Phương pháp giải:

Công thức tính hệ số công suất: \(\cos \varphi =\frac{R}{Z}=\frac{R}{\sqrt{{{R}^{2}}+{{({{Z}_{L}}-{{Z}_{C}})}^{2}}}}\)

Lời giải chi tiết:

Với \(f={{f}_{1}}=\text{60 Hz}\): \({{\text{Z}}_{\text{L1}}}\text{ = }{{\text{Z}}_{\text{C1}}}\)

Khi \(f={{f}_{2}}=120\text{ Hz = 2}{{f}_{1}}\):

\({{\text{Z}}_{\text{L2}}}\text{ =2 }{{\text{Z}}_{\text{L1}}};\text{     }{{\text{Z}}_{\text{C2}}}\text{ = }\frac{{{\text{Z}}_{\text{C1}}}}{\text{2}}\text{  }\Rightarrow \text{ }{{\text{Z}}_{\text{L2}}}=4{{\text{Z}}_{\text{C2}}}\text{ }\)

\(\Rightarrow \cos \varphi =\frac{R}{\sqrt{{{R}^{2}}+9.{{Z}_{C2}}^{2}}}=\frac{1}{\sqrt{2}}\Leftrightarrow 2{{R}^{2}}={{R}^{2}}+9.{{Z}_{C2}}^{2}\Leftrightarrow R=3{{Z}_{C2}}=\frac{3}{2}{{Z}_{C1}}\)

Khi \(f={{f}_{2}}=90\text{ Hz = }\frac{3}{2}{{f}_{1}}\):

\({{\text{Z}}_{\text{L3}}}\text{ = }\frac{3}{2}{{\text{Z}}_{\text{L1}}}\text{= }\frac{3}{2}{{\text{Z}}_{C1}};\text{     }{{\text{Z}}_{\text{C3}}}\text{ = }\frac{\text{2}{{\text{Z}}_{\text{C1}}}}{3}\text{  }\)

\(\Rightarrow \cos \varphi =\frac{R}{\sqrt{{{R}^{2}}+{{({{Z}_{L3}}-{{Z}_{C3}})}^{2}}}}=\frac{\frac{3}{2}{{Z}_{C1}}}{\sqrt{{{\left( \frac{3}{2}{{Z}_{C1}} \right)}^{2}}+{{\left( \frac{3}{2}{{\text{Z}}_{C1}}-\frac{\text{2}{{\text{Z}}_{\text{C1}}}}{3} \right)}^{2}}}}\approx 0,87\)

Chọn C