Physics Laboratory 2
last update: 2009. 8. 31
Exp. 7. Properties of R, L, C Circuits
- Transient, AC and Resonance Properties -
Purpose of Experiment
In electric circuit active device -diode, transistor or electric field effect transistor- made from semiconductor element is always used, but there is another essential device called passive device. The criterion between active and passive device is whether additional energy input to do function is needed or not. A trio of passive circuit are resistance(R), inductor(L), and capacitance(C). They have different DC or AC characteristic each other. So we can do various function using these things. In this experiment, using them comprehend where and why they are used. Especially, investigate of resonance in RLC circuit and compare with resonance in mechanical system.
Outline of Experiment
Investigate DC characteristic of RLC circuit
@ At first, investigate a characteristic of ideal RLC circuit.
@ Next investigate a real circuit. And, comprehend the internal resistance of ampermeter and voltmeter used electrical measurement.
Investigate AC characteristic of RLC circuit
@ At first, investigate a characteristic of ideal RLC circuit.
Experimental Method
These equipments are prepared in the laboratory. (Parentheses mean the number of them.)
Function Generator(source)
Resistance box(1)
Inductor(5H)
Capacitor
RLC box(1)
Analog-Digital converter(1)
Digital-Analog converter(1)
Input terminal(1)
Computer(1)
Wire with tweezer(1)
BNC cable(2, common)
Digital multimeter(1)
Digital LCR meter(1, common)
If you need more stuff, inquire to your teaching assistant or experiment preparation room (19-114), or prepare yourself.
In this experiment, instead of using AC voltmeter, amperemeter, DC voltmeter and amperemeter, we use Analog-Digital converter to observe change of voltage varying time between two point of circuit by computer. And using Digital-Analog converter, we could make any shape of wave and use this as source. So we can observe a response of RLC DC circuit corresponding arbitrary shape of wave.
Before experiment, think sufficiently about the role of switch on the RLC box.
1)Character of DC current in RC parallel circuit.
Observe the voltage charged in capacitor and current in the circuit.
- From output of Function Generator, connect ground to the GND in input terminal, signal to the #2. Then we can check the signal from function generator(i.e. reference). Using T-BNC Connector, connect output from Function generator to the RLC circuit. In other words, Function generator becomes power source of RLC circuit. Measuring signal of RLC circuit(voltage difference with ground), use #1 as measuring input.
- Seeing voltage drop by resistance, you should have connected more than two resistors parallel.
- Connect only resistance(R=300k) and capacitor(C=0.01) from RLC box(control resistance from resistance box)
(cf : R, C value can be different each box. If you have doubt, although values are recorded on the box, you can measure from digital multimeter after experiment)
- Give pulse wave as shape from Function generator. Turn on the computer and monitor, run RLC measuring program.
- Put [start] and decide data point and measuring time. Measuring time, data point and output frequency of Function Generator must be sufficiently decided to get better data.(cf : Getting better data, sufficiently adjustment of resistance at each experiment is important, set higher time and # of data from RLC measuring program and lower frequency of function from Function Generator). Although it can be different with setting, getting better data set time and # of data to be 1000, 1000, and frequency of Function Generator to be 4~6 * 10Hz, resistance to be more than 1000k(maximum value).(If you set time and # of data to be 300, 300, getting better data set frequency 12 * 100Hz, and resistance 300).
- Set gain 1 except the case which signal is too small to see.(For example, wave shape is seen 0.01V because the voltage difference from resistance is too big, although wave shape is peak to peak 2V sine wave, it can be seen as line. To magnify the amplitude of wave, set the gain higher. But, reference wave shape doesn’t magnified.)
- Do experiment what you want. Below picture is response of RC circuit when observe the voltage drop of capacitance.
- Put the Save in the File menu and save the data. This data can be loaded in excel.
- Analyzing wave shape, find the charge time constant. And compare with theoretical value solved by C and R value measured digital multimeter. Are they same? If not, what is reason? Varying R value in resistance box, measure the change of time constant.
- Think how we can measure character of R, L, C device separately, varying position of ground and power source. How can we measure voltage drop of R in RC circuit?
2)Character of DC current in RL parallel circuit.
Investigate the change of shape of voltage and current in inductor when DC voltage is applied.
- Connect only resistance(R=2~3k) and inductor(L=5) from RLC box parallel(cf : Value of L can be different. Measure the common digital LCR meter after experiment. Adjustment resistance in the resistance box).
- In the same way, choose sufficiently frequency of wave(4~6 * 10Hz), and watch signal. Getting better data set time and data as 1000, 1000.(If time and data is 300, 300, getting better data set frequency and resistance 12 * 100Hz, 3k)
- Neglect the change shown first – voltage change before delay time. Below picture shows voltage drop of R in RL circuit.
- Save the data and analyze from excel.
- Analyzing wave shape, find the charge time constant. And compare with theoretical value of time constant solved by L and R value measured digital LCR meter. Are they same? If not, what is reason? Varying R value in resistance box, measure the change of time constant.(cf : Think why connect parallel resistance box and resistance, not series.)
- How can we observe voltage drop of L?
What do you know from? And what do you say the relation between electromotive force and currents?
3)Character of AC current in RC parallel circuit.
Observe the relation between currents and voltage in capacitor applied by AC voltage.
- Do the experiment same way. But you should give the wave shape from Function generator as Sine wave. Getting better data, set the frequency sufficiently(Although it can be different, getting better data set the resistance 1000k).
- Save the data and analyzing using excel.
- Find the phase difference of voltage and current in RC circuit, impedance Z and reactance X analyzing wave shape.
- Compare the theoretical data respectively.
- Observe the change of wave shape varying R in resistance box and frequency of AC voltage.
4)Character of AC current in RL parallel circuit.
Observe the relation between currents and voltage in inductor applied by AC voltage.
- Do the experiment same way. But you should give the wave shape from Function generator as Sine wave. Getting better data, set the frequency sufficiently(Although it can be different, getting better data set the resistance 3k).
- Save the data and analyzing using excel.
- Find the phase difference of voltage and current in RL circuit, impedance Z and reactance X analyzing wave shape.
- Compare the theoretical data respectively.
- Observe the change of wave shape varying R in resistance box and frequency of AC voltage.
5)Character of AC current in RLC parallel circuit.
Observe the relation between currents and voltage in inductor applied by AC voltage and resonance.
- Do the experiment same way((Although it can be different, getting better data set the R=330k, C=0.022, L=5, frequency = 40*100Hz and time, data = 50, 50).
- Supply wave shape, and read the peak to peak voltage by voltage drop by resistance from monitor.
- Measure the change of Ipp varying frequency, and draw resonance curve.(cf : You can do 3. With digital multimeter. Set the digital multimeter in mV and choose AC putting DC/AC. Set time scale as 5s, click Run and read the voltage digital multimeter reads. This voltage is root-mean-square voltage)
- Connect terminal 1 and 2 in RLC box(short), re-measure the resonance curve. Observe the difference and explain reason.
- Compare with theoretical value and find reason if there is difference.
From this, find the self resistance of inductor in RLC box. Is it same with the value measured from digital multimeter?
(Character of DC current in RC parallel circuit.)
Check the change of voltage and currents of C after DC voltage is applied. And find the time constant and compare with theoretical value.
(Character of DC current in RL parallel circuit.)
Check the change of voltage and currents of L after DC voltage is applied. And find the time constant and compare with theoretical value.
(Character of AC current in RC parallel circuit.)
Measure the currents of the circuit after AC voltage is applied. And find the reactance X, impedance Z and phase difference and compare with theoretical value.
(Character of AC current in RL parallel circuit.)
Measure the currents of the circuit after AC voltage is applied. And find the reactance X, impedance Z and phase difference and compare with theoretical value.
(Character of AC current in RLC parallel circuit.)
Measure the currents of the circuit after AC voltage is applied. And find the phase difference, voltage in each part RLC, impedance Z and peak of currents I. Using these values draw resonance curve and find resonance frequency and Q factor. Compare with theoretical value also.
Background Theory
At first, consider ideal R, L, C circuit. In real, there is no exactly ideal R, L, C circuit. But in most case there is no problem considering them as ideal circuit. But you should careful because these characteristic of real can be more important.
The capacitor of capacitance C in Fig 1(a) is initially uncharged. If the circuit is complete, charge begins to flow between a capacitor plate and a battery on each side of the capacitor. Here we want to examine the charging process. In particular we want to know how the charge q(t) on the capacitor plates, the potential difference V(t) across the capacitor, and the current i(t) in the circuit vary with time during the charging process. We begin by applying Kirchoff’s law and we find
(1)
The charge Q and currents I are related by
(2)
Substituting this for I in (1) and rearranging, we find
(3)
This differential equation describes the time variation of the charge q on the capacitor. Integrating formula (3)
(5)
And
(6)
Or
(7a)
(7b)
Currents will be from eq (2)
(8)
The changing pattern is given in fig(2). The product RC has the dimensions of time. The product RC is called the capacitive time constant of the circuit and is represented with the symbol . If (타우) increases, change about time becomes more slow. When t = ,
(9a)
(9b)
Maximum value of the charged Q and currents I are
(10a)
(10b)
After sufficiently time applying DC voltage e, currents becomes I->0, voltage difference across the capacitor becomes.
When the switch S in Fig 3. Is closed, the currents I(t) flows. If the inductor were present, self-induced emf appears in the circuit; from Lenz’s law, this emf opposes the rise of the current, which means that it opposes the battery emf e. Thus, the current in the resistor responds to the difference between two emfs, a constant e due to the battery and a variable eL due to self-induction. eL is given by;
(11)
The loop rule gives us
(12)
And we can integrate both side like former case – RC circuit;
(14)
i.e.
(15)
Or
(16)
Changing pattern is like Fig 4.
Voltage difference across the inductor is given by;
(17)
Inductive time constant is – same dimension as capacitive time constant – given by .
When t=
(18)
The maximum voltage in the circuit is
(19)
Long after the switch is closed, the current becomes I->V/R , voltage difference between inductor becomes V->0.
Like in Fig 5. Currents and voltage have same phase in resistance, currents is pi/2 faster in capacitance, voltage is pi/2 faster in inductance.
Currents flowing each element is;
(20a , b, c)
And
(21a)
Is called a capacitive reactance and
(21b)
Is called a inductive reactance. Each corresponds the magnitude of resistance in AC circuit with angular frequency w. Eq (20 a,b,c) applies not only between peak value of AC voltage and currents but also between effective value. The effective value is defined rooted value of time average of square of AC voltage and currents. So effective value is called rms(root-mean-square) value. Common notation of AC voltage and currents indicates effective value
(22a, b)
W is angular frequency of AC, T is a period, V and I are peak value of AC voltage and currents, respectively.
In composite circuit – like RL or RC – the voltage and currents have phase difference. So we use phase diagram in Fig 6. When sum resistance, reactance. The effective resistance in this circuit is called impedance Z. Fig 6 shows the impedance of RL and RC circuit. The phase difference between currents and net voltage difference is given by;
(23)
This is also called power factor of circuit. Relation between power factor of RC, RL circuit and peak voltage is also expressed in Fig 6.
The Impedance Z of RLC serial circuit with angular frequency w is given by;
(24)
Voltage and currents are;
(25a, b)
Phase difference between them is;
(26)
Peak value of currents in circuit is;
(27)
Voltage difference between each element R,L,C is given by;
(28a,b,c)
Their peak voltage is given by;
(29a,b,c)
Their relationship is like a vector sum;
(30)
-Resonance of RLC serial circuit.
For a given resistance R, that amplitude is a maximum when the quantity wL-1/wC in the denominator is zero, that is, when
(A1)
And effective resistance of circuit is
(A2)
And angular frequency of AC currents is
(A3)
Phase difference is zero for such a circuit, So peak value of currents in RLC circuit is given by;
(A4)
In this case, currents in the circuit is only dependent with R circuit. This occurs in special frequency, so (w/2pi) is called resonance frequency. As in fig 7. Resonance shape differ as R changing.
-Resonance of RLC parallel circuit
When Voltage V(t)=() is applied, the currents flowing R,L,C is
(A5)
So net currents in the serial circuit has minimum peak value when peak value of currents flowing inductor and capacitor
(A6a, b)
Are same. That is, reactance XL and XC are same. The currents flowing circuit becomes
(A7)
And dependent only resistance R.
References
- Constant current(voltage) power supply
- Measurement of specific heat of an object - calorimeter
- Measurement of temperature by the thermocouple and the digital thermometer
- Treatment of measurement data
- Analysis method based on the graph
- James Joule - The importance of precise measurement
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