Here is a question in page 117 of the cpl aero book
Q:If power is increased during a turn while maintaining the stalling angle of attack?
A:radius of turn will ramain the same and rate of turn will increase.
just want to know this in the explanation "a constant AOA results a constant radius".
as what i am thinking as followed
Centripetal Force(CF)=w×v^2/g×r;
CF=Lift×sin(bank angle);
so can have a equation:(CL:coefficient of lift )
w×v^2/g×r=CL×1/2×ρ×v^2×s×sin(bank angle);
now constant ,in this equation,are weight g CL(as constant aoa) ρ s ;
so variable quantites are just radius and sin（bank angle）as we have the square of velocity at each side so can be deleted.
as for the bank angle is absolutely increased as for keep same AOA with power increased
SO THE RADIUS SHOULD DECREASE.

anyone can help please

A simple way to look at this is to consider the case of a minimum radius turn. That would be a turn which requires the stalling angle of attack. If you want to follow the same radius of turn at a higher speed, you will need more bank [imagine going around a corner on a road on a motor bike]. To fly around a minimum radius turn you would need to use the stalling angle of attack [a fixed angle] but more bank. It really is a purely academic argument since any normal aircraft would not have the power to carry out a level turn at the stalling angle with the increased angle of bank required at high speed.

If you would prefer a more mathematical explanation, I recommend A. C. Kermode's 'Mechanics of Flight' Page 261.

Good afternoon,sir
thank you for your answer. So as the book says the theoretical minimum radius turn is the radius with stalling AOA and 90 degree AOB which is absolutely can not come true. so may i just regard the min radius as a limit which can be approached but can never be reached?