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ECE 206L
Lecture Notes
ECE 206L
1
DC Current vs. AC Current
• Direct current (DC) flows in one direction the circuit.
• Alternating current (AC) flows first in one direction
then in the opposite direction.
• The same definitions apply to alternating voltage (AC
voltage):
• DC voltage has a fixed polarity.
• AC voltage switches polarity back and forth.
• There are numerous sources of DC and AC current and
voltage. However: Sources of DC are commonly shown
as a cell or battery, and for the AC current: Generators
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The Sinusoidal AC Waveform
• The most common AC waveform is a sine (or sinusoidal)
waveform.
• The vertical axis represents the amplitude of the AC current
or voltage, in amperes or volts.
• The horizontal axis represents the angular displacement of
the waveform. The units can be degrees or radians.
• The sine waveform is accurately represented by the sine
function of plane trigonometry: y = rsinq
where: y = the instantaneous amplitude
r = the maximum amplitude
q = the horizontal displacement
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Definition
Peak and Peak-to-Peak Voltage
•
•
•
•
•
Peak and peak-to-peak values are most often used when measuring the
amplitude of ac waveforms directly from an oscilloscope display.
Peak voltage is the voltage measured from the baseline of an ac waveform to
its maximum, or peak, level. Unit: Volts peak (Vp)
Symbol: Vp
For a typical sinusoidal waveform, the positive peak voltage is equal to the
negative peak voltage. Peak voltages are expressed without a + or - sign.
Peak-to-peak voltage is the voltage measured from the maximum positive
level to the maximum negative level. Unit: Volts peak-to-peak (Vp-p)
Symbol: Vp-p
For a typical sinusoidal waveform, the peak-to-peak voltage is equal to 2 times
the peak voltage. Peak-to-peak voltages are expressed without a + or - sign .
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Conversion
• Convert Vp to Vp-p: Vp-p = 2 Vp
• Convert Vp-p to Vp: Vp =0.5Vp-p
• What is the peak-to-peak value of a sinusoidal
waveform that has a peak value of 10 V?
• What is the peak value of a sine wave that has
a peak-to-peak value of 240 V?
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Instantaneous Current and Voltage
i = Ipsinθ
Where:
i = instantaneous current in amperes
Ip= the maximum, or peak, current in amperes
θ = the angular displacement in degrees or radians
v = Vpsinθ
Where:
v = instantaneous voltage in volts
Vp= the maximum, or peak, voltage in volts
θ = the angular displacement in degrees or radians
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Average Voltage
• Average voltage is the average value of all the values for one
half-cycle of the waveform. Unit: Volts average (Vave)
Symbol: Vave
• The average voltage of a sinusoidal waveform is equal to
0.637 times its peak value.
• Vave = 0.637Vp
• The average voltage is determined from just one half-cycle of
the waveform because the average value of a full cycle is
zero. Average voltages are expressed without a + or - sign
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Average Voltage
• Convert Vp to Vave: Vave = 0.637Vp
• Convert Vave to Vp: Vp =1.57Vave
• Determine the average value of a waveform
that measured 16 Vp. Ans: 10.2 Vave
• What is the peak value of a waveform that has
an average value of 22.4 V? Ans: 35.1 Vp
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Root-Mean-Square (RMS) Voltage
• AC levels are assumed to be expressed as RMS values unless clearly
specified otherwise.
• RMS voltage is the amount of dc voltage that is required for
producing the same amount of power as the ac waveform. Unit:
Volts (V)
Symbol: Vrms
• The RMS voltage of a sinusoidal waveform is equal to 0.707 times
its peak value.
• Vrms = 0.707Vp
• In a dc circuit, applying 2 V to a 1 W resistance produces 4 W of
power. In an ac circuit, applying 2 Vrms to a 1 W resistance
produces 4 W of power.
RMS voltages are expressed without a + or - sign.
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Conversion
Root-Mean-Square (RMS) Voltage
• Convert Vp to Vrms: Vrms = 0.707Vp
• Convert Vrms to Vp : Vp = 1.414Vrms
• Determine the RMS value of a waveform that
measures 15 Vp. Ans: 10.6 V
• Determine the peak value of 120 V.
.
(Assume 120 V is in RMS) Ans: 170 Vp
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Resistors in Series
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Resistors in Parallel
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Measuring Voltage
Total Voltage: VR1+VR2
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Voltage Dividers
The voltage is divided up in such that it is
proportional to the resistances of the resistors
in a series circuit.
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Statistical Evaluation of
Measurement Data and Errors
• Average or mean value of a set of measurement
• Deviation from the average value
• Average value of the deviation
• Standard deviation(from the concept of RMS)
• Probability of error size in one observation
15
The Decibel (dB)
The decibel, or dB, is a means of
expressing the gain of an active device
(such as an amplifier) or the loss in a
passive device (such as an attenuator or
length of cable). It is simply the ratio of
output to input expressed in logarithmic
form. The decibel was developed by the
telephone company(Bel, to express the
gain or loss in telephone transmission
systems.
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Calculating the Decibel (dB)
•
•
•
Now, imagine for a moment what it would be like to calculate the total gain of a string of amplifiers.
It would be a cumbersome task at best, and especially so if there were portions of the cascade
which were lossy and reduced the total gain, thereby requiring division as well as multiplication.
log (A x B) = log A + log B
log (A/B) = log A - log B
•
Using the Decibel:
G = 10 log (Po/Pi) ,
Where:
G = Gain in dB
Po = Power output from the device
Pi = Power input to the device
Ex. : A length of coaxial transmission line is being fed with 150 watts from a transmitter, but the
power measured at the output end of the line is only 112 watts. What is the line loss in dB?
G = 10 log (112/150)
G = 10 log 0.747
G = 10 (-0.127)
G = -1.27 dB
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Capacitors
• Capacitors consist of two plates with a dielectric material inbetween. When a potential difference is placed across the
plates, a charge builds up until it is large enough to cause a
discharge across the plates through the material.
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Reading Capacitors
Larger capacitors have the number of microfarads
written on them directly. Smaller capacitors use a
code based on the number of picofarads. We
generally use microfarads, so…
XYZ = XY * 10Z * 10-6 mF
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Capacitors in Series
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Capacitors in Parallel
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Impedance vs. Resistance
• Resistance is a property of a material that causes a
reduction in the rate of flow of electrons.
• Impedance is the reduction in the rate of flow of
electrons caused by the material (resistance) AND
other the properties of the component involved
(reactance).
• Resistors have no reactance. So the impedance of a
resistor is equal to its resistance only.
• Reactance varies with the frequency of the input.
Resistance remains the same at all frequencies.
• Both impedance and resistance are measured in
ohms.
22
Impedance
Definition
A general measure of how a component or group of
components pushes against the current flowing
through it.
• Impedance = resistance + reactance
• Impedance is used to refer to the behavior of circuits
with resistors, capacitors and other components.
• When we consider components in a theoretical
circuit diagram, the impedance of inductors and
capacitors is their reactance only. Any resistance is
modeled separately as a resistor. So theoretical
capacitors and inductors have impedance, but no
resistance.
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Capacitor Impedance
Real capacitors have effectively no resistance, so
impedance is reactance for all capacitors.
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What is Reactance
•
Reactance is the property of resisting or impeding the flow of ac current or ac
voltage in inductors and capacitors. Note particularly we speak of alternating
current only ac, which expression includes audio af and radio frequencies rf. NOT
direct current dc.
Inductive Reactance
• When ac current flows through an inductance a back emf or voltage develops
opposing any change in the initial current. This opposition or impedance to a
change in current flow is measured in terms of inductive reactance. 2 * pi * f * L
where: 2 * pi = 6.2832; f = frequency in hertz and L = inductance in Henries
Capacitive Reactance
• When ac voltage flows through a capacitance an opposing change in the initial
voltage occurs, this opposition or impedance to a change in voltage is measured in
terms of capacitive reactance. 1 / (2 * pi * f * C)
where: 2 * pi = 6.2832; f = frequency in hertz and C = capacitance in Farads
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Some examples of Reactance
• What reactance does a 6.8 uH inductor present at 7 Mhz?
Using the formula above we get:
• 2 * pi * f * L
• where: 2 * pi = 6.2832; f = 7 X 10+6 Hz and L = 6.8 X -6 Henries
• Answer: = 299 ohms
• What reactance does a 33 pF capacitor present at 7 Mhz?
Using the formula above we get:
• 1 / (2 * pi * f * C)
• where: 2 * pi = 6.2832; f = 7 X 10+6 Hz and C = 33 X -12 Farads
• Answer: = 689 ohms
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Inductors
An inductor is a coil of wire through which a
current is passed. The current can be either
AC or DC.
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Inductors
dI L
VL  L
dt
This generates a magnetic field, which
induces a voltage proportional to
the rate of change of the current.
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Combining Inductors
Inductances add like resistances
Series
L  L1  L2 ... LN
Parallel
1 1
1
1


...
L L1 L2
LN
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Inductor Impedance
Real inductors always have a small resistance (that
is not shown in these circuits). The impedance of
the theoretical inductor shown is only its
reactance.
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Comparison of Components
C
VR  I R R
IC  C
dVC
dt
L
VL  L
dI L
dt
RT  R1  R2 CT1  C11  C21 LT  L1  L2
RT1  R11  R21 CT  C1  C2 LT1  L11  L21
R
open circuit
R
short circuit open circuit
short circuit
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Impedance
Definition
A general measure of how a component or group of
components pushes against the current flowing
through it.
• Impedance = resistance + reactance
• Impedance is used to refer to the behavior of circuits
with resistors, capacitors and other components.
• When we consider components in a theoretical
circuit diagram, the impedance of inductors and
capacitors is their reactance only. Any resistance is
modeled separately as a resistor. So theoretical
capacitors and inductors have impedance, but no
resistance.
32
Equipment Impedances
•
•
•
Each measuring device changes the circuit when
you use it.
The impedance of the device helps you understand
how much.
Device Impedances
–
–
–
–
–
Function Generator: 50 ohms
Scope: 1Meg ohms
DMM (DC voltage): 10Meg ohms
DMM (AC voltage): 1Meg ohms
DMM (DC current): 5 ohms (negligible)
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Effect of Impedance on Circuit
Function generator thinks it is putting out the
same thing.
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Output is clearly different.
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Effect of Impedance on Circuit
Vout
Vout
Vout
Vout
50
Vin 

50  50
Vin

2
1106
Vin 

6
110  50
 Vin
The function generator has an output
impedance of much less than 50Ω, so we can
ignore it.
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Kirchoff’s Laws
sum of voltages in any
loop is zero
sum of currents entering a
junction is the same as the sum
of the currents leaving a
junction
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Circuit Analysis (Combination Method)
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SI Suffixes
pico
p
10-12
nano
n
10-9
micro
m (u)
10-6
milli
m
10-3
Kilo
k
103
Mega
M (Meg)
106
Giga
G
109
Tera
T
1012
1
1
n
G
G
n
1
1
m
M 
M
m
1
1
m
k
k
m
ex.
1
1 1

 0.1m
10k 10 k
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Oscilloscope Tutorial
• The oscilloscope is basically a graph-displaying
device
• It draws a graph of an electrical signal.
• In most applications the graph shows how
signals change over time:
– the vertical (Y) axis represents voltage
– the horizontal (X) axis represents time.
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Oscilloscopes
Horizontal sweeps at a constant rate. Vertical plates are
attached to an external voltage, the signal you attach to the
scope.
40
Cathode Ray Tubes
Variation in potential difference (voltage)
placed on plates causes electron beam to
bend different amounts.
“Sweep” refers to refreshing repeatedly at a
fixed rate.
41
42
Scope (Con’t)
• This simple graph can tell you many things about a signal:
–
–
–
–
–
You can determine the time and voltage values of a signal.
You can calculate the frequency of an oscillating signal.
You can see the "moving parts" of a circuit represented by the signal.
You can tell if a malfunctioning component is distorting the signal.
You can find out how much of a signal is direct current (DC) or
alternating current (AC).
– You can tell how much of the signal is noise and whether the noise is
changing with time.
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44
How does an Analog Scope work?
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How does a Digital Scope work?
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Triggering Stabilizes a Repeating Waveform
47
Waveform shapes tell you a great deal about a signal
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If a signal repeats, it has a frequency. The frequency is measured in Hertz (Hz) and
equals the number of times the signal repeats itself in one second
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Voltage, Current, & Phase
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Performance Terms
•
•
•
•
•
•
•
•
Bandwidth
– The bandwidth specification tells you the frequency range the oscilloscope accurately measures.
Rise Time
– Rise time may be a more appropriate performance consideration when you expect to measure
pulses and steps. An oscilloscope cannot accurately display pulses with rise times faster than the
specified rise time of the oscilloscope.
Vertical Sensitivity
– The vertical sensitivity indicates how much the vertical amplifier can amplify a weak signal. Vertical
sensitivity is usually given in millivolts (mV) per division.
Sweep Speed
– For analog oscilloscopes, this specification indicates how fast the trace can sweep across the screen,
allowing you to see fine details. The fastest sweep speed of an oscilloscope is usually given in
nanoseconds/div.
Gain Accuracy
– The gain accuracy indicates how accurately the vertical system attenuates or amplifies a signal.
Time Base or Horizontal Accuracy
– The time base or horizontal accuracy indicates how accurately the horizontal system displays the
timing of a signal.
Sample Rate
– On digital oscilloscopes, the sampling rate indicates how many samples per second the ADC can
acquire. Maximum sample rates are usually given in megasamples per second (MS/s). The faster the
oscilloscope can sample, the more accurately it can represent fine details in a fast signal..
ADC Resolution (Or Vertical Resolution)
– The resolution, in bits, of the ADC indicates how precisely it can turn input voltages into digital
values.
Record Length
– The record length of a digital oscilloscope indicates how many waveform points the oscilloscope is
able to acquire for one waveform record.
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Grounding
• Proper grounding is an important step when setting up to take
measurements.
• Properly grounding the oscilloscope protects you from a hazardous shock
and protects your circuits from damage.
• Grounding the oscilloscope is necessary for safety. If a high voltage
contacts the case of an ungrounded oscilloscope, any part of the case,
including knobs that appear insulated, it can give you a shock. However,
with a properly grounded oscilloscope, the current travels through the
grounding path to earth ground rather than through you to earth ground.
• To ground the oscilloscope means to connect it to an electrically neutral
reference point (such as earth ground). Ground your oscilloscope by
plugging its three-pronged power cord into an outlet grounded to earth
ground.
• Grounding is also necessary for taking accurate measurements with your
oscilloscope. The oscilloscope needs to share the same ground as any
circuits you are testing.
• Some oscilloscopes do not require the separate connection to earth
ground. These oscilloscopes have insulated cases and controls, which
keeps any possible shock hazard away from the user.
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Scope Probes
Most passive probes have some degree of attenuation factor, such as 10X,
100X, and so on. By convention, attenuation factors, such as for the 10X
attenuator probe, have the X after the factor.
In contrast, magnification factors like X10 have the X first
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Vertical Controls
• Position and Volts per Division
– The vertical position control lets you move the waveform
up or down to exactly where you want it on the screen.
– The volts per division (usually written volts/div) setting
varies the size of the waveform on the screen. A good
general purpose oscilloscope can accurately display signal
levels from about 4 millivolts to 40 volts.
– Often the volts/div scale has either a variable gain or a fine
gain control for scaling a displayed signal to a certain
number of divisions.
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Input Coupling
• Coupling means the method used to connect
an electrical signal from one circuit to another.
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Horizontal Controls
• Position and Seconds per Division
– The horizontal position control moves the waveform from
left and right to exactly where you want it on the screen.
– The seconds per division (usually written as sec/div)
setting lets you select the rate at which the waveform is
drawn across the screen (also known as the time base
setting or sweep speed). This setting is a scale factor. For
example, if the setting is 1 ms, each horizontal division
represents 1 ms and the total screen width represents 10
ms (ten divisions). Changing the sec/div setting lets you
look at longer or shorter time intervals of the input signal.
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Trigger Position
•
The trigger position control may be located in the horizontal control section of
your oscilloscope. It actually represents "the horizontal position of the trigger in
the waveform record." Horizontal trigger position control is only available on
digital oscilloscopes.
•
Varying the horizontal trigger position allows you to capture what a signal did
before a trigger event (called pretrigger viewing).
•
Digital oscilloscopes can provide pretrigger viewing because they constantly
process the input signal whether a trigger has been received or not. A steady
stream of data flows through the oscilloscope; the trigger merely tells the
oscilloscope to save the present data in memory. I
•
n contrast, analog oscilloscopes only display the signal after receiving the trigger.
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Trigger Controls (con’t)
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Pulse and Rise Time Measurements
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Multimeter tutorial
• A meter is a measuring instrument. An
ammeter measures current, a voltmeter
measures the potential difference (voltage)
between two points, and an ohmmeter
measures resistance.
• A multimeter combines these functions, and
possibly some additional ones as well, into a
single instrument.
60
To measure current, the circuit must be broken to allow
the
ammeter to be connected in series
Ammeters must have a LOW resistance
61
To measure potential difference (voltage), the circuit is not
changed: the voltmeter is connected in parallel
Voltmeters must have a HIGH resistance
62
To measure resistance, the component must be removed
from the circuit altogether
Ohmmeters work by passing a current through the
component being tested
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Digital Multimeters
Digital meters give an output in numbers, usually on a
liquid crystal display.
Most modern multimeters are digital and traditional
analogue types are destined to become obsolete.
Digital multimeters come in a wide range of sizes and
capability. Everything from simple 3 ½ digit auto ranging
pocket meters to larger 8 ½ digit bench model with
operator or computer (IEEE488 compatible) settable range
selection
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65
Function Generator
• An electronic instrument that generates
various waveforms such as
– Sine wave
– Square wave
– Pulse trains
– Sawtooth
• The amplitude, DC offset, frequency are
adjustable.
66
Function Generators (con’t)
• Like multimeters there is a wide variety of
device offering various
– Amplitude characteristics
– Bandwidth
– Adjustments of rise and fall times
– Modulation capability (AM, FM, Pulse, etc.)
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Power Supply
• This is the device that transfers electric power from a
source to a load using electronic circuits.
• Typical application of power supplies is to convert utility's
AC input power to a regulated voltage(s) required for
electronic equipment.
• Depending on the mode of operation of power
semiconductors PS can be linear or switching.
• In a switched-mode power supply, or SMPS power
handling electronic components are continuously
switching on and off with high frequency in order to
provide the transfer of electric energy. By varying duty
cycle, frequency or a phase of these transitions an output
parameter (such as output voltage) is controlled. Typical
frequency range of SMPS is from 20 kHz to several MHz.
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Power Supply (con’t)
• Power supplies like many of the other electronic
instruments, come in many varieties with a wide
range of capabilities:
• Parameters that are Power Supply specific
include:
–
–
–
–
–
–
Voltage levels
Current
Regulation
Protection
Output impedance
Noise (ripple)
• It’s the designer (or researcher) responsibility to
identify the characteristics required.
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Oscilloscope
70
Oscilloscope(continue)
DEMO…….Lab3a
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Capacitance (continue from before)
• A capacitor simply consists of two conductors
which are electrically isolated from one
another. This means that no current can
readily flow from one conductor to the other.
• the units of the capacitance must equal one
coulomb per volt, which is defined to be one
farad, F:
– One Farad= one(coulomb/volts)
– 1F=1C/V
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RC Circuits
qualitative description
V1
12 V
B J1
R1
A
1k
Key = Space
C1 Delta Vc
1uF
assume the switch is thrown to position B at the time t = 0. When the
switch is at the position B the circuit consists of the single loop which
contains, starting at point B and moving around the circuit clockwise, the
resistor R, the capacitor C, and finally the voltage source DVs. In this
configuration the voltage source attempts to push charge around the circuit
in a clockwise direction (remember that the power source tries to push
current out of its positive terminal).
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RC Circuits
qualitative description(continue)
B J1
V1
12 V
A
Key = Space
R1
1k
C1 Delta Vc
1uF
After some time at position B, we will throw the switch to position A. (The
time since the switch was thrown to position A is called a new time t’
. The prime “ ’ ” on a symbol is used to denote the fact that this is the
value of the quantity under consideration since the switch was thrown to
position A. It therefore follows that the switch is thrown to the position A
at the time t’ = 0.) After the switch has been thrown to position A the
circuit consists solely of the resistor and the capacitor, with no voltage
source. In this case there is no external energy being used to move
charges around the circuit loop.
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RC Circuits
qualitative description(continue)
B J1
V1
12 V
A
Key = Space
R1
1k
C1 Delta Vc
1uF
The behavior of the voltage across the capacitor as a function of time and the
current around the circuit (and in particular, through the resistor) as a function
of time for the case in which the switch has been thrown to position B. (This is
the case of the charging capacitor.) Then after that is the case in which the
switch has been thrown to position A (the case of the discharging capacitor).
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