6.1 RLC Circuit The storage energy Fig... 6.3 the current i and the voltage e across the resistor are in phase.. 6.4 the current in the inductor lags the voltage by 90°.. 6.5 the current
Trang 16.1 RLC Circuit
The storage energy (Fig 6.1)
U = UE + UB = Li2
2
1
c
Cv 2
1
1) Undamped Oscillation
Consider the circuit in Fig 6.1 At t < 0, the switch K is at 1 At t > 0, the switch K is at 2
If the circuit is lossless (there is no resistance)
dt
dU
= Li dt
di + Cvc
dt
dvc
i = -C
dt
dvc ⇒ dt
di = -C 2c
2
dt
v d
2
dt
v d
where
LC
1
=
2) Damped Oscillation
Consider the circuit in Fig 6.2 At t < 0, the switch K is at 1 At t > 0, the switch K is at 2
If a dissipative element R is present
dt
dU
= Li dt
di + Cvc
dt
dvc
2
dt
v d + RC
dt
dvc
Trang 2where )2
L 2
R (
LC 1 −
= ω
6.2 Alternating current circuit
1) Resitive load : (Fig 6.3) the current i and the voltage e across the resistor are in phase
The impedance of the resistor
R I
V z
m
m =
=
Im, Vm : amplitude of i and e, respectively
2) Inductive load : (Fig 6.4) the current in the inductor lags the voltage by 90°
The impedance of the inductor
ωL I
V z
m
m =
=
Im, Vm : amplitude of i and e, respectively
3) Capacitive load : (Fig 6.5) the current in the capacitor leads the voltage by 90°
The impedance of the capacitor
ωC
1
=
= m
m I
V z
Im, Vm : amplitude of i and e, respectively
4) The series RLC circuit (Fig 6.6)
The impedance of the circuit
Trang 3
+
=
=
C
1 -L R
I
V
m m The phase constant
1
-L ω
ω ) tan(ϕ =
C
1
L
ω
ω > : the circuit is more inductive than capacitive, the current i lags the voltage e
C
1
L
ω
ω < : the circuit is more capacitive than inductive, the current i leads the voltage e
C
1
L
ω
ω = : the circuit is in resonance, the current i and the voltage e are in phase
The resonance frequency
LC
1
=
o
ω
6.3 Phasor
The sinusoidal quantity i = Imcos(ωt+ϕ) is represented by a vector of length Im which rotates around the origin with the angular speed ω (Fig 6.7) At time t = 0 this vector is the phasor Im ∠ ϕ of the sinusoidal quantity
6.4 Transformer (Fig 6.8)
2
1 2
1
n
n u
u
=
1
2 2
1
n
n i
i
−
=
Trang 4Problems
6.1) Consider the circuit in Fig P6.1 with e(t) = 12sin(120πt) V When S1 and S2 are open, i leads e by 30°
When S1 is closed and S2 is open, i lags e by 30° When S1 and S2 are closed, i has amplitude 0.5A What
are R, L and C ?
6.2) Consider the circuit in Fig P6.2 with e(t) = 12sin(120πt) V, r = 10Ω Find the value of R such that the
power in R is maximized ?
6.3) Consider the circuit in Fig P6.3 with e(t) = 12sin(120πt) V, L = 0.0265mH Find the value of R such that
the power in R is maximized ?
6.4) Consider the circuits in Fig P6.4 where R = 100Ω, L = 100mH, C = 10µF, e = 100sin(ωt) volts Find iR(t),
iL(t), iC(t), V(t), the storage energy of the capacitor, the storage energy of the inductor, and the total
storage energy in 3 cases :
a) ω = 500 rad/s, b) ω = 1000 rad/s, c) ω = 2000 rad/s
Fig P6.4 6.5) Consider the circuit in Fig P6.5 where e = 100sin(ωt) volts, R = 100Ω, L = 100mH, C = 10µF Determine
i(t), vR(t), vL(t), vC(t), the storage energy of the capacitor UC(t), the storage energy of the inductor UL(t),
the average power of the resistor PR, the average power of the source Pe in 3 cases :
a) ω = 500 rad/s, b) ω = 100 rad/s, c) ω = 1000 rad/s
6.6) Consider the circuit in Fig P6.6 where R = 100Ω, C = 10µF, e = 100sin(1000t) volts The capacitor C has
circular plates of radius a, the space between the two plates is d = 0.1mm
a) Find the voltage v and the current i
Trang 5plates
6.7) A typical “light dimmer” used to dim the stage lights in a theater consist of a variable inductor L connected in series with the light bulb B as shown in the figure P6.7 The power supply is 220 V (rms) at
60 Hz; the light bulb is marked “220 V, 1000W”
a) What maximum inductance L is required if the power in the light bulb is to be varied by a factor of five? Assume that the resistance of the light bulb is independent of its temperature?
b) Could one use a variable resistor instead of an inductor? If so, what maximum resistance is required? Why isn’t this done?
Fig P6.7
Homeworks 6
H6.1 Consider the circuits in Fig H6.1 where e = 100sin(1000t) volts Find iR(t), iL(t), iC(t), V(t), the storage energy of the capacitor, the storage energy of the inductor, and the total storage energy (R in Ω, L in mH,
C in µF)
R 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100
R 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200 200
Trang 6n 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
R 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150
H6.2 Consider the circuits in Fig H6.2 where e = 100sin(1000t) volts Find i(t), the storage energy of the capacitor, the storage energy of the inductor, and the total storage energy (R in Ω, L in mH, C in µF)
R 100 100 100 100 100 100 100 100 200 200 200 200 200 200 200 200
R 100 100 100 100 100 100 100 100 200 200 200 200 200 200 200 200
R 100 100 100 100 100 100 100 100 200 200 200 200 200 200 200 200
R 100 100 100 100 100 100 100 100 200 200 200 200 200 200 200 200