W8 - Capacitors, Time Constant, Reactance PDF (EEEE-105)

EEEE-105 Freshman Practicum Capacitors (Labs 5 & 6) DC Analysis: Capacitor as a Storage Element >> Charging/Discharging/Time Constant AC Analysis: Capacitors are Reactors >> Electrical Filters Recap – Resistors & Diodes Resistors (Lab 2) I is linear in...

EEEE-105 Freshman Practicum Capacitors (Labs 5 & 6) DC Analysis: Capacitor as a Storage Element >> Charging/Discharging/Time Constant AC Analysis: Capacitors are Reactors >> Electrical Filters Recap – Resistors & Diodes Resistors (Lab 2) I is linear in V slope = 1/R I = V/R V can be dc or ac I-V relationship of the Resistor Diodes (Lab 4) I is exponential in V 𝑉 𝐼~𝑒 𝑛𝑉𝑇 Capacitors (Lab 5, 6) 𝒊 is the derivative of v 𝒅𝒗 𝒊 𝒕 = 𝑪 𝒅𝒕 I-V relationship of the Diode  Capacitor Structure/Operation Capacitors are made of two conductive sheets or plates sandwiching a thin dielectric layer 𝐴𝑟𝑒𝑎: 𝐴 − Plates can be circular or rectangular − Capacitors can be cylindrical, spherical, or other shapes A dielectric material is an electrical insulator that can 𝑑 be polarized by an electric field − A material that can support charge separation − Can be solid (mica, glass, plastic, oxides), liquid gel, or gas (e.g., dry air) Conductors sandwiching an insulator give rise to an 𝐶 = 𝜀𝐴/𝑑 interesting òperation when E-field applies to it 𝜀: 𝑃𝑒𝑟𝑚𝑖𝑡𝑡𝑖𝑣𝑖𝑡𝑦 − When a +ve DC voltage is placed across a capacitor, the positive (+ve) charge accumulates on the top plate creating an electric field in the process. The dielectric causes an opposite (-ve) charge to flow towards the bottom plate − This opposite plate charge attraction makes it appear as though current is flowing through the capacitor − This continues till the voltage across C reaches Vs Capacitors act as a storage medium. They store energy in the form of an electrical charge  Capacitors capacitors vary in value from 1 pF to about 1 mF (could be more for supercapacitors) Farad* (F) = coulombs/volt * Named after the British physicist Michael Faraday An electrolytic capacitor is a type of capacitor that uses an electrolyte to achieve a larger capacitance. Electrolyte is a liquid or gel or solid polymer containing a high concentration of ions (commonly made of tantalum or aluminum). Electrolytic capacitors are typically polarized, which means that the voltage on the +ve terminal is greater than the voltage on the -ve terminal. Supercapacitors are a special subtype of electrolytic capacitors made of double-layer electrolytic capacitors, with capacitances of hundreds of farads Used in applications requiring rapid charge/discharge cycles (buses, trains, cranes, elevators) that require burst-mode power delivery. Capacitor as a Storage Element Capacitor has the ability or “capacity” to store energy on its plates in the form of an electrical charge. Current is the time rate of change of charges 𝑞 = 𝐶𝑣 𝒅𝒒 𝒊 𝒕 = 𝒅𝒕 𝒅𝒗 𝒊 𝒕 =𝑪 𝒅𝒕 1 𝑣 𝑡 = න 𝑖 𝑡 𝑑𝑡.. 𝐴𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝐸𝑓𝑓𝑒𝑐𝑡 𝐶 1 2 𝐸 = 𝐶𝑉.. 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 2 q: Charge in Coulombs C: Capacitance in Coulombs/Volt or Farads DC Analysis: Capacitor Equation > Capacitor is Charging 𝑡=0 𝑣𝑅 (𝑡) + 𝑣𝐶 (𝑡) = 𝑉𝑠.. Kirchhoff's Voltage Law (KVL) 𝑅. 𝑖(𝑡)+𝑣𝐶 (𝑡) = 𝑉𝑠 (DC).. Solve for 𝑣𝐶 𝜏𝑣𝐶′ (𝑡) + 𝑣𝐶 (𝑡) = 𝑉𝑠  = 𝑅𝐶 (𝑡𝑖𝑚𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) This is a first-order differential equation. You’ll learn how to solve DEs in MATH-231 (2nd year) 𝑤𝑖𝑡ℎ 𝑣𝐶 𝑡 = 0 𝑓𝑜𝑟 𝑡 < 0, 𝑤𝑒 ℎ𝑎𝑣𝑒 Charging 𝑣𝐶 (𝑡) = 𝑉𝑠 (1 − 𝑒 −𝑡/𝜏 ) Equation 𝑣𝐶 (0) = 0 Switch 𝑣𝐶 () = 0.63 𝑉𝑠  is the time required for the closes capacitor to reach 63% of max 𝑣𝐶 (∞) = 𝑉𝑠 ➔ Unlike the voltage divider, voltages across R and C are not settled instantly Effect of Time Constant  = 𝑅𝐶 on Charging Rate vc(t) Examples: R = 10k Ohm C = 1mF →  = 0.01 sec R = 10k Ohm 0.63 C = 4mF →  = 0.04 sec Larger capacitors take more time to charge as expected but hold more 1 energy 𝐸 = 2 𝐶𝑉 2  = 0.01  = 0.04 t DC Analysis: Capacitor Equation > Capacitor is Discharging A similar expression can be derived when the capacitor is discharging through a resistor R C is now the power source (battery) and R is the load Assuming the voltage across the capacitor to have an initial value of 𝑉𝑖 , then 𝑣𝐶 (𝑡) = 𝑉𝑖 𝑒 −𝑡/𝜏  = 𝑅𝐶 (𝑡𝑖𝑚𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) Capacitor 𝑣𝐶 𝑡 = 𝑉𝑖 𝑎𝑡 𝑡 = 0 Here,  is the time required for the capacitor to discharge 37% of its charge Capacitor Response to a Square Wave Switching between charging and discharging can be realized using a square wave Charging/discharging can be visualized on the VSQ Oscilloscope or Pspice Input Signal Voltage Across C  𝑣𝐶 (𝑡) = 𝑉𝑆𝑄 𝑒 −𝑡/𝑅𝐶 𝑣𝐶 (𝑡) = 𝑉𝑆𝑄 (1 − 𝑒 −𝑡/𝑅𝐶 )  AC Analysis: Capacitive Reactance 𝑑𝑣 𝑿𝑪 Recall that 𝑖 𝑡 = 𝐶 𝑑𝑡 Capacitors are reactors – they react to change in signals (frequency) 1 Reactance (Impedance): 𝑋𝐶 = 𝑗2𝜋𝑓𝐶 𝑋𝐶 is inversely proportional to f f = 0, XC = , i = 0 for dc voltage (capacitor exhibits high impedance at low frequencies) o Capacitors don’t allow direct current to flow (act as open circuit to dc) 01 1 f = , XC = 0, i is high for high frequency |Xc| Current (capacitor exhibits low impedance at high (Reactance) frequencies) o Capacitors act as a short circuit at high frequency Capacitors as reactors can be used to construct electrical filters Allow high frequencies to pass through and block low frequencies f Capacitive Reactance vs Frequency Application – AC to DC Convertor Using RC Circuit as A smoothing Device through charging- discharging effect RC circuit acts as a smoothing device (or a low-pass filter) in the full-wave rectifier Smoothing can be accomplished using a proper value of time constant RC Capacitors Have a Small Resistance Most real capacitors have a small inherent resistance called Effective Serial Resistance (ESR) ESR can be significant in electrolytic capacitors o ESR for ceramic capacitors is between 0.01 and 0.1 ohms. o ESR for aluminum electrolytic capacitors can reach 30 ohms Scope & FUN GEN can be used to measure ESR ESR Can be measured at high frequency since XC ~ 0 C At high frequency, XC ~ 0, VC ~ 0, VCH2 ~ VESR, VR = VCH1 – VCH2 I = VR/R ESR = VCH2/I Additional Reading Introduction to Capacitors, Capacitance and Charge https://www.electronics-tutorials.ws/capacitor/cap_1.html Wikipedia (Extensive Coverage) https://en.wikipedia.org/wiki/Capacitor Electrolytic Capacitors https://en.wikipedia.org/wiki/Electrolytic_capacitor http://www.capacitorguide.com/electrolytic-capacitor/