Nested Circuits

Example 4.4 Impedance Expression for a Nested Circuit Derive an expression for the impedance response of the circuit shown in Figure 4.5(a), and give expressions for the dc current and the current at infinite frequency for an applied potential V. 

Figure 4.5 Circuits demonstrating the calculation of the impedance of nested circuit elements a) circuit with resistor and capacitor elements b) reconstruction of the circuit in terms of the generic impedance Zj and c) secondary reconstruction to facilitate calculation.

The next parentheses indicate a subsequent change from parallel to series connection. Inserting the capacitance in series with the resistance in a nested circuit (Fig. 2.35) is described as Rs(Cdi(RctCp)), that is, Rs in series with the parallel connection of Cd and a series connection of / ct and C . 

Let us look now at the impedance plots for a nested circuit using the same parameters as for a circuit in series. The results are presented in Fig. 2.39. 

Note that, using the same set of parameters / i, Ci, / 2> and C2, different plots are obtained for both circuits. Nevertheless, exactly the same plots as in the case of a nested circuit can be obtained using a circuit in series but with different parameters. 

Fig. 2.39 Complex plane and Bode plots for a nested circuit in Fig. 2.37, i(f 2C 2))), using...

Table 2.3 Results of fit of impedances in Fig. 2.39 to circuit containing two parallel (RC) elements in series. The parameters of the nested circuit are as follows Ro = lOSl.Ri = = 100...

In fact, plots for the nested circuit can be obtained using the circuit in series and the parameters shown in Table 2.3, where the resistance Rq is always the same, / o = 10 Cl, and other parameters are different. It should be stressed that the fit is exact, which means that the two circuits give exactly the same values of impedances. 

This effect arises fi om the fact that impedances of circuits in series and nested circuits (Fig. 2.37) can be represented by the same equation ... 

Example 14.2 Let us consider now a circuit characterized by two time constants producing two semicircles on complex plane plots. Three possible circuits explaining the impedances are shown in Fig. 14.3. Circuit (a) represents so-called a Voigt circuit consisting of the resistance in series with two (RC) circuits in parallel circuit (b) is a so-called ladder or nested circuit, and circuit (c) is a mixed circuit. The impedance of each circuit can be written as... 

Fig. 9.6 Hierarchical organisation of neuronal circuits governing the control of movement. Note the nesting of feedback loops. (After a figure in Shepherd, G.M. (1988). Neurobiology, Oxford University Press, Oxford.)...

It should be noted, however, that our application of the new interpretation does not imply that the traditional labour embodied definition of value should be completely abandoned. Foley (2000 30) is open to the possibility that there may be a role for both the new and traditional interpretations of the value of labour power. As Appendix 4 shows, the labour embodied definition of the value of labour power is nested in the input-output model of the circulation of money between departments of production, regardless of how prices are defined. The deviation of prices from values does not modify the constituent role of the labour embodied measure in the interindustry monetary circuit. It is only when a macroeconomic aggregation is developed under price-value deviations, and in the derivation of the scalar Keynesian multiplier, that a switch to the value-form definition is required. 

EXEHCI5E 1-B For the circuit below, find the nested family of transfer curves if the breakdown voltage of the Zener is swept from 1 V to 6 V. 

The nested DC Sweep can be used to generate characteristic curves for transistors. We will illustrate generating these curves using a BJT. Wire the circuit below ... 

We will now generate Hfe versus Ic curves at different temperatures. This is a typical curve found in most data sheets for BJTs. We will use almost the same Secondary Sweep that we used in Section 4.F.2, except that for the Secondary Sweep we will vary temperature rather than VCe- We will use the same circuit and setup as in Section 4.F.2, but we will modify the Nested Sweep. We will start with the circuit from the previous example ... 

The use of inductors to model low-frequency inductive loops in an impedance response is somewhat controversial. Demonstrate that the circuit presented as Figure 4.6(b) with negative R and C in the nested element can be mathematically equivalent to a circuit containing an inductor... 

The two circuits in Fig. 2.37 in series and nested are described by / o(Ci/ i) RiCi) and R(, C R C2Rt))). Other distributed circuit elements can also be used Q represents a constant phase element, CPE, W a semi-infinite Warburg element, Ws a finite length transmissive element, Wo a finite length reflecting element, and so forth. In the case of distributed elements, it is preferable to define them specifically. 

Fig. 2.37 Examples of two circuits producing two semicircles rai complex plane plots left series R( RiCi) R2C2) and right nested Ro(Cj(Rj(R2C2)))...

Until now, only circuits containing resistances and capacitances have been discussed. Inductive effects in electrical circuits appear when alternative electrical current flow creates a magnetic field interacting with the flowing current of course, in a strait wire the inductance is very small, but in looped wires or a coil it becomes larger. The inductive effects always lead to positive imaginary impedances, as will be shown in what follows. Let us first consider the circuit in Fig. 2.40, which contains inductance L in series with resistance Rq and a nested coimection of two (RQ circuits, i.e., LRo(Ci(Ri(R2C2))). The complex plane and Bode plots for this circuit without inductance were presented in Fig. 2.39. 

Electronics professionals sometimes call assemblies of wires like this "haywire circuits." A newcomer might be tempted to call this one a "rats nest."... 

A common defect associated with no-clean wave soldering is the seepage of flux between the pallet and the board. The resulting flux residue may inhibit in-circuit test probe contact. Care should be taken to ensure that the board fits well into the pallet nest, is rigidly indexed within the nest of the pallet, and is adequately retained against the pallet surface. As a pallet ages, it may shrink, take on twist, or bow and even delaminate. All of these may interfere with proper board seating. 

Output function represents the functional dependency of the output pin on the input pin. This function is formed as (functor argument-list). Any symbol can be specified as a functor, and (functor argument-list) can be nested in another argument-list. A few reserved functors are provided these include and, or, buffer, reg, and not. For these functors, OPTMAP attempts to improve the circuit by taking into account the functional meanings of its components. For other functors, OPTMAP uses function to obtain replacement candidates by pattern-matching. 

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