# Capacitor transient charging equation of an RC series circuit

• Engineering
• priya.k
In summary, the conversation discusses the attempt to derive the equation Vc=Vss+(Vi-Vss)*e^(-t/RC) using Laplace transform and the confusion regarding the difference between Vss and Vc. The circuit in question includes a DC input voltage and a capacitor, with Vc representing the capacitor voltage and Vss representing the steady state voltage.
priya.k

## Homework Statement

I would like to derive the equation Vc=Vss+(Vi-Vss)*e^(-t/RC)
Vss is the steady state voltage
Vi is the initial capacitor voltage
Vc is the capacitor voltage

## The Attempt at a Solution

I tried to find solution using laplace transform. E=iR+1/c∫idt.
Taking laplace, E/s=I(s)R+1/(cs)*I(s)-q(0+)/(cs)
Put q(0+)=cVc(0+)=cVi
Then taking inverse laplace
i(t)=(E-vi)/R *e^(-t/RC)
Vc=1/c∫idt
=(E-vi)(1-e^(-t/RC))

priya.k said:

## Homework Statement

I would like to derive the equation Vc=Vss+(Vi-Vss)*e^(-t/RC)
Vss is the steady state voltage
Vi is the initial capacitor voltage
Vc is the capacitor voltage

## The Attempt at a Solution

I tried to find solution using laplace transform. E=iR+1/c∫idt.
Taking laplace, E/s=I(s)R+1/(cs)*I(s)-q(0+)/(cs)
Put q(0+)=cVc(0+)=cVi
Then taking inverse laplace
i(t)=(E-vi)/R *e^(-t/RC)
Vc=1/c∫idt
=(E-vi)(1-e^(-t/RC))

Welcome to the PF.

It looks like your two starting equations are fundamentally different. Could you show the circuit along with labels for the variables (what is Vss versus Vc for example?).

Vc=Vss+(Vi-Vss)*e^(-t/RC)

E=iR+1/c∫idt

|-------E volt dc----------|
(+)___^^^^______||_____|(-)
.....(+)|---Vc---|(-)
Input is a dc voltage(polarities shown).
Vc=capacitor voltage
Before applying input, capacitor voltage Vc=Vi
After applying input and reaching steady state, Vc=Vss
But here in this case, steady state capacitor voltage Vss=E

## 1. What is the Capacitor Transient Charging Equation of an RC Series Circuit?

The Capacitor Transient Charging Equation is an equation that describes the behavior of a capacitor as it charges in an RC series circuit. It takes into account the resistance of the circuit (R), the capacitance of the capacitor (C), and the time (t) it takes for the capacitor to charge.

## 2. How is the Capacitor Transient Charging Equation derived?

The Capacitor Transient Charging Equation is derived from Kirchhoff's laws, which state that the sum of currents entering a node in a circuit must equal the sum of currents leaving the node, and the sum of voltages in a closed loop in a circuit must equal zero. These laws are applied to a simple RC series circuit to find a differential equation that describes the charging of a capacitor over time. This equation is then solved to get the Capacitor Transient Charging Equation.

## 3. What is the significance of the Capacitor Transient Charging Equation?

The Capacitor Transient Charging Equation is significant because it helps us understand and predict the behavior of capacitors in circuits. It allows us to calculate the amount of charge and voltage across a capacitor at any given time, which is important in the design and analysis of electronic circuits.

## 4. How is the Capacitor Transient Charging Equation used in practical applications?

The Capacitor Transient Charging Equation is used in a variety of practical applications, such as in electronic circuits, power supplies, and filters. It is also used in the design of timing circuits, signal processing circuits, and other electronic devices. Additionally, it is used in the analysis and troubleshooting of circuits to understand the behavior of capacitors and their effects on the overall circuit performance.

## 5. What are the limitations of the Capacitor Transient Charging Equation?

While the Capacitor Transient Charging Equation is a useful tool for analyzing and predicting the behavior of capacitors, it has some limitations. It assumes ideal conditions, such as zero resistance in the wires and ideal capacitors with no leakage. In practical applications, these conditions may not hold true, which can lead to errors in the equation's predictions. Additionally, the equation only applies to simple RC series circuits and may not be accurate for more complex circuits.

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