In this brief, we present two inductorless realizations of Chua's circuit in an exact form while using only two active elements, namely a current feedback operational amplifier (CFOA) for the precise implementation of the Chua circuit equations, and an op-amp for the realization of the active nonlinear resistor. First, we analyze the widely used Wien bridge based modified Chua circuit and apply a variable transformation and reveal the difference of its mathematical model from the original Chua circuit in a precise form. Second, we present an exact inductorless implementation of the Chua circuit employing the Wien bridge structure, using two active elements only. Third, we introduce another simple yet exact inductorless implementation of the Chua circuit with a featured property where the state variable corresponding to the inductor current of the original Chua circuit can be accessed as a capacitor voltage. Chaotic attractors obtained by experimental realizations match well with the result of numerical simulations, demonstrating the success of the proposed designs.