2.3 Sinusoidal Signals
and Capacitance
• many signals occur over the frequency range
of 0.1 Hz to 10 kHz
• definition of alternating current (ac), rootmean-square (rms) values, and ac power
• capacitance and capacitors
• electrical properties of combined capacitors
• voltage and current across a capacitor
• advantage of using complex waves in
calculations
• reactance relates ac current and voltage
• impedance as a vector in the complex plane
2.3 : 1/12
Time-Varying Voltages as Cosines
1.4*cos(2π0.04t+π/8)
1.5
1.0*cos(2π0.05t)
1
voltage (V)
0.5
0
0
20
40
60
80
100
-0.5
-1
-1.5
time (s)
A cosine signal is written as, V(t) = V0 cos(2πf0t + φ), where V0
is the amplitude, f0 is the frequency, and φ is the phase angle.
The period is given by, _________. The peak-to-peak voltage
is _______. +φ is called a phase lead, while –φ is a lag
2.3 : 2/12
Why Cosines?
The mathematician Fourier has shown that any temporal signal
measured in the laboratory can be written as a sum of sines and
cosines, or alternatively, as phase-shifted cosines.
frequency content of a square wave
t0
f (t ) =
4
π
∞
∑ ( −1)
n =1,3,5,"
n −1
2
cos ( 2π nt t0 )
n
When a signal is composed entirely of cosines with periods
longer than ___________ (0.1 to 0.01 Hz), it can be treated as
direct current for the purposes of electronic circuitry. When a
signal gets above _________ it begins to behave more like an
electromagnetic wave than a simple electrical voltage. We will
deal with ac signals from 0.1 Hz to 10 kHz.
2.3 : 3/12
Alternating Current and RMS Values
• alternating current can be obtained from an alternating voltage
and Ohm's law
V (t )
V0
i (t ) =
=
cos ( 2π f 0t ) = i0 cos ( 2π f 0 t )
R
R
• the average alternating voltage, Vavg, is _______
• the rms (root mean square) voltage is the square root of the
average of (V(t) - Vavg)2
T
1
2
2
cos
Vrms =
V
( 2π f0t ) dt =
0
∫
T
0
• the rms current is given by the rms voltage and Ohm's law
irms =
2.3 : 4/12
Vrms 0.707V0
=
=
R
R
AC Power
• instantaneous power is given by the product of the voltage
and current
P ( t ) = V ( t ) i ( t ) = V0i0 cos 2 ( 2π f0t )
note that the instantaneous power ranges between ____ and V0i0
• average power is given by the product of the rms voltage and
current
T
T
0
0
Vi
Vi
1
Pavg = ∫ P (t )dt = 0 0 ∫ cos 2 ( 2π f 0t ) dt = 0 0 =
T
T
2
• when the voltage and current differ by a phase angle of φ the
average power is given by
Pavg = Vrms irms cos φ
which means that the average power goes to zero when the
voltage and current differ by __________
2.3 : 5/12
Capacitance
• capacitance is the ability of two parallel conductors to hold
charge at a given electric potential, C = Q / V
• capacitance is given in units of farads, where F = CV-1
• commercially available capacitors range from 100,000 μF to
10 pF (note that the units ___ and ___ are almost never used)
• two parallel plates of area, A, and separation, d, have a
capacitance given by
C = ε0
A
d
A and d must be in meters!
for A = 1 cm2 and d = 0.1 mm, C = 8.85 pF
• when the capacitor plates are separated by an insulator, the
capacitance is given by
where κ is the dielectric constant of the insulator (note that κ =
ε/ε0)
2.3 : 6/12
Common Capacitors
• common insulating materials
material
dielectric constant
air
1.00059
polystyrene
2.56
paper
3.7
SrTiO3
233
dielectric strength (Vm-1)
3×106
24×106
16×106
8×106
• electrolytic capacitors are composed of a sheet of foil inserted
into a conducting liquid, with insulation provided by an oxide layer
• electrolytic capacitors have + and − leads, and if connected
backwards the oxide layer dissolves with explosive results
• common capacitor types
type
electrolytic (big)
electrolytic (small)
polyester (orange drop)
ceramic
2.3 : 7/12
capacitance range
100 μF - 120,000 μF
1 μF - 2,500 μF
1,000 pF - 1 μF
10 pF - 4,700 pF
Multiple Capacitors
with a parallel connection each capacitor sees the same voltage
QT = Q1 + Q2 = C1V + C2V = ( C1 + C2 )V
+
!
V
C1
+
!
C2
+
!
equivalent to increasing A
with a series connection the amount of charge separated by each
capacitor has to be the same (charge is added to the top
capacitor and removed from the bottom capacitor)
+
!
C1
+
!
2.3 : 8/12
VT = V1 + V2 =
Q Q ⎛ 1
1 ⎞
+
=⎜ +
⎟Q
C1 C2 ⎝ C1 C2 ⎠
V
+
!
C2
equivalent to
increasing d
Current and Voltage with a Capacitor
What is the alternating current through the
capacitor?
V(t)
C
i
V (t ) = V0 cos ( 2π f 0t )
Q = CV
dQ
dV
i (t ) =
=C
dt
dt
i (t ) = −2π f 0CV0 sin ( 2π f 0t )
• as f0 → 0, i → 0 and as f0 → ∞, i → -∞
• the current is -90° out of phase with the voltage
• because of the -90° phase difference no power is dissipated in
the capacitor (this is different behavior than a resistor)
• because of the phase change, the ratio of voltage divided by
current is not a constant
V0 cos ( 2π f 0t )
−2π f 0CV0 sin ( 2π f 0t )
2.3 : 9/12
=
cot ( 2π f 0t )
−2π f 0C
Capacitive Reactance
• in order to obtain a constant that relates ac voltage and
current, it is necessary to use a complex wave ( j = −1 )
V ( t ) = V0 e j 2π f0t where
e± jθ = cos θ ± j sin θ
• solve for current using complex waves
i (t ) = C
XC =
dV ( t )
dt
V (t )
i (t )
=
= j 2π f 0CV0 e j 2π f0t
V0 e j 2π f0t
j 2π f 0CV0 e j 2π f0t
=
1
−j
=
j 2π f 0C 2π f 0C
• the constant is called ________________ and has units of ohms
• as f0 → 0, XC → -∞j, and as f0 → ∞, XC → 0
• although we will not be using circuits with inductors, it is worth
noting that _______________ is given by XL = j2πf0L, where L is
the inductance of a coil
2.3 : 10/12
Impedance and the Complex Plane
• the relationship between voltage and current
amplitudes is given by the circuit impedance
• resistance, capacitance and inductance are
considered to be vectors in the complex plane
• impedance is the sum vector, Z
• the relationship between voltage and current
amplitudes is given by the magnitude |Z|
10 :H
1 MHz
100 S
1 nF
Im
Z = ⎡⎣ R + ( X L + X C ) ⎤⎦ ⎡⎣ R − ( X L + X C ) ⎤⎦
• the phase angle between
voltage and current is given by
⎛ XC + X L ⎞
⎟
R
⎝
⎠
φ = tan −1 ⎜
XL = 63j
R = 100
φ = -44E
Re
XC = -160j
Z = 139 Ω
2.3 : 11/12
Impedance of Single Components
• impedance across a resistor
Z R = RR* =
• phase angle across a resistor
⎛0⎞
⎟=
⎝R⎠
φ = tan −1 ⎜
• impedance across a capacitor
ZC =
−j
+j
1
=
2π f 0C 2π f 0C 2π f 0C
• phase angle across a capacitor
2.3 : 12/12
⎛ −1
⎜
−1 2π f 0C
φ = tan ⎜
⎜ 0
⎜
⎝
⎞
⎟
⎟=
⎟
⎟
⎠