The π and T matrices are then easily derived using simple matrix multiplication.įor the decomposition of an unknown network, examine the Z 1 of the T model and the Y 1 of the π model for proportionality to f or 1/f.
#Abcd matrix series#
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.
![abcd matrix abcd matrix](https://image3.slideserve.com/5399085/abcd-matrix-for-a-cascaded-connection-of-two-port-network-l.jpg)
The intuitive understanding of an RF network is only possible if its behavior can at least be understood in first order terms.
![abcd matrix abcd matrix](https://d2vlcm61l7u1fs.cloudfront.net/media/b0c/b0c01303-6335-46fa-9fdf-a2eb4757b306/phpimW6v4.png)
Since dimension wise D is a ratio of current to current, it’s a dimensionless parameter.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. Thus it’s implied that on applying short circuit condition to ABCD parameters, we get parameter D as the ratio of sending end current to the short circuit receiving end current. Thus B is the short circuit resistance and is given byĪpplying the same short circuit condition i.e V R = 0 to equation (2) we get Since dimension wise B is a ratio of voltage to current, its unit is Ω.
![abcd matrix abcd matrix](https://d2vlcm61l7u1fs.cloudfront.net/media/b0d/b0d5e3ac-bc49-43e5-b4ed-00b715a90f9f/phpGTfvum.png)
Thus it’s implied that on applying short circuit condition to ABCD parameters, we get parameter B as the ratio of sending end voltage to the short circuit receiving end’s current. Receiving end is short circuited meaning receiving end voltage V R = 0Īpplying this condition to equation (1) we get,