Direct Conversation of Generalized Parameters of Multicomponent Two-Terminal Networks Using Capacitive Differentiator RC-Chains

We present a device for direct conversion of generalized parameters of passive multicomponent two-terminal networks (TTN) with excitation of measuring circuit (MC), comprising a model component and the two-terminal network being tested, on applying to the measuring circuit voltage pulses which vary as the n-th power of the time and n-fold signal differentiation at measuring circuit input and output with differentiators being executed on RC-chains.


Introduction
Transformation of the parameters of bipolar electrical circuits is an important branch of modern information and measuring equipment.Areas for application of converters: transformation parameters of physical processes using parametric sensors, control elements and components electronic equipment.Converters parameters multielement passive multicomponent two-terminal networks (TTN) can be constructed using the direct conversion and equilibration.Converters with balancing voltage or current TTN compensating signal are highly accurate, but have a low speed.When monitoring processes with rapidly changing properties appropriate to use direct object parameters being tested.To convert the parameters of measuring circuit (MC) into electrical signals as MC, it is advisable to use a voltage divider into one arm which includes the measured two-terminal, and in another-defensive chain.If the impact on the use voltage pulse, the shape of which has the form of a power function where U m is the test impulse amplitude, t imp is their duration, and in a steady-set mode on completion of the transient process, the divider output response represents a total of impulses in the form of power functions with indices from n to zero: Each component of tension (2) contains information on the corresponding generalized MC parameter.Where H 0 , H 1 , H 2 ,…, H n are generalized parameters of measuring circuit transfer function, with its operator image being generally expressed as The value of each output signal component includes information on one or several electrical parameters of the two-terminal network.The exponent of n of the power impulse must correspond to the number of N-components in the two-terminal network being tested: n ≥ N-1.Thus, for example, at excitation of the measuring circuit (MC) by cubic impulses of voltage, the MC output signal becomes: Triple differentiation of signal (5) allows to define every H-parameter Results of determination of H-parameters directly on formulas ( 5)-( 8) depend on amplitude of feed-in impulse.
For the removal of this factor it is necessary to add the second channel of differentiators: and to ration the values of signals ( 5), ( 6), ( 7), (8) by signals ( 1), ( 9), ( 10), (11) accordingly.Thus, formulas for the calculation of H-parameters will look like: In publications (Ivanov, Titov, & Petrov, 2011, 2012) it was justified the method of direct conversion of H-parameters of the two-terminal network using operations of n-fold differentiation of the test impulse signal and divider output voltage comprising a single-component network Z 0 and a two-terminal network (TTN) of the object measured.When the transient process in the measuring circuit is completed, sampling of transient values of divider input and output signals is performed, as well as all differentiator outputs.The scheme of the converter of parameters of the four-element two-pole network with differentiation of signals is given on Figure 1.In the article (Ivanov, Titov, & Petrov, 2012) a method and device of direct transformation are considered with differentiators on operational amplifiers (OpAmp).For steady work of multistage differentiators the correction of frequency description of OpAmp is applied in area of high-frequencies.

Using Passive RC-Section for Differentiation Signal
Differentiating cascades on active components and operating amplifiers in particular has drawbacks, such as the tendency to stability loss, complexity of provision of cascade identity, stability, and drift elimination.In this article it is suggested to apply differentiators on the passive RC-chains consisting of condensers and resistors.Use of passive circuits is proposed as multi-cascade differentiators comprising in-series differentiating RC-chains.The scheme of converter with differentiating RC-chains is presented on Figure 2 (Ivanov, Emelyanov, Titov, & Sohan, 2011).
To obtain a n-cascade differentiator having transfer function it would be good to include buffer cascades between RC-chains complicating the device scheme.However, a transfer function close to the one described above can be provided using only passive RC circuits.To simplify analytical expressions, it is reasonable to set each cascade's time constants to the same values: R 1 C 1 = R 2 C 2 = R 3 C 3 = τ, using different capacitance and resistance values in each cascade.For example, if one takes R 2 C 2 = RC = τ in the second chain, then resistance is to be reduced and capacitance increased as much as in the first chain, and, conversely, resistance is to be increased and capacitance reduced by as much as in the third chain: where m < 1.Let us define transfer functions for the output of the first, second and third cascades of differentiators and their generalized parameters.Transfer function for the first RC-chain output is as follows Generalized parameters of transfer function ( 16) equal the following: Transfer function for the second RC-chain output is expressed as and its generalized parameters equal to the following: Finally, the transfer function for the third RC-chain output can be presented with the formula and its K-parameters equal to Using expressions for operator image of cubic-form power impulse (1) and generalized parameters ( 17), ( 19), ( 21) of transfer functions K 1RC (p), K 2RC (p) and K 3RC (p), we find signals at outputs of the first differentiator's three cascades: Likewise, using operator image of voltage on the two-terminal network being tested (5), transfer functions K 1RC (p), K 2RC (p) and K 3RC (p) and their generalized parameters ( 17), ( 19), ( 21), we find signals at outputs of each of the second differentiator's three cascades on completion of the transient process in the measuring circuit: The output signal ratios of the third differentiating cascades ( 27) and ( 24), second differentiating cascades ( 26) and ( 23), first differentiating cascades ( 25) and ( 22), as well as voltages on the two-terminal network and power impulse at time t, respectively, equal the following: Using Equations ( 28)-( 31), one can calculate the values of H-parameters of the measuring circuit:

Numerical Simulation
To check an adequacy for the converter's theoretical model parameters, the modeling of devices' generalized parameters measurement for four-element RLC two-pole networks was executed.The scheme of one of them is represented by R 1 , C 1 , R 2 , L 1 on Figure 3. Modeling is executed by means of the MultiSim program, and calculation -MathCad. .

The Conclusions
It is shown that in the device for definition of generalized parameters of multicomponent passive two-terminal networks with differentiating signals, the measuring system can be used differentiators, consisting of series-connected passive RC-circuits.Simple scheme on stable and adjustment-free components (capacitors and resistors) allows you to measure the generalized parameters of a wide class of objects that have a schema of replacement RC-, RL-and RLC-two-terminal device.Received the analytical relations for calculation of the generalized parameters for the totality of the measured values of input and output voltages of the measuring circuit and the outputs of both channels differentiation.

Figure 1 .
Figure 1.Parameter converter scheme with differentiators

Figure 2 .
Figure 2. Scheme of parameter converter of two-terminal network with differentiating RC-chains u

Figure 3 .
Figure 3.The scheme of four-element RCL two-pole network

Figure 4 .
Figure 4. Diagrams of voltages in converter parameters