By virtue of the negative electric parameter concept,i.e.negative lumped resistance,inductance,conductance and capacitance（N-RLGC）,the lumped equivalent models of transmission line systems,including the circuit model,two-portπ-network and T-network,are given.We start from the N-segment-ladder-like equivalent networks composed distributed parameters,and achieve the input impedance in the form of a continued fraction. Utilizing the continued fraction theory,the expressions of input impedance are obtained under three kinds of extreme cases,i.e.the load impedances are equal to zero,infinity and characteristic impedance,respectively.When the number of segment N is limited to infinity,they are transformed to lumped elements.Comparison between the distributed model and lumped model of transmission lines,the expression of tanh yd,which is the key term in the transmission line equations,are obtained by RLGC,furthermore,according to input admittance,admittance matrix and ABCD matrix of transmission lines,the lumped equivalent circuit models,π-networks and T-networks have been given.The models are verified in the frequency and time domain,respectively,showing that the models are accurate and efficient.
Based on Smith-Beljers theory and classical laminate theory, an explicit model is proposed for the ferromagnetic resonance （FMR） frequency shift of a stress-mediumed laminated magnetoelectric structure tuned by an electric field. This model can effectively predict the experimental phenomenon that the FMR frequency increases under a parallel magnetic field and decreases under a perpendicular magnetic field when the electric field ranges from - 10 kV/m to 10 kV/m. Besides, this theory further shows that the FMR frequency increases monotonically as the angle between the direction of the external magnetic field and the outside normal direction of the laminated structure increases, and the frequency will increase as great as 7 GHz. In addition, when the angle reaches a certain critical value, the external electric field fails to tune the FMR frequency. When the angle is above the critical value, the increase of the electric field induces the FMR frequency to increase, and the opposite scenario happens when it is below the critical value. When the angle is 90~ （parallel magnetic field）, the FMR frequency is the most sensitive to the change of the electric field.
