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AC components
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Alterna(ng Currents and Components
Alterna(ng Currents
• Alterna(ng currents flow back and forth. • They are preferred over direct currents due to their ease of genera(on and distribu(on.
Alterna(ng Currents
• DC
• AC
• Watch h@p://www.youtube.com/watch?v=JZjMuIHoBeg
AC Voltages
• If we pass an alterna(ng current through a resistor, we can observe across the resistor an AC voltage whose instantaneous value obeys Ohm’s law.
• This voltage can be wri@en as
• Vp is the peak value, ω is the angular frequency (rad/s), and t is (me. €
v t( ) =Vp sinωt
AC Voltages
• Note that • f is the frequency in Hz. • What about the power?
€
ω = 2πf
AC Voltages
-‐2.5
-‐2
-‐1.5
-‐1
-‐0.5
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5
I
V
P
AC Voltages
• Let us solve for the power
€
V =Vp sinωt
I =Vp
Rsinωt
P =VI =Vp2
Rsin2ωt =
Vp2
2R1− cos 2ωt( )[ ]
Pave =Vp2
2R≡Vrms2
R
Vrms ≡Vp
2
Capacitance
• If we align two conduc(ve plates parallel to each other, separate them with an insulator, we have formed a capacitor.
• Capacitors stored balanced charge. • The variable C is used to denote capacitance and the unit is Farads.
€
Q = CV
Capacitance
Capacitance
Capacitance
• Using the defini(on of current
• For a parallel plate capacitor,
• ε0 is the dielectric constant of the material and is 8.85 X 10-‐12 F/m.
€
I =dQdt
= C dVdt
€
C =εrε0Ad
Capacitance
• εr is the rela(ve dielectric constant
Material Rela*ve Dielectric Constant
Dielectric Strength (kV/cm)
Vacuum 1.00000 ∞
Air 1.00054 8
Paper 3.5 140
Polystyrene 2.6 250
Teflon 2.1 600
Titanium Dioxide 100 60
Capacitance
• For a nice anima(on of ac circuits with resistors or capacitors, please see h@p://www.magnet.fsu.edu/educa(on/tutorials/java/ac/
Capacitance
• The water model for a capacitor is a water filled cylinder with a movable piston or a tank divided by a rubber diaphragm.
Capacitance
• For the water tank analogy, please see the anima(on: h@p://www.wisc-‐online.com/objects/ViewObject.aspx?ID=ACE4803
Capacitance
-‐6
-‐4
-‐2
0
2
4
6
0 0.5 1 1.5 2 2.5
I
V
P
Capacitance
• The overall power dissipa(on over (me is zero.
• Capacitors do not dissipate power, they store energy when charging and return it to the circuit when discharging.
• The energy stored in a capacitor is given by
€
U =12CV 2
RC Circuits
• Imagine the circuit below with the capacitor charged to V0.
C R
RC Circuits
• How will the voltage across the capacitor look like?
• It will start from V0.
• It will decrease un(l all the charge is dissipated and will drop to 0V.
• As current is flowing, an opposite voltage will appear across the resistor, slowing down the discharge.
RC Circuits
• We can write the following equa(ons:
• The voltage is a func(on whose deriva(ve is similar to itself.
• What can this func(on be? • An exponen(al!!!
€
C dVdt
= I = −VR
dVdt
= −1RC
V ≡ −1τV
RC Circuits
• Thus, V(t) turns out to be
• V0 is the ini(al voltage, whereas τ is called the (me constant and is given by τ=RC.
• C determines how much charge is stored, R determines how fast it is dissipated.
• Their product determines the rate of decay.
€
V (t) =V0e− tτ
RC Circuits
time
volt
age
XXX
0.0 2.0 4.0 6.0 8.0 10.0ms
0
2
4
6
8
10
V v(2)
RC Circuits
• This func(on is called the exponen(al decay. • It is very common in many natural processes: – Radioac(ve decay – Newton’s law of cooling – Chemical reac(on rates depending on concentra(on of reactant.
– …
RC Circuits
• Now, let us take the following circuit
+-
VDC:5V
R
C
RC Circuits
• What does the voltage across the capacitor look like?
• We expect the capacitor to charge to the value of the voltage source.
• We expect that it charges fast in the beginning, slowing down as the capacitor voltage increases.
RC Circuits
• We can write the following equa(ons:
€
−VS +VR +VC = 0−VS + IR +VC = 0
I = C dVCdt
VC t( ) =VS 1− e− tτ( )
RC Circuits
time
volt
age
XXX
0.0 2.0 4.0 6.0 8.0 10.0ms
0
2
4
6
8
10
V v(2)
RC Circuits
• What if the input were a pulse? • The capacitor would repeatedly charge and discharge.
RC Circuits
time
volt
age
XXX
0.0 2.0 4.0 6.0 8.0 10.0ms
0.0
2.0
4.0
6.0
8.0
10.0
V v(1) v(2)
Inductance
• When an electric current passes through an inductor, it creates a magne(c field.
• Energy is stored in space around the inductor as magne(c field builds up.
• This opposes any change in current. • It is like momentum or iner(a.
• In our water model, it is like a heavy paddle wheel placed in the current.
Inductance
• We can write the following equa(on for inductance:
€
V = L didt
Inductance
Transformers
• When two or more inductors share a common magne(c core, the resul(ng device is a transformer.
• When an AC voltage is applied to one of the windings of the transformer, it will create a magne(c field propor(onal to the number of turns.
• This magne(c field will be coupled to the next winding, crea(ng an AC voltage depending on its number of turns.
Transformers
• Therefore,
€
V1V2
=N1N2
Transformers
• Since an ideal transformer cannot create or dissipate power,
€
P =V1I1 =V2I2
Transformers
Electrical Quan((es
Quan*ty Variable Unit Unit Symbol
Typical Values
Defining Rela*ons
Important Equa*ons
Charge Q Coulomb C 10-‐18 – 1 Mag of 6.24X10-‐18 charges
I = dq/dt
Current I Ampere A 10-‐6 – 103 1A = 1C/s KCL
Voltage V Volt V 10-‐6 – 106 1V=1N-‐m/C KVL
Power P Wa@ W 10-‐6 – 106 1W = 1J/s P = IV
Energy U Joule J 10-‐15 – 1012 1J = 1N-‐m U = QV
Force F Newton N 1N=1kg-‐m/s2
Time t Second s
Resistance R Ohm Ω 1 – 107 V = IR
Capacitance C Farad F 10-‐15 – 10 Q = CV
Inductance L Henry H 10-‐6 – 1 V = L di/dt