In polar coordinates, what is the impedance of a network composed of a 100-picofarad capacitor in parallel with a 4000-ohm resistor, at 500 KHz?

To determine the impedance of a network composed of a capacitor and a resistor in parallel, it's essential to first calculate the impedance of each component and then find their combined effect.

The impedance of the resistor (R) is straightforward and is simply 4000 ohms. The impedance of the capacitor (Xc) can be calculated using the formula:

[

X_c = \frac{1}{2 \pi f C}

]

Where:

• ( f ) is the frequency (500 KHz = 500,000 Hz)

• ( C ) is the capacitance (100 picofarads = 100 x 10^{-12} F)

Substituting in these values:

[

X_c = \frac{1}{2 \pi (500,000)(100 \times 10^{-12})

]

Calculating this gives:

[

X_c ≈ -3183 , \text{ohms}

]

This negative value indicates a capacitive reactance because capacitors lead the current.

Next, we need to find the total impedance of the parallel combination of the resistor and the capacitor. The formula for calculating the total impedance ( Z ) of two components in parallel is given by: