Chapter 6: RC Circuit PDF

# Chapter 6: RC Circuit ## 3rd Part: RC Circuit ### Level: **Key Concepts of Electrical Circuits (Recap)** * **Dipoles:** Any electrical device containing two terminals is called a dipole. * **Electric Current Direction (Convention):** Conceptually, electric current flows from the positive termin...

# Chapter 6: RC Circuit ## 3rd Part: RC Circuit ### Level: **Key Concepts of Electrical Circuits (Recap)** * **Dipoles:** Any electrical device containing two terminals is called a dipole. * **Electric Current Direction (Convention):** Conceptually, electric current flows from the positive terminal to the negative terminal outside of the power source. * **Nature of Electric Current:** In metals, current is caused by the movement of electrons from the negative terminal to the positive terminal. * **Generator-Receiver Convention:** * **Generator:** Current and voltage have the same direction. * **Receiver:** Current and voltage have opposite directions. * **Ohm's Law:** The current *I* flowing through a conductor is proportional to the voltage *U* applied across its terminals. * *U* = *R* *I* * *R* is the resistance (Ω) * **Oscilloscope:** A measuring device that displays the voltage across the terminals of an electrical component. * **Capacitor:** * **Definition:** A capacitor is an electrical component composed of two conductive plates, separated by an insulating material. * **Symbol:** ``` ----- | | | | | | ----- ``` * **Charging a Capacitor:** * **Activity 1:** 1. Connect the following components in series: a capacitor, LED, a switch, and a DC voltage source. 2. Close the switch, the LED will light up for a short time, then, after some time, the LED will be turned off. * **Explanation:** The capacitor charges through the movement of electrons from one conductor plate to the other, creating a voltage difference. This flow of electrons constitutes the current, leading to the LED turning on. Once the capacitor is charged, the current flow ceases, causing the LED to be turned off. * **Charge:** The charge on a capacitor is represented by "q". * *q* = *C* *Uc* * *C* is the capacitance of the capacitor, measured in Farads (F), and *Uc* is the voltage across its terminals. * **Activity 2:** Build a circuit using a capacitor and a current source. Observe the voltage (Uc) across the capacitor over time. The graph shows a linear relationship between Uc and t. * **Relation between Charge, Current, and Voltage:** * *I* = d*q*/d*t* * *I* is the current, *q* is the charge, and *t* is time. * *I* = *C* *d*Uc*/d*t* * *C* is the capacitance, *Uc* is the voltage, and *t* is time. * **Association of Capacitors:** * **Series:** * Consider two capacitors, C1 and C2, in series. * If the same current passes through both capacitors, the charges of each capacitor are equal: *q* = *q1* = *q2*. * The voltage across each capacitor adds up to the total voltage, *UAB* = *UA* + *UB*. * 1/ *C* = 1/ *C1* + 1/ *C2* * **Parallel:** * Consider two capacitors, C1 and C2, in parallel. * The total current entering the parallel circuit is equal to the sum of the currents in each branch, *i* = *i1* + *i2*. * *C* = *C1* + *C2* * **Generalization:** The capacitance **C** of an array of capacitors in series or parallel is given by the following: * **Series:** * 1/ *C* = 1/ *C1* + 1/ *C2* + ... + 1/ *CN* * **Parallel:** * *C* = *C1* + *C2* + ... + *CN* * The parallel connection allows you to obtain larger capacitance values. ## Response of an RC Circuit to a Step in Voltage * **RC Circuit:** A circuit consisting of a resistor (R) and a capacitor (C) connected in series. * **Step Input:** A sudden change in the voltage applied across the circuit. ### Response to a Rising Step 1. **Differential Equation:** When the switch is closed and the voltage across the capacitor is *Uc*, the differential equation is as follows: * *RC* *d*Uc*/d*t* + *Uc* = *E* 2. **Solution:** The voltage across the capacitor *Uc* is * *Uc (t)* = *E* (1 - *e*^-*t/τ*) * *τ* is the time constant. 3. **Charge and Current:** * *q(t)* = *CE* (1 - *e*^-*t/τ*) * *i(t)* = (*CE*/ *τ*) *e*^-*t/τ* ### Response to a Falling Step 1. **Differential Equation:** When the switch is closed, the voltage across the capacitor *Uc* is: * *RC* *d*Uc*/d*t* + *Uc* = 0 2. **Solution:** The voltage across the capacitor *Uc* is: * *Uc* (t) = *E* *e*^-*t/τ* 3. **Charge and Current:** * *q(t)* = *CE* *e*^-*t/τ* * *i(t)* = -(*CE*/ *τ*) *e*^-*t/τ* ## Time Constant * **Definition:** The time constant *τ* of an RC circuit quantifies how quickly the capacitor charges or discharges. It is determined by: * *τ* = *RC* * **Determining the Time Constant:** * **Charging:** The time it takes for the capacitor to charge to approximately 63% of the final voltage. * **Discharging:** The time it takes for the voltage to drop to approximately 37% of its initial value. * **Energy Storage:** A capacitor can store energy during charging. * The power delivered to the capacitor is given by: *P* = *Uc** *i*. * The energy stored in the capacitor is: *E* = (1/2) *C* *Uc*² This document explains the basic concepts related to RC circuits, including the definition and operation of a capacitor, the charging and discharging processes, the time constant, and the energy stored in a capacitor.