Slide 2.0CE164 – 2024 F. Sepulveda - CSEE - Essex University
Foundations of Electronics – II
Lecture 2 (Week 17)
RC Circuits – Part 1:
 Steady state response
 (Graphical) Phasor analysis
Francisco Sepulveda
E-mail: f.sepulveda
CE164
moodle.essex.ac.uk/course/view.php?id=3644
Slide 2.1CE164 – 2024 F. Sepulveda - CSEE - Essex University
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?
643 
C( )
C1 C2 C3 C
1
1 1 1 1tot
n
X
X X X X

    
The reactance of each capacitor is 1.93 k
C( )
C1 C2 C3
1 1
1 1 1 1 1 1
+ +
1.93 k 1.93 k 1.93 k
tot
X
X X X
  
 
  
Capacitive reactance
Slide 2.8CE164 – 2024 F. Sepulveda - CSEE - Essex University
When a sine wave is
applied to a capacitor,
there is a phase shift
between voltage and
current such that current
always leads the capacitor
voltage by 90o.
VC
0
I
0
90
o
Capacitive phase shift
Q = CV  I = dQ/dt = C dV/dt
AC Steady State
VS
C
Slide 2.9CE164 – 2024 F. Sepulveda - CSEE - Essex University
When both resistance and capacitance are in a series circuit, the phase
angle between the applied voltage and total current is between 0 and
90, depending on the values of resistance and reactance.
R
VR
C
VR leads VS VC lags VS
I leads VS
I
VS
VC
Sinusoidal phase response of RC circuits
Compared to Vs:
R alone: = 0o
C alone: = – 90o
KVL applies at all times: vR(t) = vs(t) – vC(t)
so, R = – C
The current’s phase shift (i)
will be determined by VR
Slide 2.10CE164 – 2024 F. Sepulveda - CSEE - Essex University
In a series RC circuit, the total impedance Z is the phasor
sum of R and XC :
R is plotted along the (real) positive x-axis.
R
XC is plotted along the (imaginary) negative y-axis.
XC
1tan C
X
R
     
 
Z
Z is the
diagonal
(use Pythagoras’
theorem)

It is convenient to reposition the phasors into the impedance triangle
R
XC
Z

Impedance of series RC circuits
Impedance Z = overall opposition to current
 includes R and X
Z = R – j XC Compared to Vs:
R alone: = 0o
C alone: = – 90o
Slide 2.11CE164 – 2024 F. Sepulveda - CSEE - Essex University
Sketch the impedance triangle showing the values
for R = 1.2 k and XC = 960 and find the
impedance and the phase shift.
Impedance of series RC circuits
R = 1.2 k
XC = 960 

   2 21.2 k + 0.96 k
1.33 k
Z   
 
1 0.96 ktan
1.2 k
39
 



 
Z = 1.54 k
39o
NB: in most calculators the result for arctan will be in radians,
so you will need to convert to theta to degrees.
Slide 2.12CE164 – 2024 F. Sepulveda - CSEE - Essex University
Ohm’s law applies to series RC circuits using Z, V, and I.

V V
V IZ I Z
Z I
  
Because I is the same everywhere in a series circuit, you can obtain the
voltages across different components by multiplying the impedance of that
component by the current as shown in the following example.
Analysis of series RC circuits
Remember: to be able to use Ohm’s law (and Watt’s law) with AC signals,
we must use rms for I and V .
For AC signals in general, unless we have additional information, we assume
that voltages and currents are already in rms and that the DC offset is zero.
Slide 2.13CE164 – 2024 F. Sepulveda - CSEE - Essex University
Assume the current in the previous example is 10mArms. Sketch
the voltage phasor diagram. The impedance triangle from the
previous example is shown for reference.
The voltage phasor diagram can be found from Ohm’s law.
Multiply each impedance phasor by 10 mA.
R = 1.2 k
XC =
960 Z = 1.54 k
39o

VR = 12 V
VC = 9.6 V
x 10 mA
=
VS = 15.4 V
39o

Analysis of series RC circuits
Slide 2.14CE164 – 2024 F. Sepulveda - CSEE - Essex University
Phasor diagrams that have reactance phasors can only be
drawn for a single frequency because XC is inversely
proportional to the frequency.
As frequency changes, the impedance
triangle for an RC circuit changes as
illustrated here because XC decreases
with increasing f. This determines the
frequency response of RC circuits.

Z3
XC1
XC2
XC3
Z2
Z1
1
2
3
1
2
f
f
f
3
Increasing f


R
Variation of phase angle and Z with frequency
1
2π
CX
fC

The lower the frequency, the more capacitors
will reduce a signal’s amplitude and will
increase its phase shift magnitude.
Slide 2.15CE164 – 2024 F. Sepulveda - CSEE - Essex University
10 V dc
VoutV in
100 
1 F
10 V dc
0
10 V dc
0
When a signal is applied to an RC circuit, and the output is taken across
the capacitor as shown, the circuit acts as a low-pass filter.
Low–pass: As the frequency increases, the output amplitude decreases.
Plotting the
response:
Vout (V)
9.98
8.46
1.57
0.79
0.1 1 10 20 100
f (kHz)
9
8
7
6
5
4
3
2
1
1ƒ = 1 kHz
8.46 V rms10 V rms 100

1.57 V rms
10 V rms
0 kHz
100
F
0.79 V rms
10 V r s
20
Frequency Response of RC Circuits
steady state
Slide 2.16CE164 – 2024 F. Sepulveda - CSEE - Essex University
Vin
10 V dc
0
Vout
0 V dc10 V dc 100 
1 F
Reversing the components, and taking the output across the resistor as
shown, the circuit acts as a high-pass filter.
High–pass: as the frequency increases, the output amplitude also increases.
Plotting the
response:
ƒ = 100 Hz
0.63 V rms
10 V rms
100 
1 F
ƒ = 1 kHz
5.32 V rms
10 V rms
ƒ = 10 kHz
9.87 V rms10 r s
100 

Vout (V)
f (kHz)
9.87
5.32
0.63
0
0.01 0.1 1
10
9
8
7
6
5
4
3
2
1
10
Frequency Response of RC Circuits
steady state
Slide 2.17CE164 – 2024 F. Sepulveda - CSEE - Essex University


(phase lead)
(phase lead)
V
R
VC
Vout
C
VoutVin
Vin
Vout
Vin
As seen in the previous slides, RC circuits work as high-pass and low–
pass filters.
But, they will also produce a phase shift. For example:
Application: phase shift
useful for producing spatial effects, e.g., delay-related audio
signals that produce 3D effects such as surround without
needing more speakers.