Post on 28-Oct-2015
description
transcript
L211 Series Resonance and Time/Frequency Response of Passive Networks
-1-
1. Abstract
In this experiment, the characteristics of series resonance RLC circuit are studied.
Resonance occurs at the point when inductor voltage and capacitor voltage are same
and at the same time, current in the circuit reaches its maximum. The relationship
between half-power frequencies, bandwidth, quality factor and selectivity are also
studied in the experiment by comparing the difference of High-Q series and Low-Q
series resonance circuit.
The time response and frequency response of RC circuit are also discussed. The
importance of time constant (τ) is studied in part B. Time constant will determine the
shape of the time response of the RC circuit. Reducing the value of τ (i.e. reducing R
or C) means that the output will change faster and that any given voltage will be
reached sooner. In terms of frequency response, the output voltage will differ with
respect to the change of the input frequency and RC circuit will act as low pass filter.
2.
L211 Series Resonance and Time/Frequency Response of Passive Networks
-2-
1. Content
1. Abstract ................................................................................ Error! Bookmark not defined.
2. Content ........................................................................................................................................... 2
3. Introduction .................................................................................... Error! Bookmark not defined.
4. Objectives ..................................................................................................................................... 4
5. Equipment and Component ........................................................... Error! Bookmark not defined.
6. Procedure ...................................................................................................................................... 5
6.1 Low-Q and High-Q series resonance circuit ........................................................................... 5
6.2 Time/Frequency response of RC networks ............................................................................. 6
7. Results and discussion(log sheet questions) ............................................................................. 8
8. Conclusion .................................................................................................................................. 19
9. Referance ..................................................................................................................................... 20
10. Appendix ..................................................................................................................................... 21
L211 Series Resonance and Time/Frequency Response of Passive Networks
-3-
3. Introduction:
I. Resonant circuit
Resonance in AC circuits implies a special frequency determined by the values of
the resistance, capacitance, and inductance. For series resonance the condition of
resonance is straightforward and it occurs when the inductive and capacitive
reactances are equal in magnitude but cancel each other because they are 180
degrees apart in phase. The sharp minimum in impedance which occurs is useful
in tuning applications. The sharpness of the minimum depends on the value of R
and is characterized by the quality factor "Q" of the circuit.
Resonant circuits are used to respond selectively to signals of a given frequency
while discriminating against signals of different frequencies. If the response of
the circuit is more narrowly peaked around the chosen frequency, we say that the
circuit has higher "selectivity". A "quality factor" Q, as described below, is a
measure of that selectivity, and we speak of a circuit having a "high Q" if it is
more narrowly selective.
The quality factor of a circuit is dependent upon the amount of resistance in the
circuit. The smaller the resistance, the higher the "Q" for given values of L and C.
The quality factor Q is defined by
𝑄 =𝜔0
𝛥𝜔
where Δω is the width of the resonance power curve at half maximum.
The Q is a commonly used parameter in electronics, with values usually in the
range of Q=10 to 100 for circuit applications.
II. Time/frequency response of RC networks
When we applied a dc voltage to a resistor and capacitor in series, the
capacitor charged to the applied voltage along an exponential curve, and
then just sat there. In this experiment, a square wave function is used to
study the step response of the RC circuit. Here, the input voltage will
change direction during each cycle, so the capacitor will constantly charge
and discharge as it continually tries to oppose the changes.
RC circuits, like other types of circuits, are used to "filter" a signal
waveform, changing the relative amounts of low-frequency and high-
frequency information in their output signals relative to their input signals.
There are high-pass filter and low-pass filter and band-pass filter versions.
A common application is for smoothing a signal, using a low-pass version
as discussed in this experiment.
L211 Series Resonance and Time/Frequency Response of Passive Networks
-4-
4. Objectives
The objectives of the experiment are:
(a) To study the characteristics of series resonant circuits
(b) To investigate the step response of and RC network
(c) To study the frequency response characteristics of RC circuits
5. Equipment and components
5.1 Equipment and components for part (A)
Resisters: 47Ω ,220 Ω
Inductor: 10 mH
Capacitor: 0.1 µF
Digital Multimeter(DMM)
Function Generator
Breadboard
Digital Oscilloscope(CRO)
5.2 Equipment and components for part (B)
Resisters: 1kΩ ,100Ω
Capacitors: 0.1µF, 0.01µF
Digital multimeter(DMM)
Function generator
Breadboard
Digital oscilloscope(CRO)
d
Figure 1:Digital Multimeter(DMM) Figure 2:Function Generator
Figure 3: Breadboard Figure 4: Digital Oscilloscope (CRO)
L211 Series Resonance and Time/Frequency Response of Passive Networks
-5-
6. Procedure:
6.1 Part(A) Low & high Q series resonance circuit
I. Low Q series resonance circuit
1) Construct the circuit of Figure 5. Measure the resistances of R and
RL(resistance of inductor). Using the nominal values of L and C and
measured resistance values, compute the radian frequency fs and quality
factor Qs of the series circuit at resonance.
2) Energize the circuit and vary the function generator from 1 kHz to 9 kHz.
One of the frequencies must be set at the resonance frequency, fs. At each
frequency, reset the input to 1 V (rms) (with the circuit connected) and
measure the rms values of the voltages VC, VR and VL with the digital
multimeter (DMM). Calculate 𝐼𝑟𝑚𝑠 =𝑉𝑅 (𝑟𝑚𝑠 )
𝑅 for each frequency.
Figure 5: Series resonant circuit
II. High Q series resonance circuit
3) Replace the 220-Ω resister with resistor of 47-Ω in the circuit of figure 5.
Repeat 1) and 2) above.
4) Draw the curves of current Irms versus frequency for low and high –Q
circuits.
5) Plot VL (rms) and VC (rms) versus frequency for the two cases.
L
Function
generator R=220Ω
VR VL
INDUCTOR
C=0.1µF
VC E=1V(RMS)
(Sine wave)
10MH
RL
L211 Series Resonance and Time/Frequency Response of Passive Networks
-6-
6.2 (Part B &C) Time/frequency response of RC networks
1) Connect an RC circuit as shown in Figure 6.
Figure 6: RC circuit
2) Apply a square waveform of the form shown in figure 7 to the input of
figure 6.
Figure 7: Square waveform
3) Sketch the observed output waveform V0(t) on the same time scale as Vi(t)
for the valued of R and C in Table 1.
Table 1: Values of R, C and τ
R C τ=RC
(i) 1kΩ 0.1µF 0.10ms
(ii) 1kΩ 0.01µF 0.01ms
(iii) 100kΩ 0.1µF 0.01ms
L211 Series Resonance and Time/Frequency Response of Passive Networks
-7-
4) With R=1kΩ, C=0.1µF in figure 2, measure the time constant of the circuit
from the observed step response.
5) With r=1kΩ, C=0.1µF in figure 2, apply a 5-V peak to peak sinusoidal
input with various frequency to the network. Plot the magnitude of V0/Vi
against frequency (from 100Hz to 1MHz). At the same time measure and
plot the phase angle different between Vo and Vi.
6) Draw V0 curve against frequency for network of figure 2.
L211 Series Resonance and Time/Frequency Response of Passive Networks
-8-
7. Results and Discussion
7.1.1) (Q5.1 (a)) For low –Q series resonant circuit the nominal values of L and
C, measured resistance values, and the computed the radian frequency fs
and quality factor Qs of the series circuit at resonance are as shown in
table 2.
Table 2: Values of R,L,C fs ,Qs in low-Q circuit
Quantity Value
Measured resistance R 219.40
Measured resistanceRL 38.80
Nominal value of L 10mH
Nominal value ofC 0.1F
Radian frequency ωs ωs=1 𝐿𝐶 =31.6×103 rad/s
Resonant frequency fs fs=1 2𝜋 𝐿𝐶 =5KHz
Quality factor Qs Qs=
1
𝑅𝑇 𝐿
𝐶 =
1
𝑅+𝑅𝐿 𝐿
𝐶 =1.22
7.1.2) (Q5.1(b)) The rms values of the voltages VC, VR and VL measured
for various frequencies for the Low-Q circuit are as shown in the
following table.
Table 3: Values of voltages and calculated current in low-Q circuit
f(kHz) vc(V) vL(V) vR(V) Irms(A)
1 1.046 0.049 0.144 0.000656
2 1.129 0.186 0.31 0.001413
3 1.251 0.455 0.515 0.002347
4 1.352 0.869 0.743 0.003387
5 1.23 1.241 0.846 0.003856
6 0.958 1.366 0.785 0.003578
7 0.688 1.33 0.657 0.002995
8 0.513 1.29 0.559 0.002548
9 0.39 1.237 0.478 0.002179
L211 Series Resonance and Time/Frequency Response of Passive Networks
-9-
7.1.3) (Q5.2(a)) For High –Q series resonant circuit the nominal values of
L and C, measured resistance values, and the computed the radian
frequency fs and quality factor Qs of the series circuit at resonance
are as shown in table 4.
Table 4: Values of R,L,C fs ,Qs in high-Q circuit
Value
Measured resistance R 46.90
Measured resistanceRL 38.80
Nominal value of L 10mH
Nominal value ofC 0.1F
Radian frequency ωs ωs=1 𝐿𝐶 =31.6×103 rad/s
Resonant frequency fs fs=1 2𝜋 𝐿𝐶 =5KHz
Quality factor Qs Qs=
1
𝑅𝑇 𝐿
𝐶 =
1
𝑅+𝑅𝐿 𝐿
𝐶 =
7.1.4) (Q5.2(b)) The rms values of the voltages VC, VR and VL measured
for various frequencies for the Low-Q circuit are as shown in the
following table.
Table 5: Values of voltages and calculated current in high-Q circuit
f(kHz)) vc(V) vL(V) vR(V) Irms(A)
1 1.057 0.05 0.031 0.000661
2 1.206 0.201 0.071 0.001514
3 1.526 0.555 0.134 0.002857
4 2.39 1.545 0.279 0.005949
5 3.57 3.628 0.521 0.011109
6 1.854 2.678 0.324 0.006908
7 0.997 1.938 0.203 0.004328
8 0.645 1.625 0.15 0.003198
9 0.459 1.457 0.12 0.002559
L211 Series Resonance and Time/Frequency Response of Passive Networks
-10-
7.1.5) (5.2(d)) The curve of current Irms versus frequency for the low and
high –Q circuit are as shown in figure 8.
Figure 8: Irms verses frequency for Low- and high-Q series resonant
7.1.6) (Q5.2(e)) The plot of VL (rms) and VC (rms) versus frequency for
high-Q and low-Q circuit are as shown in figure 9.
Figure 9: VL (rms) and VC (rms) versus frequency for high-Q and low-Q circuit
3.86
11.109
0
1
2
3
4
5
6
7
8
9
10
11
12
0 1 2 3 4 5 6 7 8 9 10
low Q
HIGH Q
f1L fsf2Hfsf2Hfs
f1H f2L
IH-MAX
IL-MAX
0.707×IL-MAX
0.707×IH-MAX
frequency (kHZ)
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4 5 6 7 8 9 10
volt
age
(V
)
FREQUENCY(kHZ)
vc
vl
vc
vL
VC-
HIGH
VL-
HIGH
VC-
BW
BW
Irms(mA)
L211 Series Resonance and Time/Frequency Response of Passive Networks
-11-
Figure 10: Comparison between High-Q and Low-Q circuit
7.1.7) (Q5.2(f)) Compare with theory, the above curve have the following
characteristics:
I. For both Low-Q and High-Q series resonance circuits, the VL VC and
VR versus frequency curves agree well with the theory. According to
the theory, VL and VC are equal at resonance since XL=XC, but they
are not maximum at the resonant frequency fs and this can be clearly
shown in figure 10. As the figures show, when VL=VR, Irms reaches
the maximum, which means that it is at resonance.
II. As shown in the figure, VC reaches a maximum just before resonance
and VL reaches a maximum just after maximum which correspond to
the theory closely. For the curve of Irms versus frequency, maximum
occur at fs. So according to theoretical calculation, the fs=5 kHZ.as
shown in the figure, resonance occur at f=5 kHz which is
approximately equal to theoretical value.
III. For High-Q circuit, which has a low resistance, the curve of Irms and
frequency has a narrow bandwidth as compared to a high resistance,
low Q circuit. It will decay more quickly as the frequency moves
away from the resonant frequency.
7.1.8) (Q5.2(g)) The High-Q circuit will be more selective.
Bandwidth is measured between the 0.707 current amplitude points. In
Figure 8, for the high-Q circuit, the 100% current point is 11.109 mA.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1 2 3 4 5 6 7 8 9 10
volt
age
FREQUENCY
LOW Q
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4 5 6 7 8 9 10
Vo
ltag
e
FREQUENCY
HIGH -Q
vc
vr
vl
L211 Series Resonance and Time/Frequency Response of Passive Networks
-12-
The 70.7% level is 0707(111.109 mA)=7.85 mA. The upper and lower
band edges read from the curve are 4.4k Hz for fl-H and 5.8k Hz for f2-H.
The bandwidth is 1.4k Hz. Similarly in the figure of the Low-Q curve,
the bandwidth of low-Q circuit will be 4 kHZ. Thus, for high Q circuit,
it has smaller bandwidth.
Actually, according to theory, BW=S
S
Q
ff 12f , which means that
BW is negatively proportional to quality factor. For RLC circuit, the
smaller the BW, the higher the circuit selectivity, at the same time, it
also means that the larger Qs, the higher the circuit selectivity.
Thus, for High-Q circuit has smaller bandwidth, it will has a larger Qs
and also has a higher selectivity.
7.1.9) (Q5.2(h)(i))Using the equation Qs = |Vc|s/E ,the calculated Qs is as
shown below:
For the Low-Q circuit: Qs = |Vc|s/E = 1.23/1 =1.23
For the High-Q circuit: Qs = |Vc|s/E = 3.69/1 = 3.69
Using the equation Qs = fs/BW, the values of Qs are also as shown:
For the Low-Q circuit: Qs = fs/BW = fs/( LOWLOW f _1_2f )=5.033/(7.2-3.2) = 1.26
For the High-Q circuit: Qs = fs/BW = fs/( HIGHHIGH ff _1_2 )5.033/(4.4-5.8) = 3.60
According to the above calculation, the values of Qs for each circuit are
approximately the same. Actually, the theoretical values of the circuit are Q
223.1_ LOWS and Q 69.3_ HIGHS correspondingly and the above values
calculated almost consist with them respectively.
7.1.10) (Q5.2(j)) At frequencies below the resonant frequency, the current
leads the voltage, which is characteristic of an RC circuit. As the
power factor of the RC circuit are less than 1 and it is leading power
factor, the power factor of the circuit are pf<1 (leading).
At the resonant frequency, the voltage of the capacitor and inductor
are the same. And the impedance of the capacitor and inductor are
also the same. So the total apparent power is equal to the average
L211 Series Resonance and Time/Frequency Response of Passive Networks
-13-
power dissipated by the resistor and the power factor is equal to 1.
This is a maximum power factor. Thus, at resonant frequency, pf=1.
At frequency above resonance, the current lags the voltage, and the
series RLC circuit looks like a series RL circuit. Thus, the power
factor at high frequency pf<1 (lagging).
7.1.11) (Q5.2(k)) A network is in resonance when the voltage and current at
the network input terminals are in phase and the input impedance of
the network is purely resistive.
Figure 11: Parallel Resonance Circuit
Consider the Parallel RLC circuit of figure 1. The steady-state
admittance offered by the circuit is:
Y = 1/R + j( ωC – 1/ωL)
Resonance occurs when the voltage and current at the input terminals
are in phase. This corresponds to a purely real admittance, so that the
necessary condition is given by
ωC – 1/ωL = 0
The resonant condition may be achieved by adjusting L, C, or ω.
Keeping L and C constant, the resonant frequency ωo is given by:
𝜔0=
1
𝐿𝐶
OR
𝑓𝑠 =1
2𝜋 𝐿𝐶
From above, we find that the conditions for resonance to occur are the
same for both parallel- or series-RLC circuit. However, when the
resonance of the parallel occurs, the current gets its minimum value
instead of maximum.
L211 Series Resonance and Time/Frequency Response of Passive Networks
-14-
7.2.1) (Q6.3)The output waveform v0(t) for various values of R and C in table
1 are as shown in the following figure:
Figure 12: time response of RC circuit with τ=0.10ms(R = 1kΩ and C = 0.1μF)
Figure 13: time response of RC circuit with τ=10µs (R = 1kΩ and C = 0.01μF)
L211 Series Resonance and Time/Frequency Response of Passive Networks
-15-
Figure 14: time response of RC circuit with τ=10µs (R = 100Ω and C = 0.1μF)
7.2.2) (Q6.2.1) To study the step response of RC circuit, step input should be
used. However, if only one step is used, the step response will just
occur at the time when the step input is injected into the system and it
will last a very short time. (Actually when the time exceeds five time
constants, the capacitor voltage will be nearly the same as the final
voltage.)
When a square wave is used, the capacitor will be charged and
discharged continuously during each cycle of the square wave. Each
positive and negative cycle of the square wave can be viewed as a step
function and hence the step response of the system can be studied.
7.2.3) (Q6.5) The output voltage (Vc) reaches 63.2% of its final value in 1
time constant (1 second in this case). In general, the time taken to reach
a particular value is related to the number of time constants given in the
table below.
Table 6: Number of time constants required to reach a proportion of the final value
τ 2 τ 3τ 4 τ 5 τ
63.2% 86.5% 95.0% 98.2% 99.3%
L211 Series Resonance and Time/Frequency Response of Passive Networks
-16-
Figure 15: time constant versus percentage of charge
As shown in the above table and figure, when the time is 5 τ, the output
voltage (VC) reaches more than 99% of its final value.
For each positive and negative half cycle of the square waveform, the
capacitor will get charged for each half cycle. In order to get the
capacitor fully charged for each charging cycle, the time shall be greater
than 5τ. Thus, period of the square wave function shall be at least 10τ.
7.2.4) (Q6.6)With R=1, C=0.1µF in Figure 6, apply a 5-V peak-to-peak
sinusoidal input with various frequencies to the network. The values of
V0, Vi as well as the phase angle between V0 and Vi are shown in the
following table. The plot the magnitude of V0/Vi against frequency (say
from 100HZ to 100 kHz) is as shown in figure 16. The plot the phase
angle difference between V0 and Vi is as shown in figure 17.
Table 7: Frequency response of RC circuit
f/Hz i0 /VV )(reslo )(idealo Error/%
100 0.994 0 3.6 -5.56
500 0.940 -16.2 17.4 -6.90
1000 0.831 -32.4 32.1 0.93
f 5.1591c 0.700 -45.3 45 0.67
5k 0.306 -73.8 72.3 2.07
10k 0.163 -79.6 81.0 1.73
50k 0.035 -88.64 88.2 -5.32
100k 0.02 -88.92 89.1 -3.03
L211 Series Resonance and Time/Frequency Response of Passive Networks
-17-
Figure 16: Plot of V0/Vi versus frequency
Figure 17: phase angle between V0 and Vi versus frequency
7.2.5) (Q6.7) V0 curve against frequency for network of figure 6 can be draw
from table 5 as i
0
i
0
o 5*V
V
V
VVV i and the value of
i
0
V
V can be got from table
5 directly. The graph is shown below:
0.9940.9376
0.8312
0.7
0.306
0.1630.035
0.02
0
0.2
0.4
0.6
0.8
1
1.2
1 10 100 1000 10000 100000
V0/
VI
frequency (Hz)
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
1 10 100 1000 10000 100000
ph
ase
an
gle
frequency(Hz)
L211 Series Resonance and Time/Frequency Response of Passive Networks
-18-
Figure 18: V0 versus frequency
According to the plot and the theory, it is shown that the RC circuit in the
experiment is Low-pass filter as for low frequency the output voltage is much
higher than that at high frequency.
7.2.6) (Q6.8) Comment on phase angle of the transfer function:
According to the step 5, we know that the phase angle keeps be larger
while the transfer function keeps be larger. What’s more, the changing
rate of the phase angle and the transfer function both first become larger
and then smaller. Therefore, it is reasonable that we can assume the
relation between the phase angle and the transfer function is negatively
proportional.
0
1
2
3
4
5
6
1 10 100 1000 10000 100000
V0
frequency (Hz)
Series1
L211 Series Resonance and Time/Frequency Response of Passive Networks
-19-
8 Conclusion:
I. For the resonance of a series RLC circuit occurs when the inductive and
capacitive reactance are equal in magnitude but cancel each other
because they are 180 degrees apart in phase. At the same time, VC=VL
and Irms reaches its maximum value.
II. The selectivity of a circuit depends on the quality factor of the circuit. As
for higher Qs, the selectivity is higher. The selectivity can also be told
from the curve of the response of the circuit. If the curve is more
narrowly peaked around some certain frequency, thus it will have a small
bandwidth and we call that the circuit has higher selectivity.
III. For the time response of a RC circuit, the shape of the curve depend
highly on the time constant of the circuit which τ=RC. Time constant can
also be got from the curve of the step response of the circuit which
equals to the time for the voltage to become 63% of its final value.
IV. The frequency response of the circuit can tell the properties of the RC
circuit. For example, the experiment show that the circuit in figure is
Low-pass circuit which means that it will give high output at low
frequency while low output at high frequency.
L211 Series Resonance and Time/Frequency Response of Passive Networks
-20-
9 Reference
1) Paul A. Tipler, ―Physics for Scientists and Engineers‖,3rd edition,Extend version
2) R. A. Serway & J. W. Jewett, Jr. “Physics for Scientists and Engineers with Modern
Physics‖, 6th Edition,
3) Retrieved September/20/2009 http://en.wikipedia.org/wiki/RLC_circuit
4) Laboratory Experiment EE2071 Laboratory Manual for Experiment L211
L211 Series Resonance and Time/Frequency Response of Passive Networks
-21-
10 Appendix
11
Quality Factor and Bandwidth (BW)
The quality factor sQ of a series resonant circuit is defined as the ratio of the reactive
power of either the inductor or the capacitor to the average power of the resistor at
resonance, i.e.,
sQ =C
L
RL
LCRR
Lf
R
L
RI
XI
TTT
s
T
S
T
L 1)
2
1(
22
*
*
power average
power reactive2
2
Therefore, at resonance the voltage across the inductor V L can be written as:
T
L
S
LSL
R
EX
Z
EXV
**|| EQE
R
LV S
T
sL
s||
Similarly, EQV SSC || and where R RRLT
What’s the physical meaning of the quality factor Q S ? Actually the quality
factor Q S affects the circuit selectivity—high-Q S circuits are said to be more
selective. There is a certain range of frequencies at which the current is near
its maximum(impedance is at its minimum). The frequencies corresponding to
1/ 2 or 0.707 times the maximum current are called the cut-off or half-power
frequencies f 1 and 2f , as
shown in figure
Figure :I vs f for a series resonant circuits
BW
0.707I MAX
I MAX =E/R
1f 2f
L211 Series Resonance and Time/Frequency Response of Passive Networks
-22-
The frequency range between 1f and 2f is called the bandwidth (BW) of the resonant
circuit, i.e., BW=f 2 -2f =
s
s
Q
f
The smaller the BW, the higher the circuit selectivity. For circuits where Q S >=10, the
resonant frequency sf approximately bisects the BW and the resonant curve is
symmetrical about sf .
Now consider the plots of the voltages across the resistor (VR
), the capacitor (V C )
versus the frequency (see figure 3.4.2).
From the figure, we know that V R
has exactly the same shape as the circuit current,
and is a maximum at resonance. What’s more, LV
and CV are equal at resonance since CL XX ,
however, they are not maximum at the resonant
frequency sf . In fact, CV reaches a maximum
Figure 3.4.2 V R , LV , CV ,I vs f for a series
resonance circuit
just before resonance and LV reaches a maximum just before resonance and V L reaches
a maximum just after resonance. In other words, the series R-L-C circuit is
predominantly capacitive from zero to sf , and predominantly inductive for any
frequency above sf , i.e.,
for f< sf , lVV c and
for f> sf , CL VV
The above characteristic about RLC circuit can be proved as below:
We treat capacitor first:
CLCLR
CE
LCR
CE
Z
XEV C
C
/2/1
/
)/1(
/1**
22242222
And then, let f( )= CLCLR /2/1 222422 . In order to get the CV maximum,
the f( ) gets minimum. Therefore,
L211 Series Resonance and Time/Frequency Response of Passive Networks
-23-
0)(d
d
f LRCL // 22 LRCL // 2 < CLS / ,
Which means that SMAXC f_f .
Similarly, it is easy to prove that sL ff max_
Time constant
When a capacitor (C) is connected to a dc voltage source like a battery, charge builds
upon its plates and the voltage across the plates increases until it equals the voltage (V)
of the battery. At any time (t), the charge (Q) on the capacitor plates is given by Q =CV.
The rate of voltage rise depends on the value of the capacitance and the resistance in
the circuit. Similarly, when a capacitor is discharged, the rate of voltage decay depends
on the same parameters.
Both charging and discharging times of a capacitor are characterized by a quantity
called the time constant τ, which is the product of the capacitance (C) and the resistance
(R), i.e. τ= RC.
Figure 1
When a capacitor is charged through a resistor by a dc voltage source by putting the switch
to position B in the figure, the charge in the capacitor and the voltage across the capacitor
increase with time. The voltage V as a function of time t is given by:
where the exponential e=2.718 is the base of natural logarithm and Vo is the voltage of
the source. The quantity τ=RC is called the time constant. The curve of the exponential
rise in voltage with time during the charging process is illustrated in Figure 2.
L211 Series Resonance and Time/Frequency Response of Passive Networks
-24-
At time t=τ=RC (one time constant), the voltage across the capacitor has grown to a value:
It will take an infinite amount of time for the capacitor to fully charge to its maximum
value. For practical purposes we will assume the after five time constants the capacitor is
fully charged.
When a fully charged capacitor is discharged through a resistor by putting the switch to
position A in Figure 1, the voltage across the capacitor decreases with time. The voltage
V as a function of time t is given by:
The exponential decay of the voltage with time is also illustrated in Figure 2. After a time
t=τ=RC (one time constant), the voltage across the capacitor has decreased to a value: