EE250Lab7transients_2a

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University of AlbertaDepartment of Electrical and Computer Engineering

EE 250 Laboratory

Experiment # ! "ransient Analysis

b$ective%

To study the transient responses of series RC, RL, and RLC circuits following theapplication of a step voltage.

&ntroduction%

The transient response is the fluctuation in current and voltage in a circuit (after theapplication of a step voltage or current) before it settles down to its steady state. This lab

will focus on series RL (resistor-inductor), RC (resistor-capacitor), and RLC (resistor-inductor-capacitor) circuits to demonstrate transient analysis.

"'eory%

"ransient (esponse of Circuit Elements%

A) (esistors% s has been studied before, the application of a voltage * to aresistor (with resistance ( ohms), results in a current &, according to theformula!

I = V

R

"""""""""""..E+uation ),

The current response to voltage change is instantaneous# a resistor has notransient response.

-) &nductors% change in voltage across an inductor (with inductance L$enrys) does not result in an instantaneous change in the current through it.The i-v relationship is described with the e%uation!

v= L di

dt """"""."""".E+uation )2

This relationship implies that the voltage across an inductor approaches &ero

as the current in the circuit reaches a steady value. This means that in a 'Ccircuit, an inductor will eventually act lie a short circuit.

C) Capacitors% The transient response of a capacitor is such that it resistsinstantaneous change in the voltage across it. ts i-v relationship is described by!

i=C dv

dt """"""""""E+uation ).

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This implies that as the voltage across the capacitor reaches a steady value,the current through it approaches &ero. n other words, a capacitor eventuallyacts lie an open circuit in a 'C circuit.

eries Combinations of Circuit Elements% olving the circuits shown below involves the solution of first and second orderdifferential e%uations. /nly the solutions have been included, as that is all that isneeded for the lab.

A) eries (C Circuits%

1igure ),% A eries (C Circuit

f the switch in this circuit was initially open, and then closed at time t0, thecurrent in this circuit is!

i t = I O

e0p −t τ ""..""""".E+uation )3

where! I O=

V O

R 1 the initial current in the circuit

2 1 RC 1 the time constant for the circuit

nother definition of 2 is obtained by setting t 4 into E+uation )3. 'oingso gives i(2) 1 /3(45e). The time constant of an RC circuit is the timere%uired for the current in the circuit to fall to 45e of its initial value.

-) eries (L Circuits

1igure )2% A eries (L Circuit

f the switch in this circuit is initially open, and then closed at time t0, thecurrent in this circuit will be described as!

i t = I O 4−e−t / τ """""""..E+uation )5

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where! I O=

V O

R 1 the limiting value of the current in the circuit

τ = L

R 1 the time constant for the circuit

2 can also be described by noting what happens when t 4 is substituted intoE+uation )5. 'oing so gives i(2) 1 /3(4-45e). n other words, 2 is the timere%uired in an RL circuit for the current to grow to (4-45e) of its limitingvalue.

C) eries (LC Circuits

1igure ).% A eries (LC Circuit

n theory, there are three cases for the way a series RLC circuit will respondwhen the switch is closed at time t0. n this lab, only the underdamped casewill be dealt with. 6or this case, the current in the circuit is described by!

i=V

O

ωd L

e0p−αt sin ωd

t """""""E+uation )

where! ωd = ω,

+ − α+ #

ω, =

4

LC and α =

R

+L

E+uation ) describes a current that is both fluctuating and deteriorating, asshown in 1igure )3!

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1igure )3% "'e Current in an Underdamped eries (LC Circuit

The current in the circuit oscillates due to the sine component in E+uation), but the ma0imum value it can reach is decaying due to the negativee0ponential. The 7envelope8 that the current must fall within is described by!

i=±V

O

ωd L e0p−αt or ∣i∣ =

V O

ωd L e0p−αt

The %uantity 9 is referred to as the time constant of the envelope. t is

determined by taing the natural logarithm of both sides of the abovee%uation!

ln∣i∣ = ln V Oωd L − αt """"..""E+uation )

which is a linear e%uation. f a graph of ln∣i∣ vs . t is plotted, its slope will be :9.

(esonance in eries (LC Circuits%The resonant fre%uency of an RLC Circuit is the fre%uency at which current isa ma0imum. This occurs when the impedance of the capacitor e%uals theimpedance of the inductor.

∣ Z C ∣ = ∣ Z

L∣

∣ 4 jωO

C ∣=∣ jωO L∣t this fre%uency, the impedance of the capacitor (negative) e0actly cancelsthe impedance of the inductor (positive). The only impedance felt by the



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source is the resistor. t follows that the current at this fre%uency will be in phase with the voltage source, and at a ma0imum magnitude.

/ractical Considerations for &mplementing t'ese Circuits%

The input voltage described in all the above has been described as a step voltage. This ismost easily obtained by using the function generator set to deliver s%uare waves. The rateof delivery of the step voltages is set to a time long enough that the circuit is allowed toreach a steady state before the voltage changes again. This effectively simulates a voltagesource in series with an on-off switch.

(eal 6orld *oltage ource% real world voltage source is not an ideal voltage source.Real voltage sources will produce very large short circuit currents but will never produceinfinite currents when shorted. The real world imperfections or limitations of the voltagesource limits the available current. This current limiting effect of the real world source isoften described as the internal source impedance. real voltage source can be modelledas an ideal voltage source in series 7it' an internal source impedance. f the load

impedance is much greater than the internal source impedance, then the source impedancecan be ignored. $owever, if the load impedance is comparable to the internal sourceimpedance, it may not be ignored. "'e internal source impedance of t'e functiongenerator is 508 resistive)

ll of the theory above described the transient current response. The measurement ofcurrent can be achieved with the oscilloscope in these circuits, because all are seriescircuits, and all have a resistor in them. To get the current, measure the voltage across theresistors, then divide by their resistance.

Experimental /rocedure%

Throughout this e0periment, results should be recorded in the report section of thishandout.

/art ,% eries (C CircuitThe circuit shown in 1igure )5 will be constructed to illustrate the transient currentresponse, and the transient voltage response across the capacitor, to an applied voltagesource.

1igure )5



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,), et Up t'e Circuit4.) Connect the circuit as shown in 1igure )5, with channels 4 and + of the

oscilloscope set up to measure the input and output voltages, respectively.+.) et the function generator to deliver a s%uare wave!

a) ;ush in the button to select s%uare wave. b)


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2), et up t'e Circuit4.) Connect the circuit as shown in 1igure ), with channels 4 and + of the

oscilloscope set up to measure the input and output voltages, respectively.+.) et the function generator to deliver a s%uare wave, the same way as in /art ,.?.) 'isplay the input and output voltages on the oscilloscope.

Due to t'e load on t'e function generator? t'e input voltage 7ill appear similarto t'e 7aveform s'o7n in 1igure )) (emember to include t'e sourceimpedance in t'e ( for calculations)

1igure )% 6aveform of t'e Loaded 1unction @enerator

2)2 "ransient (eaction bservations4.) 'isplay the function generator and output voltage on channels 4 and + of the

oscilloscope. et the voltage and time scales for ma0imum resolution.+.) /bserve and record the input and output waveforms on the grid provided.?.) Tabulate the values of the output voltage as a function of time for one of the

e0ponential curves on the oscilloscope. Dote the similar form to part ,)..B.) 2 can be measured from the oscilloscope as follows!

a) Line the forward edge of a s%uare pulse with s on the display. b) =easure the limiting voltage across the resistor, @o.c) ince i(2) 1 o3(4-45e), v(2) 1 @o3(4-45e). Calculate v(2).d) 6ind the first point past the &ero mar where @1v(2).e)


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1igure )

.), et Up t'e Circuit4.) Connect the circuit shown in 1igure ), with channels 4 and + of the

oscilloscope set to measure input voltage and output voltage, respectively.+.) et the function generator to deliver a s%uare wave, the same way as in /art ,.

.)2 "ransient (eaction bservations

4.) 'isplay the input and output voltages on the oscilloscope. et the voltage andtime scales for ma0imum resolution. dust the time scale so that a complete7ringing8 waveform is displayed on the screen.

+.) /bserve and record the input and output waveforms.?.) =easure at least three positive and three negative peas of the waveform, and the

time at which each pea occurs. =easure the times at which the current is &eroand determine the fre%uency f d of these damped oscillations.

.). "'e (esonant 1re+uency4.) witch the function generator to give a sine wave output. dust the fre%uency

until the current (represented by the voltage across the resistor) is in phase with

the input voltage. Under t'is condition? t'e current is a maximum)+.) =easure this fre%uency with the '==. This is the resonant fre+uency f o of thecircuit.

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Experiment # ! "ransient Analysis/re;lab Assignment

Read the lab over and answer the following %uestions. ll wor must be handed in with

the pre-lab hand in sheet, and your answers should be copied into the lab data sheet for useduring the lab.

4.) Refer to 1igures )5 and ) in the lab. Calculate the time constant for each circuit.

+.) Refer to 1igure ) in the lab. Calculate the resonant fre%uency for this circuit.

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Experiment # ! "ransient Analysis/ost Lab uestions

nswer the %uestions below, using measured values from the lab. how at least one

sample calculation per %uestion, and give all answers to + decimal places. (emember toinclude t'e source impedance in cases 7'ere it is comparable to t'e circuit resistance)

/art ,% A eries (C Circuit

4.) 'raw a graph showing the time relationship between the input signal and the outputas a function of time for part ,)2)..

+.) 'etermine the time constant from the graph produced in %uestion 4.).

?.) To the theoretical value of 2 determined in the pre-lab, compare!a) The value determined from the graph above.

b) The value determined from the oscilloscope in part ,)2)3.c) The value determined from the oscilloscope in part ,)..

B.) 'erive the e%uation for the output voltage across the capacitor @C(t) for the circuit of part ,).. 'oes this function describe the waveform seenF Test by graphing measuredand calculated values.

/art 2% A eries (L Circuit

G.)


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4.) Compare the fre%uency f d of the damped oscillations as measured in .)2). with theresonant fre%uency f o, measured in .).)2. how why you would not e0pect there to beany significant difference between these two fre%uencies.

44.) The resonant fre%uency, f o, of the series RLC circuit is given by!

f O = 4

+ LC or ωO =

4

LC


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EE 250 Laboratory Pre- Lab Report

Experiment #7 – Transient Analysis

Name:

Lab Section: Lab Date:

Nuestion 4!

2RC 2RL

Nuestion +!

f o

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EE 250 Laboratory Lab (eport

Experiment # ! "ransient Analysis TT/D D


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4.+.?)Tabulation of @/


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+.+.+) Qraph of the nput and /utput Maveforms (not necessarily to scale)

+.+.?) Tabulation of @/


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+.?.B) Time constant of the RL circuit

2RL

?.+.+) Qraph of the input and output waveforms. (Dot necessarily to scale)

?.+.?) =easurement of the positive and negative peas, and the &eros, of the outputwaveform.

?.?.+) The resonant fre%uency

f o

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