Lossy Capacitor in Series Circuit

A lossy capacitor in a series circuit consists of a resistance (R) and a capacitance (C). When a voltage is applied to the circuit, the capacitor charges up and the current flows through the resistor. However, due to the resistance of the capacitor, some of the energy is lost in the form of heat.

Equivalent Parallel Model

To find the equivalent parallel model of the series circuit, we need to calculate the impedance of the series circuit at a given frequency. The impedance of a series circuit is given by:

Z = R + jXc

Where R is the resistance, Xc is the capacitive reactance given by:

Xc = 1/2πfC

Where f is the frequency and C is the capacitance.

Calculating Impedance

At 50kHz, the impedance of the series circuit is:

Z = 20 + j(1/(2π(50kHz)25pF))
Z = 20 - j127.3Ω

Converting to Equivalent Parallel Model

To convert the series circuit to an equivalent parallel model, we need to calculate the resistance and capacitance values of the parallel circuit. The resistance is given by:

Rp = R/(1 + R*Xc)

And the capacitance is given by:

Cp = C/(1 + R*Xc)

Substituting the values we get:

Rp = 20/(1 + 20*(-127.3))
Rp = 0.156Ω

Cp = 25pF/(1 + 20*(-127.3))
Cp = 3.156nF

Therefore, the equivalent parallel model at 50kHz will have values of R = 0.156Ω and C = 3.156nF.