In general, your construction loan rate will either be the first year of an intermediate ARM like a 2/1, 5/1 or 7/1 ARM, or it will be an adjustable rate unrelated to the permanent financing rate. It you get a 2/1 ARM, for example, the initial rate is fixed 2 years, and then would adjust annually the next 28 years. The construction period would be 12 months, during which time the loan is interest only on the amount that has been disbursed (not the entire loan amount). In the case of a 2/1 ARM (adjustable rate mortgage), after 12 months the house would be done (assuming all goes as planned) and the loan will be fixed one more year at the inital rate, and the loan will begin 29 year amortization from that point. This would be the situation if you got a single close construction loan.
If your loan is only an interim construction loan (not the permanent financing), or if the rate during construction is adjustable, based on some index and margin over that index, with a promise of a note modification (without additional documentation) when the house is done, your construction loan rate will not be the same as your permanent loan rate.