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ECE 206L

Lecture Notes

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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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DefinitionPeak 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 = IpsinθWhere: 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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ConversionRoot-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 10ECE 206L

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Resistors in Series

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Resistors in Parallel

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Measuring Voltage

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Total Voltage: VR1+VR2

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Voltage Dividers

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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

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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 deviceEx. : 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 in-

between. 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

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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 F

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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.

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ImpedanceDefinition

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

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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

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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

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dt

dILV L

L

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 resistancesSeries

Parallel

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L L L L N 1 2 . . .

1 1 1 1

1 2L L L L N . . .

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Inductor Impedance

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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

RIV RR

21 RRRT

12

11

1 RRRT

R

R

C

dt

dVCI C

C

12

11

1 CCCT

21 CCCT

circuitshort

circuitopen

dt

dILV L

L

L

21 LLLT

12

11

1 LLLT

circuitshort

circuitopen

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ImpedanceDefinition

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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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

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Function generator thinks it is putting out the same thing. Output is clearly different.

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Effect of Impedance on Circuit

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5050

50

inout

inout

VV

VV

The function generator has an output impedance of much less than 50Ω, so we can ignore it.

inout

inout

VV

VV

50101

1016

6

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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 Suffixespico p 10-12

nano n 10-9

micro (u) 10-6

milli m 10-3

Kilo k 103

Mega M (Meg) 106

Giga G 109

Tera T 1012

mk

km

MM

nG

Gn

11

11

11

mkk

ex 1.01

10

1

10

1.

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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.

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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.

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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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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

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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.

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To measure current, the circuit must be broken to allow theammeter to be connected in series

Ammeters must have a LOW resistance

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To measure potential difference (voltage), the circuit is not changed: the voltmeter is connected in parallel

Voltmeters must have a HIGH resistance

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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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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.

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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

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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 Circuitsqualitative description

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 Vs.  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).

C11uF

R1

1k

J1

Key = SpaceV112 V Delta Vc

B

A

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RC Circuitsqualitative description(continue)

C11uF

R1

1k

J1

Key = SpaceV112 V Delta Vc

B

A

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 Circuitsqualitative description(continue)

C11uF

R1

1k

J1

Key = SpaceV112 V Delta Vc

B

A

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).