Introduction - Electrical and Computer Engineering

Download Report

Transcript Introduction - Electrical and Computer Engineering

Transient Analysis
DC Steady-State
ELEC 308
Elements of Electrical Engineering
Dr. Ron Hayne
Images Courtesy of Allan Hambley and Prentice-Hall
Transient Analysis
 Scope of study:

Circuits that contain sources, switches, resistances,
inductances, and capacitances
 Transients:

Time-varying currents and voltages resulting from
sudden application of sources, usually due to switching
 Transient Analysis:



Involves using circuit concepts from Chapters 1 & 2
Current-voltage relationships for inductances and
capacitances involve derivatives and integrals
Create circuit equations that are differential equations
ELEC 308
2
First-Order RC or RL Circuits
 First-Order RC Circuits

Contains DC sources, resistances, and a
SINGLE capacitance
 First-Order RL Circuits

Contains DC sources, resistances, and a
SINGLE inductance
 Frequently used in timing applications

Due to time constant
ELEC 308
3
First-Order Circuit Algorithm
1. Apply KCL, KVL, and/or Ohm’s Law to write
the circuit equation.
2. If equation contains integrals, differentiate
each term to produce a PURE differential
equation.
3. Assume a solution of the form K1+K2est.
4. Substitute the solution into the diff. eqn. to
determine the values of K1 and s.
5. Use the initial conditions to determine the
value of K2.
6. Write the final solution.
ELEC 308
4
Discharge of a Capacitance
ELEC 308
5
Discharge of a Capacitance
Writinga KCL equat ion at topnode aft erswitch closed, we have:
dvC t  vC t 
C

0
dt
R
Multiplying by theresist anceR :
dvC t 
RC
 vC t   0
dt
Generalformof solutionis given by
vC t   Ke st
ELEC 308
6
Discharge of a Capacitance
Substitute our generalsolution:
RCKse st  Ke st  0
Solving for s :
1
RCs  1  s 
RC
ELEC 308
7
Discharge of a Capacitance
Use our initialconditionsto solve for K :
 
vC 0   Vi  Ke s 0  K  Vi
Substituing thisin our solutiongives us our completesolution:
vC t   Vi e t RC
ELEC 308
8
Time Constant
 Time interval τ = RC is called the time constant of
the circuit
 After t=5τ, vC(t)≈0
ELEC 308
9
Charging a Capacitance
ELEC 308
10
Charging a Capacitance
WritingKCL equation at node between resistance& capacit ance,
dvC t  vC t   Vs
C

0
dt
R
Rearranging,
RC
dvC t 
 vC t   Vs
dt
Generalformof solutionis given by
vC t   K1  K 2 e st
ELEC 308
11
Charging a Capacitance
Substitute our generalsolutionand solve for s and K1 :
1
s
RC
and
K1  Vs
ELEC 308
12
Charging a Capacitance
Use our initialconditionsto solvefor K 2 :
 
vC 0   0  Vs  K 2e s 0  K 2  Vs
T hus thecompletesolution:
vC t   Vs  Vs e t RC  Vs  Vs e t 
 First term is STEADY-STATE RESPONSE

Or FORCED RESPONSE
 Second term is TRANSIENT RESPONSE
ELEC 308
13
Charging a Capacitance
ELEC 308
14
DC Steady State
 Transient terms in the expressions for current
and voltages in RLC circuits decay to zero
with time
 For DC sources, steady-state currents and
voltages are CONSTANT
 For steady-state conditions with DC sources:


CAPACITANCES behave like OPEN circuits
INDUCTANCES behave like SHORT circuits
ELEC 308
15
Capacitance in DC Steady-State
 Remember current through a capacitance:
dv C t 
iC t   C
dt
 If voltage is constant, current is _________.

 CAPACITANCE behaves just like an ____
circuit
ELEC 308
16
Inductance in DC Steady-State
 Remember voltage across an inductance:
di L t 
v L t   L
dt
 If current is constant, voltage is _________.

 INDUCTANCE behaves just like a ______
circuit
ELEC 308
17
Steady-State DC Analysis
 Find vx and ix for t >> 0
ELEC 308
18
Exercise 4.3
 Find va and ia for t >> 0
ELEC 308
19
RL Transient Analysis
 Find i(t) and v(t)
ELEC 308
20
RL Transient Analysis
 Time interval τ = L/R is called the time constant of
the circuit
 After t=5τ, i (t)≈2
ELEC 308
21
Example 4.3
 Find i(t) and v(t)
ELEC 308
22
Example 4.3
ELEC 308
23
Exercise 4.6
 Find i(t) and v(t)
ELEC 308
24
Summary
 Transient Analysis


First Order RC Circuits
First Order RL Circuits
 DC Steady State
ELEC 308
25