Lecture 28 lc, rlc circuits.

Jan 14, 2014Download as ppt, pdf9 likes5,268 views

1) This lecture discusses energy storage in capacitors and inductors, as well as RC, RL, LC, and RLC circuits. 2) In an LC circuit without resistance, the charge oscillates back and forth between the capacitor and inductor at the characteristic frequency. 3) When resistance is added to an RLC circuit, the charge still oscillates but decays exponentially over time, governed by the damping ratio.

Energy in E and B fields
Energy stored in a capacitor
1
U = CV 2
2

++++++
−−−−−−

E

→ energy of the electric field
Energy stored in a inductor

U =

1
LI 2
2

B
→ energy of the magnetic field

RC circuits (review)
Capacitor is initially charged with Q0. What happens when switch is closed?
++

C −−

( )

+
Current starts at I 0 =

R

Q0 −t
e τ
And then decays: I ( t ) =
RC

I

Q

Q0
RC

t

τ = RC

U

2
Q0
2C

Q0

Current through R

VC
Q
= 0
R RC

t
Charge in capacitor

t
Energy in
capacitor

RL circuit
Capacitor is initially charged with Q0. What happens when switch is closed?

++
C −−

L

• Right after it is closed: I = 0. Then
current builds up.
• Magnetic field is created in inductor
while charge in capacitor is decreasing
• Energy is being transferred form C to L
(from electric to magnetic energy)

As C discharges, current should decrease and stop, but inductor keeps it
going, so at the end the capacitor is charged again but with the positive
charge on the bottom plate.
Then we start all over again…

→ Oscillatory system

Note: No dissipation mechanism (no R) ⇒ total energy is constant

I=0
Q = Q0
++++
E
C
----

I = I0
Q=0

L

⇒

I=0

I = -I0

C

B

⇓

⇑

Q=0

L

C

L

⇐
B

Q = -Q0
---E
C
++++

L

ACT: LC circuit
At t =0, the capacitor in the LC circuit shown has a total charge Q0. At t =
t1, the capacitor is uncharged. What is the sign of Vb-Va, the voltage across
the inductor at time t1?
A. Vb-Va > 0
B. Vb-Va = 0
C. Vb-Va < 0

Q = 0 ⇒ VC = 0
But VC = −VL at all times!

++++
----

C

L

•a
•b

So at t =t1, VL = 0

Compare the energy stored in the capacitor and the inductor at time t1:
A. UC < UL
B. UC = UL
C. UC > UL

Q = 0 ⇒ UC = 0
UL is then maximum.

LC circuit: math
Kirchhoff:

VC +VL = 0

Q
dI
+L
=0
C
dt
Q
d 2Q
+L
=0
2
C
dt
d 2Q
1
+
Q =0
2
LC
dt

C

L

This is the SHM equation!!

d 2Q
+ ω 2Q = 0
dt 2

with

ω=

1

LC

Solution:

Q = Q0 cos ( ωt + ϕ )

Q = Q0 cos ( ωt + ϕ )
VC =

I = −Q0ω sin ( ωt + ϕ )

Q Q0
=
cos ( ωt + ϕ )
C
C

VL = L

2
Q 2 Q0
UC =
=
cos2 ( ωt + ϕ )
2C
2C

VC

VL

UL =

dI
= −LQ0ω 2 cos ( ωt + ϕ )
dt

1
1
LI 2 = LQ02ω 2 sin2 ( ωt + ϕ )
2
2
Q02
=
sin2 ( ωt + ϕ )
2C
Q

2
LQ02ω 2 LQ02 1
=
= 0
2
2 LC 2C

0

UL

0

UC
t

Energy is continuously transformed:
electric ↔ magnetic
Compare to mass and spring (SHM):
kinetic ↔ potential

Real inductor (with resistance)

Then, our LC circuit becomes an RLC circuit…

R
C

L

RLC circuit
R

We’ve seen this already:

L

C

Simple Harmonic Motion + Damping

R=0

R >0

Q

Q

0

0

t

t

RLC circuit: math
R

Kirchhoff:

Q
dI
− IR + L
=0
C
dt
Q dQ
d 2Q
−
R +L
=0
C dt
dt 2

L

C

Q = Q0e

− βt

′
cos(ω0t + ϕ )

 1
R2 
′
ω0 = 
 LC − 4L2 ÷
÷



β=

R
2L

Frequency is a
little lower

Indicates how
fast is the decay

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The document discusses AC circuits using phasors to represent voltages and currents. It introduces the concepts of resistive reactance (R), capacitive reactance (XC), and inductive reactance (XL). R causes voltage to be in phase with current, while XC causes voltage to lag current by 90° and XL causes voltage to lead current by 90°. Together, R, XC and XL determine the impedance (Z) and phase angle (φ) of the circuit. The document uses an example of an L-C circuit to show how to calculate the frequency where XC and XL are equal.

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Thyristor commutation techniques can be natural or forced. Natural commutation occurs in AC circuits when the thyristor turns off as the AC voltage passes through zero. Forced commutation is used in DC circuits using methods like self, resonant pulse, complementary, and impulse commutation. Self commutation uses an underdamped LC circuit to create oscillating current that turns off the thyristor when current reaches zero. Resonant pulse commutation uses an auxiliary thyristor to discharge the capacitor and create reverse voltage across the main thyristor. Complementary commutation uses one thyristor to turn off the other in alternating fashion. Impulse commutation uses an inductor to create oscillating current for

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This document contains solutions to tutorial exercises on electronics circuits involving capacitors and other circuit elements. The first exercise involves determining currents and equivalent capacitance in a circuit with two capacitors. The second involves calculating capacitor voltage over time when a switch changes positions. The third determines capacitor voltage over time for virtually parallel capacitors driven by a voltage source. Subsequent exercises involve determining voltages, currents, and impedances in various RC, RL, and RLC circuits.

Initial and final condition for circuitvishalgohel12195

Initial and final condition for circuit Explain the transient response of a RC circuit As the capacitor stores energy when there is: a transition in a unit step function source, u(t-to) or a voltage or current source is switched into the circuit. Explain the transient response of a RL circuit As the inductor stores energy when there is: a transition in a unit step function source, u(t-to) or a voltage or current source is switched into the circuit. RC Circuit RL Circuit

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This document contains definitions and concepts related to electricity. It defines key terms like potential difference, electric current, volt, ampere, and resistance. It explains that potential difference is the work done to move a charge between two points, and current is the rate of flow of electric charge. It presents Ohm's law, which states that current is directly proportional to voltage. The document also provides examples of calculations using these concepts and definitions.

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Lecture 29 ac circuits. phasors.Albania Energy Association

Chapter7 response of first order rl and rc circuit.pptxssuser073206

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Initial and final condition for circuitvishalgohel12195

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Electric circuit analysis 1 series RC circuit explanation presentationBilalHassan413756

Electricity 10_2.pptxAdityaKumar62443

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Lecture 28 lc, rlc circuits.

1. Lecture 28 LC and RLC circuits.

2. Energy in E and B fields Energy stored in a capacitor 1 U = CV 2 2 ++++++ −−−−−− E → energy of the electric field Energy stored in a inductor U = 1 LI 2 2 B → energy of the magnetic field

3. RC circuits (review) Capacitor is initially charged with Q0. What happens when switch is closed? ++ C −− ( ) + Current starts at I 0 = R Q0 −t e τ And then decays: I ( t ) = RC I Q Q0 RC t τ = RC U 2 Q0 2C Q0 Current through R VC Q = 0 R RC t Charge in capacitor t Energy in capacitor

4. RL circuit Capacitor is initially charged with Q0. What happens when switch is closed? ++ C −− L • Right after it is closed: I = 0. Then current builds up. • Magnetic field is created in inductor while charge in capacitor is decreasing • Energy is being transferred form C to L (from electric to magnetic energy) As C discharges, current should decrease and stop, but inductor keeps it going, so at the end the capacitor is charged again but with the positive charge on the bottom plate. Then we start all over again… → Oscillatory system Note: No dissipation mechanism (no R) ⇒ total energy is constant

5. I=0 Q = Q0 ++++ E C ---- I = I0 Q=0 L ⇒ I=0 I = -I0 C B ⇓ ⇑ Q=0 L C L ⇐ B Q = -Q0 ---E C ++++ L

6. ACT: LC circuit At t =0, the capacitor in the LC circuit shown has a total charge Q0. At t = t1, the capacitor is uncharged. What is the sign of Vb-Va, the voltage across the inductor at time t1? A. Vb-Va > 0 B. Vb-Va = 0 C. Vb-Va < 0 Q = 0 ⇒ VC = 0 But VC = −VL at all times! ++++ ---- C L •a •b So at t =t1, VL = 0 Compare the energy stored in the capacitor and the inductor at time t1: A. UC < UL B. UC = UL C. UC > UL Q = 0 ⇒ UC = 0 UL is then maximum.

7. LC circuit: math Kirchhoff: VC +VL = 0 Q dI +L =0 C dt Q d 2Q +L =0 2 C dt d 2Q 1 + Q =0 2 LC dt C L This is the SHM equation!! d 2Q + ω 2Q = 0 dt 2 with ω= 1 LC Solution: Q = Q0 cos ( ωt + ϕ )

8. Q = Q0 cos ( ωt + ϕ ) VC = I = −Q0ω sin ( ωt + ϕ ) Q Q0 = cos ( ωt + ϕ ) C C VL = L 2 Q 2 Q0 UC = = cos2 ( ωt + ϕ ) 2C 2C VC VL UL = dI = −LQ0ω 2 cos ( ωt + ϕ ) dt 1 1 LI 2 = LQ02ω 2 sin2 ( ωt + ϕ ) 2 2 Q02 = sin2 ( ωt + ϕ ) 2C Q 2 LQ02ω 2 LQ02 1 = = 0 2 2 LC 2C 0 UL 0 UC t Energy is continuously transformed: electric ↔ magnetic Compare to mass and spring (SHM): kinetic ↔ potential

9. Real inductor (with resistance) Then, our LC circuit becomes an RLC circuit… R C L

10. RLC circuit R We’ve seen this already: L C Simple Harmonic Motion + Damping R=0 R >0 Q Q 0 0 t t

11. RLC circuit: math R Kirchhoff: Q dI − IR + L =0 C dt Q dQ d 2Q − R +L =0 C dt dt 2 L C Q = Q0e − βt ′ cos(ω0t + ϕ )  1 R2  ′ ω0 =   LC − 4L2 ÷ ÷   β= R 2L Frequency is a little lower Indicates how fast is the decay

Lecture 28 lc, rlc circuits.

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Lecture 28 lc, rlc circuits.