Question

An alternating source with a variable frequency, a capacitor with capacitance $$C$$, and a resistor with resistance $$R$$ are connected in series. Above figure, gives the impedance $$Z$$ of the circuit versus the driving angular frequency $$\omega _{d}$$; the curve reaches an asymptote of $$500\Omega$$, and the horizontal scale is set by $$\omega _{ds}=300rad/s$$. The figure also gives the reactance $$X_{C}$$ for the capacitor versus $$\omega _{d}$$. What are (a) $$R$$ and (b) $$C$$?

Answer 1

(a) The circuit has a resistor and a capacitor (but no inductor). Since the capacitive reactance decreases with frequency, then the asymptotic value of $$Z$$ must be the resistance: $$R = 500 Ω$$.
(b) We describe three methods here (each using information from different points on the graph):
Method 1: At $$\omega _{d}=50rad/s$$, we have $$Z ≈ 700 Ω$$, which gives $$C=\left ( \omega _{d}\sqrt{Z^{2}-R^{2}} \right )^{-1}=41\mu F$$.
Method 2: At $$\omega _{d}=50rad/s$$, we have $$X_{C}\approx 500\Omega$$, which gives $$C=\left ( \omega _{d}X_{C} \right )^{-1}=40\mu F$$.
Method 3: At $$\omega _{d}=250rad/s$$, we have $$X_{C}\approx 100\Omega$$, which gives $$C=\left ( \omega _{d}X_{C} \right )^{-1}=40\mu F$$.