# consider an open switch rc series circuit with r the capacitor is charged with a 24.0 v battery as in figure below, obtain the equation of the charge growth in the capacitor as a function of time. if the capacitor is at a voltage 18.0 v, when the switch is transferred from point a to point b, how, long does it take the capacitor to discharge to 90.0% of its original voltage? 4. 17.5 k? and c-231 ? f. if

### James

Guys, does anyone know the answer?

get consider an open switch rc series circuit with r the capacitor is charged with a 24.0 v battery as in figure below, obtain the equation of the charge growth in the capacitor as a function of time. if the capacitor is at a voltage 18.0 v, when the switch is transferred from point a to point b, how, long does it take the capacitor to discharge to 90.0% of its original voltage? 4. 17.5 k? and c-231 ? f. if from EN Bilgi.

## RC Charging Circuit Tutorial & RC Time Constant

Electronics Tutorial about the RC Charging Circuit and Resistor Capacitor Networks along with the RC Charging Circuit time constant description

## RC Charging Circuit

When a voltage source is applied to an RC circuit, the capacitor, C charges up through the resistance, R

All Electrical or Electronic circuits or systems suffer from some form of “time-delay” between its input and output terminals when either a signal or voltage, continuous, ( DC ) or alternating ( AC ), is applied to it.

This delay is generally known as the circuits **time delay** or **Time Constant** which represents the time response of the circuit when an input step voltage or signal is applied. The resultant time constant of any electronic circuit or system will mainly depend upon the reactive components either capacitive or inductive connected to it. Time constant has units of, **Tau – **τ

When an increasing DC voltage is applied to a discharged Capacitor, the capacitor draws what is called a “charging current” and “charges up”. When this voltage is reduced, the capacitor begins to discharge in the opposite direction. Because capacitors can store electrical energy they act in many ways like small batteries, storing or releasing the energy on their plates as required.

The electrical charge stored on the plates of the capacitor is given as: Q = CV. This charging (storage) and discharging (release) of a capacitors energy is never instant but takes a certain amount of time to occur with the time taken for the capacitor to charge or discharge to within a certain percentage of its maximum supply value being known as its **Time Constant** ( τ ).

If a resistor is connected in series with the capacitor forming an RC circuit, the capacitor will charge up gradually through the resistor until the voltage across it reaches that of the supply voltage. The time required for the capacitor to be fully charge is equivalent to about **5 time constants** or 5T. Thus, the transient response or a series RC circuit is equivalent to 5 time constants.

This transient response time T, is measured in terms of τ = R x C, in seconds, where R is the value of the resistor in ohms and C is the value of the capacitor in Farads. This then forms the basis of an RC charging circuit were 5T can also be thought of as “5 x RC”.

## RC Charging Circuit

The figure below shows a capacitor, ( C ) in series with a resistor, ( R ) forming a **RC Charging Circuit** connected across a DC battery supply ( Vs ) via a mechanical switch. at time zero, when the switch is first closed, the capacitor gradually charges up through the resistor until the voltage across it reaches the supply voltage of the battery. The manner in which the capacitor charges up is shown below.

### RC Charging Circuit

Let us assume above, that the capacitor, C is fully “discharged” and the switch (S) is fully open. These are the initial conditions of the circuit, then t = 0, i = 0 and q = 0. When the switch is closed the time begins at t = 0 and current begins to flow into the capacitor via the resistor.

Since the initial voltage across the capacitor is zero, ( Vc = 0 ) at t = 0 the capacitor appears to be a short circuit to the external circuit and the maximum current flows through the circuit restricted only by the resistor R. Then by using Kirchhoff’s voltage law (KVL), the voltage drops around the circuit are given as:

The current now flowing around the circuit is called the **Charging Current** and is found by using Ohms law as: i = Vs/R.

### RC Charging Circuit Curves

The capacitor (C), charges up at a rate shown by the graph. The rise in the RC charging curve is much steeper at the beginning because the charging rate is fastest at the start of charge but soon tapers off exponentially as the capacitor takes on additional charge at a slower rate.

As the capacitor charges up, the potential difference across its plates begins to increase with the actual time taken for the charge on the capacitor to reach 63% of its maximum possible fully charged voltage, in our curve 0.63Vs, being known as one full Time Constant, ( T ).

This 0.63Vs voltage point is given the abbreviation of 1T, (one time constant).

The capacitor continues charging up and the voltage difference between Vs and Vc reduces, so too does the circuit current, i. Then at its final condition greater than five time constants ( 5T ) when the capacitor is said to be fully charged, t = ∞, i = 0, q = Q = CV. At infinity the charging current finally diminishes to zero and the capacitor acts like an open circuit with the supply voltage value entirely across the capacitor as Vc = Vs.

So mathematically we can say that the time required for a capacitor to charge up to one time constant, ( 1T ) is given as:

### RC Time Constant, Tau

This RC time constant only specifies a rate of charge where, R is in Ω and C in Farads.

Since voltage V is related to charge on a capacitor given by the equation, Vc = Q/C, the voltage across the capacitor ( Vc ) at any instant in time during the charging period is given as:

Source : **www.electronics-tutorials.ws**

## Lab 4

## Lab 4 - Charge and Discharge of a Capacitor

Lab 4 - Charge and Discharge of a Capacitor INTRODUCTION

Capacitors are devices that can store electric charge and energy. Capacitors have several uses, such as filters in DC power supplies and as energy storage banks for pulsed lasers. Capacitors pass AC current, but not DC current, so they are used to block the DC component of a signal so that the AC component can be measured. Plasma physics makes use of the energy storing ability of capacitors. In plasma physics short pulses of energy at extremely high voltages and currents are frequently needed. A capacitor can be slowly charged to the necessary voltage and then discharged quickly to provide the energy needed. It is even possible to charge several capacitors to a certain voltage and then discharge them in such a way as to get more voltage (but not more energy) out of the system than was put in. This experiment features an RC circuit, which is one of the simplest circuits that uses a capacitor. You will study this circuit and ways to change its effective capacitance by combining capacitors in series and parallel arrangements.

## DISCUSSION OF PRINCIPLES

A capacitor consists of two conductors separated by a small distance. When the conductors are connected to a charging device (for example, a battery), charge is transferred from one conductor to the other until the difference in potential between the conductors due to their equal but opposite charge becomes equal to the potential difference between the terminals of the charging device. The amount of charge stored on either conductor is directly proportional to the voltage, and the constant of proportionality is known as the capacitance. This is written algebraically as

( 1 ) Q = CΔV.

The charge C is measured in units of coulomb (C), the voltage

ΔV

in volts (V), and the capacitance C in units of farads (F). Capacitors are physical devices; capacitance is a property of devices.

### Charging and Discharging

In a simple RC circuit, a resistor and a capacitor are connected in series with a battery and a switch. See Fig. 1.

**Figure 1**

**:**A simple RC circuit

When the switch is in position 1 as shown in Fig. 1(a), charge on the conductors builds to a maximum value after some time. When the switch is thrown to position 2 as in Fig. 1(b), the battery is no longer part of the circuit and, therefore, the charge on the capacitor cannot be replenished. As a result the capacitor discharges through the resistor. If we wish to examine the charging and discharging of the capacitor, we are interested in what happens immediately after the switch is moved to position 1 or position 2, not the later behavior of the circuit in its steady state. For the circuit shown in Fig. 1(a), Kirchhoff's loop equation can be written as

( 2 ) ΔV − Q C − R dQ dt = 0.

The solution to is Eq. (2) is

( 3 ) Q = Qf 1 − e(−t / RC) where Qf

represents the final charge on the capacitor that accumulates after an infinite length of time, R is the circuit resistance, and C is the capacitance of the capacitor. From this expression you can see that charge builds up exponentially during the charging process. See Fig. 2(a). When the switch is moved to position 2, for the circuit shown in Fig. 1(b), Kirchhoff's loop equation is now given by

( 4 ) Q C − R dQ dt = 0.

The solution to Eq. (4) is

( 5 ) Q = Q0e(−t / RC) where Q0

represents the initial charge on the capacitor at the beginning of the discharge, i.e., at

t = 0.

You can see from this expression that the charge decays exponentially when the capacitor discharges, and that it takes an infinite amount of time to fully discharge. See Fig. 2(b).

**Figure 2**

**:**Change versus time graphs

### Time Constant τ

The product RC

(having units of time) has a special significance; it is called the time constant of the circuit. The time constant is the amount of time required for the charge on a charging capacitor to rise to 63% of its final value. In other words, when

t = RC, ( 6 ) Q = Qf 1 − e−1 and ( 7 ) 1 − e−1 = 0.632.

Another way to describe the time constant is to say that it is the number of seconds required for the charge on a discharging capacitor to fall to 36.8%

(e−1 = 0.368)

of its initial value. We can use the definition

(I = dQ/dt)

of current through the resistor and Eq. (3) and Eq. (5) to get an expression for the current during the charging and discharging processes.

( 8 )

charging: I = +I0e−t/RC

( 9 )

discharging: I = −I0e−t/RC

where I0 = ΔV0 R

in Eq. (8) and Eq. (9) is the maximum current in the circuit at time t = 0. Then the potential difference across the resistor will be given by the following.

( 10 )

charging: ΔV = + ΔVfe−t/RC

( 11 )

discharging: ΔV = − ΔV0e−t/RC

Note that during the discharging process the current will flow through the resistor in the opposite direction. Hence I and

## RC Circuits – University Physics Volume 2

## 66 RC CIRCUITS

### Learning Objectives

By the end of the section, you will be able to:

Describe the charging process of a capacitor

Describe the discharging process of a capacitor

List some applications of RC circuits

When you use a flash camera, it takes a few seconds to charge the capacitor that powers the flash. The light flash discharges the capacitor in a tiny fraction of a second. Why does charging take longer than discharging? This question and several other phenomena that involve charging and discharging capacitors are discussed in this module.

### Circuits with Resistance and Capacitance

An RC circuit is a circuit containing resistance and capacitance. As presented in Capacitance, the capacitor is an electrical component that stores electric charge, storing energy in an electric field.

(Figure)(a) shows a simple RC circuit that employs a dc (direct current) voltage source , a resistor R, a capacitor C, and a two-position switch. The circuit allows the capacitor to be charged or discharged, depending on the position of the switch. When the switch is moved to position A, the capacitor charges, resulting in the circuit in part (b). When the switch is moved to position B, the capacitor discharges through the resistor.

(a) An RC circuit with a two-pole switch that can be used to charge and discharge a capacitor. (b) When the switch is moved to position A, the circuit reduces to a simple series connection of the voltage source, the resistor, the capacitor, and the switch. (c) When the switch is moved to position B, the circuit reduces to a simple series connection of the resistor, the capacitor, and the switch. The voltage source is removed from the circuit.

### Charging a Capacitor

We can use Kirchhoff’s loop rule to understand the charging of the capacitor. This results in the equation This equation can be used to model the charge as a function of time as the capacitor charges. Capacitance is defined as so the voltage across the capacitor is . Using Ohm’s law, the potential drop across the resistor is , and the current is defined as

This differential equation can be integrated to find an equation for the charge on the capacitor as a function of time.

Let , then The result is

Simplifying results in an equation for the charge on the charging capacitor as a function of time:

A graph of the charge on the capacitor versus time is shown in (Figure)(a). First note that as time approaches infinity, the exponential goes to zero, so the charge approaches the maximum charge and has units of coulombs. The units of RC are seconds, units of time. This quantity is known as the time constant:

At time , the charge is equal to of the maximum charge . Notice that the time rate change of the charge is the slope at a point of the charge versus time plot. The slope of the graph is large at time and approaches zero as time increases.

As the charge on the capacitor increases, the current through the resistor decreases, as shown in (Figure)(b). The current through the resistor can be found by taking the time derivative of the charge.

At time the current through the resistor is . As time approaches infinity, the current approaches zero. At time , the current through the resistor is

(a) Charge on the capacitor versus time as the capacitor charges. (b) Current through the resistor versus time. (c) Voltage difference across the capacitor. (d) Voltage difference across the resistor.

(Figure)(c) and (Figure)(d) show the voltage differences across the capacitor and the resistor, respectively. As the charge on the capacitor increases, the current decreases, as does the voltage difference across the resistor The voltage difference across the capacitor increases as

### Discharging a Capacitor

When the switch in (Figure)(a) is moved to position B, the circuit reduces to the circuit in part (c), and the charged capacitor is allowed to discharge through the resistor. A graph of the charge on the capacitor as a function of time is shown in (Figure)(a). Using Kirchhoff’s loop rule to analyze the circuit as the capacitor discharges results in the equation , which simplifies to . Using the definition of current and integrating the loop equation yields an equation for the charge on the capacitor as a function of time:

Here, Q is the initial charge on the capacitor and is the time constant of the circuit. As shown in the graph, the charge decreases exponentially from the initial charge, approaching zero as time approaches infinity.

The current as a function of time can be found by taking the time derivative of the charge:

The negative sign shows that the current flows in the opposite direction of the current found when the capacitor is charging. (Figure)(b) shows an example of a plot of charge versus time and current versus time. A plot of the voltage difference across the capacitor and the voltage difference across the resistor as a function of time are shown in parts (c) and (d) of the figure. Note that the magnitudes of the charge, current, and voltage all decrease exponentially, approaching zero as time increases.

Guys, does anyone know the answer?