A：${\left(x-\frac{1}{2}\right)}^2+{\left(y-1\right)}^2=1$

B：${\left(x-\frac{1}{2}\right)}^2+{\left(y\pm 1\right)}^2=1$

C：${\left(x-\frac{1}{2}\right)}^2+{\left(y\pm \frac{1}{2}\right)}^2=\frac{1}{4}$

D：${\left(x-\frac{1}{2}\right)}^2+{\left(y+1\right)}^2=1$