natural and step responses of rlc circuitsxuanqi-net.com/circuit/chapter8.pdfnatural and step...

1 Electric Circuits

Natural and Step Responses of RLC Circuits

Qi Xuan Zhejiang University of Technology

Nov 2015

Structure

•  Introduc)on to the Natural Response of a Parallel RLC Circuit

•  The Forms of the Natural Response of a Parallel RLC Circuit

•  The Step Response of a Parallel RLC Circuit •  The Natural and Step Response of a Series RLC Circuit

•  A Circuit with Two Integra)ng Amplifiers

2 Electric Circuits

An Igni)on Circuit

Electric Circuits 3

The rapidly changing current in the primary winding induces via magne)c coupling (mutual inductance) a very high voltage in the secondary winding.

Introduc)on to the Natural Response of a Parallel RLC Circuit

Electric Circuits 4

The ini)al voltage on the capacitor, V0 represents the ini)al energy stored in the capacitor. The ini)al current through the inductor, I0 represents the ini)al energy stored in the inductor.

Second-ordered circuit

General Solu.on for the Second-‐Order Differen.al Equa.on

Electric Circuits 5

Two roots:

Some Useful Nota.ons

Electric Circuits 6

Deno)ng the two corresponding solu)ons v1 and v2, respec)vely, we can show that their sum also is a solu)on.

(Proof from Eq. 8.10 to 8.12)

Denoting

We have

Only when s1 ≠ s2

Unit: radians per second (rad/s)

Electric Circuits 7

Over damped: ω02 < α2 , both roots will be real and dis)nct

Under damped: ω02 > α2, two roots will be complex and

conjungates Cri.cal damped: ω0

2 = α2, two roots will be real and equal

Example #1 a)  Find the roots of the characteris)c equa)on that governs the

transient behavior of the voltage if R = 200 Ω, L=50 mH,and C = 0.2 µF.

b)  Will the response be overdamped, underdamped, or cri)cally damped?

c)  Repeat (a) and (b) for R = 312.5 Ω. d)  What value of R causes the response to be cri)cally damped?

Electric Circuits 8

Solu.on for Example #1

Electric Circuits 9

a) For the given values of R, L, and C, we have

= 5000 rad/s = 20000 rad/s

b) In this case, α2 > ω02, thus the voltage response is overdamped.

c) Increase of R will result in decrease of α, thus the voltage response will be from overdamped to underdamped.

Electric Circuits 10

d) For cri)cal damping, α2 = ω02, so

The Forms of the Natural Response of a parallel RLC Circuit

•  Calculate the two roots s1 and s2, based on the given R, L and C;

•  Determine whether the response is over-‐, under-‐, or cri)cally damped;

•  Find the unknown coefficients, such as A1 and A2


Overdamped Voltage Response


= V0


We summarize the process for finding the overdamped response, v(t), as follows: 1.  Find the roots of the characteris)c equa)on, s1 and

s2, using the values of R, L, and C. 2.  Find v(0+) and dv(0+)/dt using circuit analysis. 3.  Find the values of A1 and A2 by solving the following

Equa)ons simultaneously: v(0+) = A1 + A2 dv(0+)/dt = iC(0+)/C = s1A1 + s2A2 4.  Subs)tute the values for s1, s2, A1, and A2 into Eq.

8.18 to determine the expression for v(t) for t > 0.

Underdamped Voltage Response


B1 B2 Real!


Because α determines how quickly the oscilla)ons subside, it is also referred to as the damping factor (or damping coefficient).

The oscillatory behavior is possible because of the two types of energy-‐ storage elements in the circuit: the inductor and the capacitor.

Rè∞，αè0

The oscilla)on at ωd = ω0 is sustained! No Dissipa)on on R.

Overdamped or Underdamped? •  When specifying the desired response of a second order

system, you may want to reach the final value in the shortest .me possible, and you may not be concerned with small oscilla.ons about that final value. If so, you would design the system components to achieve an underdamped response.

•  On the other hand, you may be concerned that the response not exceed its final value, perhaps to ensure that components are not damaged. In such a case, you would design the system components to achieve an overdamped response, and you would have to accept a rela.vely slow rise to the final value.


Cri.cally Damped Voltage Response


In the case of a repeated root, the solu)on involves a simple exponen)al term plus the product of a linear and an exponen)al term.

You will rarely encounter cri)cally damped systems in prac)ce, largely because ω0 must equal α exactly. Both of these quan))es depend on circuit parameters, and in a real circuit it is very difficult to choose component values that sa)sfy an exact equality rela)onship.

Example #2

In the given circuit, V0 = 0 and I0 = -12.25 mA. a)  Calculate the roots of the characteris)c equa)on. b)  Calculate v and dv/dt at t = 0. c)  Calculate the voltage response for t ≥ 0. d)  Plot v(t) versus t for the )me interval 0 ≤ t ≤ 11 ms.




Underdamped

a)

b)


= 0

= 98,000 V/s

Step Response of a Parallel RLC Circuit


The Indirect Approach


The Direct Approach


= I

= 0

Example #3 The ini)al energy stored in the given circuit is zero. At t = 0, a dc current source of 24 mA is applied to the circuit. The value of the resistor is 400 Ω. a)  What is the ini)al value of iL? b)  What is the ini)al value of diL/dt? c)  What are the roots of the characteris)c equa)on? d)  What is the numerical expression for iL(t) when t > 0?


Solu.on for Example #3 a)  No energy is stored in the circuit prior to the applica)on of

the dc current source, so the ini.al current in the inductor is zero. The inductor prohibits an instantaneous change in inductor current; therefore iL(0) = 0 immediately a`er the switch has been opened.

b)  The ini.al voltage on the capacitor is zero before the switch has been opened; therefore it will be zero immediately a`er. Now, because v = LdiL/dt, we have


c)  From the circuit elements, we obtain

d) 


Overdamped

If = I = 24 mA

The Natural Response of a Series RLC Circuit


Differen)ate

Rearrange

The Step Response of a Series RLC Circuit


Compare Eq. 8.66 and Eq. 8.41: the procedure for finding vC parallels that for finding iL

Example #4

No energy is stored in the 100 mH inductor or the 0.4 µF capacitor when the switch in the given circuit is closed. Find vC(t) for t > 0.




Calculate the two roots:

The roots are complex, so the voltage response is underdamped. Thus, we have

No energy is stored in the circuit ini)ally, so both vC(0) and dvC(0+)/dt are zero, then

A Circuit with Two Integra)ng Amplifier


Two integra)ng amplifiers connected in cascade: the output signal of the first amplifier is the input signal for the second amplifier.

For the first integrator:

For the second integrator:

Second-‐order

Example #5 No energy is stored in the given circuit when the input voltage vg jumps instantaneously from 0 to 25 mV. a)  Derive the expression for vo(t) for 0 < t < tsat. b)  How long is it before the circuit saturates?




Integrate

a)

When t = 3 s, the second amplifier saturates, at that time we get:

= -3 V > -5 V

The first amplifier doesn’t saturates!

b)

Summary

•  characteris)c equa)on •  Second-‐order differen)al equa)on •  Overdamped, underdamped, and cri)cal damped

•  Natural Step responses


natural and step responses of rlc circuitsxuanqi-net.com/circuit/chapter8.pdfnatural and step...

Documents