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Frequency response of a capacitor - Eclecticon
frequency range 0 to 3,000Hz 5 Also record and plot the RMS current vs frequency 6 Fixing the frequency at 200Hz, find the RMS current for the full range of capacitances You should find that current is proportional to capacitance 7 For a 4 7mF capacitor, keep the frequency at 3,000Hz and switch to a square wave, and
Effect of various capacitors on frequency response - RCET
drops the signal voltage across the capacitor and reduces the circuit gain As signal frequencies decrease, capacitor reactance increase and gain continues to fall, reducing the output voltage 2 Effect of Bypass capacitors: At lower frequencies, bypass capacitor C E is not a short So emitter is not at ac ground X c in parallel with R E
How to Test a Capacitor - Learning about Electronics
A capacitor is a device that stores electric charge (memory devices) A capacitor is a device that stores energy Capacitors are easy to fabricate in small sizes (µm), use in chips How to combine capacitance: capacitors in parallel adds like resistors in series: € C tot=C 1+C 2=?C i € E= Q2 2C = CV2 2 L2: Resistors and Capacitors
Circuits in the frequency domain - Stanford University
Recall that, in a capacitor, i = Cdv dt:What happens if the voltage across the capacitor happens to be sinusoidal with amplitude V and frequency f, that is, with v(t) = Vsin(2?ft+ ?)? We would then have i(t) = C dv dt = 2?fCVcos(2?ft+ ?) = 2?fCV {z } I sin 2?ft+ ?+ ? 2 : That is, the current is also sinusoidal with the same frequency
Frequency Response - Docceptornet
Jan 10, 2021 · I-V signal in the circuit have the frequency of ?(rad/s) •That is, any voltage and current can be represented as •Suppose the voltage across a cap is v C (t) = V 0 cos(?t) then, •We can see the higher the frequency is, the larger the current flow in a capacitor is 7 t) 0 ZT V) t ZT I () C C t C dt qZt 00)+ 90°in phase ×?Cin magnitude
Searches related to frequency across a capacitor filetype:pdf
and a capacitor together as a voltage divider We will put the resistor in first, so we can connect the capacitor to ground By applying Kirchhoff’s Laws to this circuit, we can see that: 1 The same current flows through both the resistor and the capacitor, and 2 The sum of the voltage drops across the two
113325_3Lec2.pdf K.K. Gan1
Lecture 2: Capacitors and Inductors
Capacitance:
●Capacitance (C) is defined as the ratio of charge (Q) to voltage (V) on an object: ◆ C = Q/V = Coulombs/Volt = Farad ☞ Capacitance of an object depends on geometry and its dielectric constant. ◆ Symbol(s) for capacitors: ◆ A capacitor is a device that stores electric charge (memory devices). ◆ A capacitor is a device that stores energy ◆ Capacitors are easy to fabricate in small sizes (µm), use in chips
●How to combine capacitance: ◆ capacitors in parallel adds like resistors in series: € C tot =C 1 +C 2 =C i ∑ € E= Q 2 2C = CV 2 2L2: Resistors and Capacitors
Total capacitance is more than individual capacitance!◆ capacitors in series add like resistors in parallel: ●Energy and Power in Capacitors ◆ How much energy can a "typical" capacitor store? ■ Pick a 4 µF Cap (it would read 4 mF) rated at 3 kV ■ This is the same as dropping a 2 kg weight (about 4 pounds) 1 meter ◆ How much power is dissipated in a capacitor?
■dV/dt must be finite otherwise we source (or sink) an infinite amount of power!K.K. Gan2
€ 1 C tot = 1 C 1 + 1 C 2 = 1 C i ∑ €Power=
dE dt = d dt CV 2 2 " # $ % & ' P=CV dV dt € E= 1 2 CV 2 = 1 24×10
-6 •3000 2 =18 JL2: Resistors and Capacitors
Total capacitance is less than individual capacitance!THIS WOULD BE UNPHYSICAL.■ the voltage across a capacitor cannot change instantaneously ☞ a useful fact when trying to guess the transient (short term) behavior of a circuit ■ the voltage across a resistor can change instantaneously ❑ the power dissipated in a resistor does not depend on dV/dt: P = I
2R or V
2/R ◆ Why do capacitors come in such small values? ■ Example: Calculate the size of a 1 Farad parallel capacitor with air between the plates. ☞square plate of 6.5 miles per side❑ breakthroughs in capacitor technologies (driven by laptop/cell phone industries)☞10 µF capacitor in 0402 package (1.0 mm × 0.5 mm × 0.5 mm) costs only 3 pennies ◆ How small can we make capacitors?
■A wire near a ground plane has C ~ 0.1 pf = 10 -13 F.K.K. Gan3
€ C= kε 0 A d k=dielectric constant (=1 for air) ε 0 =8.85×10 -12 N -1 m -2 d=distance between plates (assumed 1 mm)A=area of plates = 1.1×10
8 m 2 !!!L2: Resistors and Capacitors
"Stray capacitance" slow down signal or induces cross talk by capacitive coupling!K.K. Gan4
◆ Some words to the wise on capacitors and their labeling.■ Typical capacitors are multiples of micro Farads (10
-6 F) or picoFarads (10 -12F).☞Whenever you see mF it almost always is micro, not milli F and never mega F.☞picoFarad (10
-12F) is sometimes written as pf and pronounced puff.■ no single convention for labeling capacitors❑Many manufacturers have their own labeling scheme (See Horowitz and Hill lab manual).● Resistors and Capacitors ◆ Examine voltage and current vs. time for a circuit with one R and one C. ■ Assume that at t < 0 all voltages are zero, V
R = V C = 0. ■ At t ≥ 0 the switch is closed and the battery (V 0 ) is connected. ■ Apply Kirchhoff's voltage rule: ■ Solve the differential equation by differentiating both sides of above equation: € V 0 =V R +V C =IR+ Q C =R dQ dt + Q C € dV 0 dt = 1 C dQ dt +R d 2 Q dt 2 0= I C +R dI dt dI dt =- I RCL2: Resistors and Capacitors
Why we care about the DC behavior?Want to know how a circuit (characterizeas R & C) responses to a pulse!
K.K. Gan5
■ This is just an exponential decay equation with time constant RC (sec). ■ The current as a function of time through the resistor and capacitor is:
◆ What's V R (t)? ■ By Ohm's law: ■ At t = 0 all the voltage appears across the resistor, V R (0) = V 0 . ■ At t = ∞, V R (∞) = 0. € V R (t)=I R ⋅R =I 0 R e -t/RC =V 0 e -t/RC €I(t)=I
0 e -t/RCL2: Resistors and Capacitors
K.K. Gan6
◆ What's V C (t)? ■ Easiest way to answer is to use the fact that V 0 = V R + V C is valid for all t. ☞At t = 0 all the voltage appears across the resistor so V C (0) = 0. ☞At t = ∞, V C (∞) = V 0. ◆ Suppose we wait until I = 0 and then short out the battery. ■ Solving the exponential equation yields,
■ We can find V C using V = Q/C, ■ Finally we can find the voltage across the resistor using V R = -V C, € V C =V 0 -V R V C =V 0 1-e -t/RC () € 0=V R +V C V R =-V C R dQ dt =- Q C dQ dt =- Q RC €Q(t)=Q
0 e -t/RC € V C (t)=V 0 e -t/RC € V R (t)=-V 0 e -t/RCL2: Resistors and Capacitors
K.K. Gan7
◆Suppose V C (t) = V 0 sinωt instead of DC ☞What happens to V C and I C ? ☞ current in capacitor varies like a sine wave too, but 90 0 out of phase with voltage. €Q(t)=CV(t)
=CV 0 sinωt I C =dQ/dt =ωCV 0 cosωt =ωCV 0 sin(ωt+π/2)L2: Resistors and Capacitors
K.K. Gan8
■We can write an equation that looks like Ohm's law by defining V*: ☞the relationship between the voltage and current in C looks like:
☞ 1/ωC can be identified as a kind of resistance, capacitive reactance: X C ≡ 1/ωC (Ohms) ❑X C= 0 for ω = ∞ ☞high frequencies: a capacitor looks like a short circuit ☞"bypass capacitor" on power supply "short circuit" ripple (noise) to ground ❑X
C= ∞ for ω = 0 ☞low frequencies: a capacitor looks like an open circuit (high resistance).
☞look at AC output of a circuit via a capacitor (use AC coupling on scope) Inductance:● Define inductance by: V = LdI/dt ◆Unit: Henry ◆Symbol: ◆Inductors are usually made from a coil of wire ■ tend to be bulky and are hard to fabricate in small sizes (µm), seldom used in chips. ◆Two inductors next to each other (transformer) can step up or down a voltage ■no change in the frequency of the voltage ■provide isolation from the rest of the circuit
€ V*=V 0 sin(ωt+π/2) € V*=I C /ωC =I C R*L2: Resistors and Capacitors
L2: Resistors and Capacitors
10 Gb/s VCSEL (Laser) Array Driver
●4-channel test chip (65 nm CMOS)2 mm240 µm●Capacitance and inductance onlong transmission lines degrade signal
K.K. Gan10
●How much energy is stored in an inductor? ☞ ●How much power is dissipated in an inductor? ☞❑dI/dt must be finite as we can't source (or sink) an infinite amount of power in an inductor!
☞the current across an inductor cannot change instantaneously. € dE=VdQ I= dQ dt dE=VIdt V=L dI dt dE=LIdIE=LIdI
0 I ∫ E= 1 2 LI 2 €Power=
dE dt = d dt LI 2 2 " # $ % & ' P=LI dI dtL2: Resistors and Capacitors
THIS WOULD BE UNPHYSICAL.
K.K. Gan11
◆Two inductors in series: ■ Apply Kirchhoff's Laws, ☞ Inductors in series add like resistors in series. € V=V 1 +V 2 =L 1 dI dt +L 2 dI dt ≡L tot dI dt L tot =L 1 +L 2 =L i ∑L2: Resistors and Capacitors
The total inductance is greater than the individual inductances. ◆Two inductors in parallel: ■ Since the inductors are in parallel, V 1 = V 2 = V ■ The total current in the circuit is ☞If we have more than 2 inductors in parallel, they combine like: ❑Inductors in parallel add like resistors in parallel.K.K. Gan12
€ I=I 1 +I 2 dI dt = dI 1 dt + dI 2 dt = V L 1 + V L 2 ≡ V L tot 1 L tot = 1 L 1 + 1 L 2 L tot = L 1 L 2 L 1 +L 2 € 1 L tot = 1 L i ∑L2: Resistors and Capacitors
The total inductance is less than the individual inductances.K.K. Gan13
●Resistors and Inductors ◆ Examine voltage and current versus time for a circuit with one R and one L. ■ Assume that at t < 0 all voltages are zero, V
R = V L = 0. ■ At t ≥ 0 the switch is closed and the battery (V 0 ) is connected. ■ Like the capacitor case, apply Kirchhoff's voltage rule:■ Solving the differential equation, assuming at t = 0, I = 0: ☞ This is just an exponential decay equation with time constant L/R (seconds). ◆ What's V
R (t)? ■ By Ohm's law V R = I RR at any time:
■ At t = 0, none of the voltage appears across the resistor, V R (0) = 0. ■ At t = ∞, V R (∞) = V 0 . € V 0 =V R +V L =IR+L dI dt € I(t)= V 0 R 1-e -tR/L () € € V R =I(t)R=V 0 1-e -tR/L ()L2: Resistors and Capacitors
K.K. Gan14
◆ What's V L (t)? ■ Easiest way to answer is to use the fact that V 0 = V R + V L is valid for all t. ■ At t = 0, all the voltage appears across the inductor so V L (0) = V 0 . ■ At t = ∞, V L (∞) = 0.Pick L/R = 1 ms
€ V L =V 0 -V R V L (t)=V 0 e -tR/LL2: Resistors and Capacitors
K.K. Gan15
◆ Suppose V L (t) = V 0 sinωt instead of DC, what happens to V L and I L ? ☞ The current in an inductor varies like a sine wave too, but 90 0out of phase with the voltage. ■We can write an equation that looks like Ohm's law by defining V*: V
* = V 0 sin(ωt - π/2) ☞ V * = I LωL = I
L R * ❑ ωL can be identified as a kind of resistance, inductive reactance: X L ≡ ωL (Ohms) ❍ X L= 0 if ω = 0 ☞ low frequencies: an inductor looks like a short circuit (low resistance). ❍ X
L= ∞if ω = ∞ ☞ high frequencies: an inductor looks like an open circuit. ☞ connect a circuit to the power supply via an inductor to filter out ripple (noise).
! ! V=L dI L dt I L = 1 L Vdt 0 t " =# V 0 $L cos$t I L (t)= V 0 $L sin($t#%/2)L2: Resistors and Capacitors
Choke on power cord
K.K. Gan16
● Some things to remember about R, L, and C's. ◆ For DC circuits, after many time constants (L/R or RC): ☞ Inductor acts like a wire (0 Ω). ☞Capacitor acts like an open circuit (∞ Ω).
◆ For circuits where the voltage changes very rapidly or transient behavior:☞ Capacitor acts like a wire (0 Ω). ☞ Inductor acts like an open circuit (∞ Ω). ◆ Example, RLC circuit with DC supply: ■At t = 0, voltages on R, C are zero and V
L = V 0. ■At t = ∞, voltages on R, L are zero and V C = V 0.