In this paper, we propose a new method to derive a family of regular rate-compatible low-density parity-check(RC-LDPC) convolutional codes from RC-LDPC block codes. In the RC-LDPC convolutional family, each extended sub-matrix of each extended code is obtained by choosing specified elements from two fixed matrices H_(E1)~K and H_(E1)~K, which are derived by modifying the extended matrices H_(E1) and H_(E2) of a systematic RC-LDPC block code. The proposed method which is based on graph extension simplifies the design, and prevent the defects caused by the puncturing method. It can be used to generate both regular and irregular RC-LDPC convolutional codes. All resulted codes in the family are systematic which simplify the encoder structure and have maximum encoding memories which ensure the property. Simulation results show the family collectively offer a steady improvement in performance with code compatibility over binary-input additive white Gaussian noise channel(BI-AWGNC). In this paper, we propose a new method to derive a family of regular rate-compatible low-density parity-check (RC-LDPC) convolutional codes from RC-LDPC block codes. In the RC-LDPC convolutional family, each extended sub- matrix of each extended code is obtained by choosing specified elements from two fixed matrices H_ (E1) ~K and H_ (E1) ~K, which are derived by modifying the extended matrices H_ (E1) and H_ (E2) of a systematic RC The proposed method which is based on graph extension simplifies the design, and prevent the defects caused by the puncturing method. It can be used to generate both regular and irregular RC-LDPC convolutional codes. All induced codes in the family are systematic code that assures the property. The simulation results show the family collectively offer a steady improvement in performance with code compatibility over binary-input additive white Gaussian noise channel (BI-AWGNC).