How Do You Calculate Parameters and Analyze a Leaky Capacitor Circuit?

[dpd069]

Homework Statement

a. A leaky parallel plate capacitor has an area A. Its dielectric has thickness d, dielectric constant k(Kappa), and resistivity p(rho). Find its resistance R and its capacitance C, and its time constant for internal discharge t(tau).
b. A second leaky capacitor is mostly the same except it has dielectric constant K. Find its capacitance D and resistance Z.

c. Two capacitors of type (a.) and one of type (b.) are connected as shown (see attached picture). Write formulas for their charges and leak currents in terms of node voltages v1 and v2.
d. write Kirchoff's current law for each numbered node.

Homework Equations

The Attempt at a Solution

I believe i have solutions for parts a and b. I am struggling to figure out part c. The layout of the circuit is that of a pi-network and i have no idea how to follow the circuit from the node to write a formula for the capacitors charges and leak currents in terms of the node voltages.

any help/advice would be appreciated.

 

[kamerling]

I think you can assume that the wire at the bottom is at 0 V initially.

For the charges you can use Q = CV. The leakage current is I = V/R.

Where V is the potential difference across the resistor or capacitor.

 

[Moetasim]

I would first like to acknowledge that the leaky capacitor problem is a common challenge in electrical engineering and can have significant impacts on the performance of electronic circuits. I would approach this problem by first understanding the underlying principles of capacitors and their behavior in circuits.

For part a, the resistance R of a leaky capacitor can be calculated using the formula R = d/(k*p*A), where d is the thickness of the dielectric, k is the dielectric constant, p is the resistivity, and A is the area of the capacitor. The capacitance C can be calculated using the formula C = k*(epsilon0*A)/d, where epsilon0 is the permittivity of free space. The time constant for internal discharge can be calculated using the formula t = R*C.

For part b, the capacitance D of the second leaky capacitor can be calculated using the same formula as in part a, but with the dielectric constant K instead of k. The resistance Z can be calculated using the same formula as in part a, but with the new capacitance D.

For part c, the circuit shown appears to be a pi-network, with two capacitors of type (a) and one capacitor of type (b). In order to write formulas for the charges and leak currents in terms of the node voltages v1 and v2, I would use Kirchoff's current law to analyze the circuit and determine the voltage drops across each capacitor. From there, I can use the formulas for capacitance and resistance to calculate the charges and leak currents for each capacitor.

For part d, Kirchoff's current law states that the sum of currents entering and leaving a node must be equal to zero. So for each numbered node in the circuit, I would write an equation that equates the sum of currents entering the node to the sum of currents leaving the node. This would help me to analyze the circuit and determine the unknown node voltages v1 and v2.

Overall, the key to solving this problem is to first understand the principles of capacitors and their behavior in circuits, and then use Kirchoff's laws and relevant equations to analyze and solve the circuit.