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A series of termination circuits for unbalanced composite right/left-handed transmission lines (CRLH TLs) are proposed to enable simultaneous excitation of the zeroth-order series and shunt resonances. The proposed termination circuits are very simple and consist merely of a capacitor or inductor in series/parallel with a parallel/series LC tank. Their operational mechanisms are explained in terms of the equivalent circuit models for CRLH TLs at the two zeroth-order resonances, which can also be used to justify the absence of one of the two zeroth-order resonances in conventional open- or short-ended CRLH TLs. Design formulas to estimate the element values of the termination circuits are derived, and some design considerations are discussed as well. Then, the proposed termination circuits are connected to unbalanced CRLH coplanar waveguides (CPWs) to realize two dual-band compact zeroth-order resonator (ZOR) antennas. Unlike conventional dual- or multi-band ZOR antennas formed by open- or short-ended CRLH TLs, in which only one of the zeroth-order resonances can be excited, our proposed ZOR antennas can operate simultaneously at the zeroth-order series and shunt resonances. It is demonstrated that the proposed antennas have similar input impedance responses, slot-like radiation patterns, and peak gains within the two bands. The antenna size is very compact (of about at the upper resonance) since both designs are formed by only two CRLH CPW unit cells.