equivalent T model derived from the ABCD parameters of an amplifier IC can be examined to determine the bond wire inductance, which can be tuned out with a judicious selection of coupling capacitor.įor reference, the ABCD parameters of a shunt admittance Y are given by: Finding this relationship drives the choice of a π or T network. Applying this concept to an unknown component, the ABCD parameters can be examined to see if the reactance of the first series element of the T network or the admittance of the first shunt element of the π network is directly proportional to f or 1/f. With a little mathematics, it was straightforward to derive a π network model of this inductor. The actual value of Z is:ĭividing the imaginary part by 2 π*200 MHz gives 12.4 nH, which is close to the expected inductance of 12.5 nH.įigure 6 Adding the inter-winding capacitance to the π model for the inductor. One might expect symmetry, yet with more digits of accuracy, S 11 is not precisely equal to S 22, and S 21 is not precisely equal to S 12. The inductor’s S-parameters at 200 MHz, available from Coilcraft’s website, are:Ĭonverting the S-parameters to the ABCD (T) parameters: Using parameter extraction, we would like to know the equivalent circuit for the inductor, including its parasitics. For RF applications, the Spring series are high Q inductors, although the footprint may be too large for some designs. To apply the concept, consider a 12.5 nH air core inductor, such as the Coilcraft A04T MiniSpring. The conversion from 50 Ω S-parameters to the equivalent ABCD matrix is given by: For a two-port network, the ABCD parameters are defined as 1:įor the T network, the ABCD parameters are:Īgain, the Y term is obvious, and the Z terms are easily derived. ABCD parameters, which are also known as cascade, chain or T parameters, are particularly useful for this purpose. While this simple example assumes ideal components, it is reasonable to say that stray capacitance between the nodes of a lowpass filter is detrimental - which is why high isolation, lumped-element lowpass filters have shields between sections, where the capacitors to ground are implemented with a feedthrough capacitor in the wall of each shield.Īctual components are never ideal, making mathematical extraction useful to understand their limitations. Above 4 GHz, the isolation is clearly compromised by the stray capacitance. Figure 1 plots the response of a three element, 500 MHz Butterworth filter, showing the effect of 0.1 pF stray capacitance across an ideal inductor. To illustrate, knowing that a 0.1 pF effective capacitance exists between the nodes of a lowpass filter might lead to a superior design: shielding between the nodes, resulting in greater stopband isolation. To aid this understanding, RLC parameter extraction can be very enlightening. The intuitive understanding of an RF network is only possible if its behavior can at least be understood in first order terms. To transform to different port impedances, the following computation mustīe applied to the resulting S-parameter matrix.Figure 1 |S 21| response of a 500 MHz Butterworth lowpass filter, showing the effect of a 0.1 pF stray capacitance in parallel with the ideal inductor. The field of high frequency techniques this is usually. But calculations can onlyīe performed with all ports being normalized to the same impedance. In a circuit normalized to the same impedance. Renormalization of S-parameters to different port impedancesĭuring S-parameter usage it sometimes appears to have not all components The following matrices and notations are used in the Other matrix representations used in electrical engineering. When dealing with n-port parameters it may be necessary or Two-Port matrix conversion based on signal waves.Two-Port matrix conversion based on current and voltage.Renormalization of S-parameters to different port impedances.Next: Solving linear equation systems Up: Mathematical background Previous: Mathematical background
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