Optimization of RFID Tags Coil's System Stability under Delayed Electromagnetic Interferences

This article discusses the very crucial subject of RFID TAG's stability. RFID equivalent circuits of a label can be represented as Parallel circuits of Capacitance (Cpl), Resistance (Rpl), and Inductance (Lpc). We define V(t) as the voltage that develops on the RFID label therefore making dV(t)/dt the voltage-time derivative. Due to electromagnetic interference, there are different time delays with respect to RFID label voltages and voltage time derivatives. We define V1(t) as V(t) and V2(t) as dV(t)/dt. The delayed voltage and voltage derivative are V1(t-τ1) and V2(t-τ2) respectively (τ***1=τ2). The RFID equivalent circuit can be represented as a delayed differential equation that depends on variable parameters and delays. The investigation of RFID's differential equation is based on bifurcation theory [1], which is the study of possible changes in the structure of the orbits of a delayed differential equation as a function of variable parameters. This article first illustrates certain observations and analyzes local bifurcations of an appropriate arbitrary scalar delayed differential equation [2]. RFID label stability analysis is done under different time delays with respect to label voltage and voltage derivative. Additional analysis of the bifurcations of RFID's delayed differential equation on the circle. Bifurcation behavior of specific delayed differential equations can be condensed into bifurcation diagrams. This serves to optimize dimensional parameters analysis of RFID TAGs under electromagnetic interferences to get ideal performances.


Introduction
This article discusses a very critical and useful subject of passive RFID TAGs system stability analysis under electromagnetic interferences. RFID TAG system has two main variables-TAG-voltage and TAG-voltage derivative with respect to time which may be subject to delay as a result of electromagnetic interferences. We define τ1 as time delay respect to TAGʹs voltage and τ2 as time delay respect to TAGʹs voltage derivative. RFID Equivalent circuits of a Label can be represented as parallel circuits of Capacitance (Cpl), Resistance (Rpl), and Inductance (Lpc). Our RFID TAG system delay differential and delay differential model can be utilized for analysis of the dynamics of delay differential equations. Incorporation of a time delay is often necessary during some stage. It is often difficult to analytically study models with delay dependent parameters, even if only a single discrete delay is present. Practical guidelines exist that combine graphical information with analytical work to effectively study the local stability of models involving delay dependent parameters. The stability of a given steady state is determined simply through the use of the graphs of a function of τ1, τ2, which can be expressed distinctly and thus can be easily depicted by Matlab and other popular software--we need only look at one such function and locate the zeros. This function often has only two zeros, providing thresholds for stability switches. As time delay increases, the stability fluctuates. We emphasize the local stability aspects of certain models with delaydependent parameters. In addition there is a general geometric criterion that, theoretically speaking, may be applied to models that include many delays or even distributed delays. The simplest case is that of a first order characteristic equation which provides more user-friendly geometric and analytic criteria for stability switches. The analytical criteria provided for the first and second order cases can be used to obtain insightful analytical statements and can be helpful for conducting simulations.

RFID Equivalent Circuit and Representation of Delay Differential Equations
RFID TAG can be represented as a parallel Equivalent Circuit of Capacitor and Resistor in Parallel. For example, see NXP/PHILIPS ICODE IC Parallel equivalent circuit and simplified complete equivalent circuit of the label (L1 is the antenna inductance) [6].
This gives us the differential equation of RFID TAG sys which describes the evolution of the sys in continuous time. V = V(t). Now I define the following Variable settings: The RFIDʹs coil calculation inductance is Due to electromagnetic interferences we get RFID TAGʹs voltage and voltage derivative with delays τ1 and τ2 respectively V1(t)→ V1(t-τ1) ; V2(t)→ V2(t-τ1). We consider no delay effect on dV1/dt and dV2/dt. The RFID TAGʹs differential equations under the effects of electromagnetic interferences (we consider electromagnetic interferences (delay terms) influence only RFID TAG voltage V1(t) and the voltage derivative V2(t) with respect to time. There is no influence on dV1(t)/dt and dV2(t)/dt) : To find the Equilibrium points (fixed points) of the RFID TAG system is by We get two equations and the only fixed point is: We choose the above expressions for our 1 2 ( ), ( ) V t V t as small displacement [v1 v2] from the system's fixed points at time t=0.
The speeds of flow toward or away from the selected fixed point for RFID TAG system voltage and voltage derivative respect to time are (14) e e e e and the time derivative of the above equations: First we take the RFID TAGʹs voltage (V1) differential equation: Second we take the RFID TAGʹs voltage (V2) differential equation: then we have saddle fixed point otherwise it is an unstable node (both eigenvalues are positive). We define (22) which gives us two delayed differential equations adding arbitrarily small increments of exponential form

Nc
In the equilibrium fixed point The small Jacobian increments of our RFID TAG system We have three stability analysis cases: 1 Just as in all of the above stability analysis cases, we need to identify characteristic equations. We study the occurrence of any possible stability switching resulting from the increase in value of the time delay  for the general characteristic equation ( , ) The expression for The expression for 3. RFID Tag System Second Order The first case we analyze involves a delay in RFID Label voltage with no delay in voltage time derivative [4] [5].
The expression for The expression for Our RFID system second order characteristic equation is : And its roots are given by Therefore the following holds true: Which jointly with 4 The second case we analyze involves no delay in RFID Label voltage but does have a delay in voltage time derivative [5].
The expression for The expression for Our RFID system second order characteristic equation: Then And like in our previous case analysis: 2 2 ( , ) Therefore ( , ) 0 And its roots are given by Therefore the following holds: Which, along with 4 Defines the maps The expression for The expression for Our RFID system second order characteristic equation: And much like our previous case analysis : (99) Hence ( , ) 0 F    implies that: The expressions for U and V can be derived easily [BK] : and we get the expression: We use different parameter terminology in this case: Unless absolutely necessary, the designation of the variation arguments 1 1 ( , , antenna parametrs) R C will subsequently be omitted from P, Q, aj, cj. The coefficients aj, cj are continuous, and differentiable functions of their arguments, and direct substitution shows that Indeed, in the limit Has at most a finite number of zeros. Indeed, this is a bicubic polynomial in  (second degree in 2  ). In addition, since the coefficients in P and Q are real, we have , , antenna parametrs C R and delay , Reλ may, at the crossing ,change its sign from (-) to (+), i.e. from a stable focus to an unstable one, or vice versa.
This feature may be further assessed by examining the sign of the partial derivatives with respect to 1 1 , C R and antenna parameters.
For the first case 1 2 = ; 0     we get the following always and additional for R   ; There are two options: first, it is always true that Not exist and always negative for any RFID TAG overall parameters values.
When writing into RFID characteristic equation ,  must satisfy the following :       as the solution of , , w,g,B0,A0,A , , ., and that on the surface Upon separating this into real and imaginary parts, with We define U and V: We choose our specific parameter as time delay x = τ. ; Differentiating with respect to  , we get Result : And its roots are given by Therefore the following holds:      Re ( ) ( ) The stability switch occur only on those delay values ( ) that fit the equation: ( 1 ) ( 1 )