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CE164 Foundations of Electronics

创建日期：2024-06-18 15:39:33

所属分类：电气工程electric engineering

标签： CE164

Foundations of Electronics

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Foundations of Electronics – II
Lecture 2 (Week 17)
RC Circuits – Part 1:
 Steady state response
 (Graphical) Phasor analysis
CE164
RC Circuits:
Sinusoidal
Steady State Response
Reading:
Chapter 10 in Floyd & Buchla (2014, 8th ed.)
Slide 2.2CE164 – 2024 F. Sepulveda - CSEE - Essex University
Definitions: (reminder from CE163)
Transient response:
 In general: a short-lived, fast response immediately after a sudden
change in a system’s input (e.g., step input, delta function, turning a
circuit ON or OFF etc.)
 Strict definition: system response when changing from one equilibrium
state to another
Steady-state response
 After the transient response settles
 Longer lasting
 Note: not necessarily a constant value (e.g., sinusoidal responses can
be steady state, after an initial transient)
transient
steady state
t
V
Example:
Slide 2.3CE164 – 2024 F. Sepulveda - CSEE - Essex University
Specific values for
current and voltage can
be read from a universal
curve. For an RC circuit,
the time constant is
τ RC
100%
80%
60%
40%
20%
0
0 1t 2t 3t 4t 5t
99%
98%
95%
86%
63%
37%
14%
5%
2% 1%
Number of time constants
P
e
r
c
e
n
t

o
f

f
i
n
a
l

v
a
l
u
e
Rising exponential
Falling exponential
Reminder: RC Transient Response
(covered in CE163)
Universal exponential curves:
response to step input
The curves reach the final
value (within 1% of max)
whenever t > 5t
Slide 2.4CE164 – 2024 F. Sepulveda - CSEE - Essex University
VC
t
Sinusoidal Transient and Steady State
VS
VS
VCtransient steady state
Closing the switch causes a
transient response first:
exponential + sinusoidal
for a detailed mathematical discussion see https://www.youtube.com/watch?app=desktop&v=fsntT_pd_xI
notice the phase
difference
Slide 2.5CE164 – 2024 F. Sepulveda - CSEE - Essex University
Capacitive reactance is the opposition to current by a capacitor. The
equation for capacitive reactance for sinusoidal signals is:
1
2π
CX
fC

The reactance of a 0.047 F capacitor when a frequency
of 15 kHz is applied is 226 
Capacitive reactance & AC Steady State
(i.e., after the transient response finishes)
where f is the frequency of the sine voltage going through and C is
the capacitance.
notice the units!
 Reactance:
 frequency-dependent opposition to current
 produces a phase shift
 a property of capacitors and inductors
 Impedance:
 Reactance + Resistance
Slide 2.6CE164 – 2024 F. Sepulveda - CSEE - Essex University
When capacitors are in series, the total reactance is the sum of the
individual reactances. That is,
Assume three 0.033 F capacitors are in series with a 2.5 kHz
AC source. What is the total reactance?
5.79 k
C( ) C1 C2 C3 Ctot n
X X X X X    
The reactance of each capacitor is
  
1 1
1.93 k
2π 2π 2.5 kHz 0.033 μF
C
X
fC
   
C( ) C1 C2 C3
1.93 k 1.93 k 1.93 k
tot
X X X X  
    
Capacitive reactance
Equivalent reactance
Slide 2.7CE164 – 2024 F. Sepulveda - CSEE - Essex University
When capacitors are in parallel, the total reactance is the reciprocal
of the sum of the reciprocals of the individual reactances. That is,
If the three 0.033 F capacitors from the last example are
placed in parallel with the 2.5 kHz ac source, what is the total
reactance?

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