capacitive :
L
diode
C
LC-circuits reactance inductive reactance Minimal Radio: runs without a battery!
Physics 122B: Lecture 25, Pg 1
ap ec R
Q
LC- circuit, Oscillations, analysis:
Q(t ) d 2Q(t ) + =0 LC dt 2
differential equation
Plug an equation into this differential equation:
dQ I =- dt
Q(t ) = Q0 cos t ,
=
1 2
1 LC
1 LC
I (t ) = Q0 sin t = I max sin t
f =
Physics 122B: Lecture 25, Pg 2
ap ec R
LC Circuit Plumber Analogy
Valve Constriction
P2
Flywheel Rubber Diaphragm
P3
Rubber Diaphragm= Capacitor
Flywheel = Inductor Pressure = Potential Water Flow = Current
Physics 122B: Lecture 25, Pg 3
AC Electricity
Voltage from a generator:
Sinusoidally varying EMF can be described using a "phasor diagram." The "phasor" is a vector of length 0 that rotates counterclockwise around the origin with angular velocity , so that the angle it makes with the horizontal axis at any time is t. The horizontal component of the phasor gives the EMF.
Physics 122B: Lecture 25, Pg 4
Clicker Question 1
The instantaneous value of the EMF e represented by this phasor is: (a) Decreasing; (b) Increasing; (c) Constant; (d) It is not possible to tell without knowing t.
Physics 122B: Lecture 25, Pg 5
AC Resistor Response
Current : iR =
vR R
Voltage : vR = iR R
Kirchoff's loop law: Vsoruce + VR = E - vR = 0
E (t ) = E0 cos t = vR
iR = vR E0 = cos t = I R cos t R R
vR and iR are parallel in the phasor diagram.
Voltage and current through a resistor are "in phase".
Physics 122B: Lecture 25, Pg 6
Example: Resistor Voltages
In the circuit shown, find (a) the peak voltage across each resistor, and (b) the instantaneous resistor voltages at t=20 ms.
Req = R1 + R2 = (5 ) + (15 ) = 20
iR = I R cos t =
vR E0 cos t (100 V) cos 2 (60 Hz)t = = = (5.0 A) cos 2 (60 Hz)t Req Req (20 )
25 V for R1 =5 VR = I 0 R = 75 V for R 2 =15
iR (t = 20 ms) = (5.0 A) cos 2 (60 Hz)(2.0 10-2 s) = 1.545 A 7.7 V for R1 =5 vR = iR R = 23.2 V for R 2 =15
Physics 122B: Lecture 25, Pg 7
Capacitor AC Circuits (1)
The capacitor voltage applied to the capacitor: vC = 0 cos t = VC cos t The charge on the capacitor is: q = CvC = CVC cos t dq d iC = = ( CVC cos t ) = -CVC sin t iC = CVC cos(t + / 2) dt dt The AC current through a capacitor leads the capacitor voltage by 900.
Physics 122B: Lecture 25, Pg 8
Capacitor AC Circuits (2)
vC = VC cos t
iC = - C VC sin t
= C VC cos t + 2
The AC current through a capacitor leads the capacitor voltage by 900.
Physics 122B: Lecture 25, Pg 9
Capacitive Reactance
XC 1 1 = C 2 f C
is called capacitive reactance. is a form of resistance. has the same units: Ohm (W).
XC is quite different from "ohmic
resistance" because: it shifts the phase of the current by 900. it is frequency dependent. the peak voltage VC, the peak current IC, are related to each other like in Ohm's Law:
VC XC = IC
Physics 122B: Lecture 25, Pg 10
Example: Capacitive Reactance
What is the capacitive reactance of a 0.10 mF capacitor at a 100 Hz audio frequency and at a 100 MHz FM radio frequency?
1 X C (100 Hz) = = 15,900 -1 -7 2 (100 s )(1.0 10 F) X C (100 MHz) = 1 2 (1.0 10 s )(1.0 10 F)
8 -1 -7
= 0.0159
Physics 122B: Lecture 25, Pg 11
Inductor AC Circuits (1)
The voltage applied to the inductor: vL = = 0 cos t = VL cos t With the changing current: vL = L
diL dt
diL = iL =
vL V cos t dt = L dt L L
V V iL = L cos t dt = L sin t L L
VL cos t - 2 L
The AC current through an inductor lags the inductor voltage by 900.
Physics 122B: Lecture 25, Pg 12
Inductor Circuits AC (2)
vL = VL cos t VL iL = sin t L VC = cos t - 2 L
The AC current through a inductor lags the inductor voltage by 900.
Physics 122B: Lecture 25, Pg 13
Inductive Reactance
f X L L = L 2
is called inductive reactance. is a form of resistance. has the same units: Ohm (W).
XL
is quite different from "ohmic resistance" because: it lags the phase of the current by 900 behind the voltage. it is proportional to the frequency. the peak voltage VL, the peak current IC, are related to each other like in Ohm's Law:
VL XL = IL
Physics 122B: Lecture 25, Pg 14
RLC Circuit:
The instantaneous current through all three elements is the same: i = i = i = i
R L C
The sum of all three instantaneous voltages must equal:
= vR + vL + vC
Physics 122B: Lecture 25, Pg 15
Analyzing an RLC Circuit
Draw the current vector I at some arbitrary angle. All elements of the circuit will have this current.
VR is in phase with
the current. Draw the capacitor voltage VC 900 behind the current and the inductor voltage VL 900 ahead . Make sure all phasor lengths scale properly.
Draw the emf as the vector sum of VR ,VC and VL. VC and VL are in opposite directions.
The phasors VR and (VL VC) form the sides of a right triangle, with the magnitude of the EMF given by:
02 = VR2 + (VL - VC )2
Physics 122B: Lecture 25, Pg 16
RLC Circuit:
02 = VR2 + (VL - VC )2
The total "resistance" is called impedance and is defined as:
Z=
0
I
02 = VR2 + (VL - VC )2 = R 2 + ( X Z2 =
I2 I2
VL - VC = arctan V R
L - XC )
2
The phase angle (angle between i and
in the phasor diagram) is:
V -V tan = L C VR
Physics 122B: Lecture 25, Pg 17
RC-Filter Circuit
The RC-filter is a special case of the RLC-circuit. The impedance (resistance) of the RCcombination is:
low-pass filtered
1 Z = R +X = R + 2 2 C
2 2 C 2
Voltage over the resistor: Voltage over the capacitor:
VR =
R
0
R 2 + 1 / 2C 2
VC =
/ 2C 2 0
R 2 + 1 / 2C 2
Physics 122B: Lecture 25, Pg 18
Filters and Transmission
An RC filter is a circuit that passes a signal with attenuation of some frequencies. The transmissio...