OBJECTIVE:
APPARATUS:
INTRODUCTION:
The rms concept allows us to describe any AC voltage as having a particular rms voltage and a particular phase.
In this experiment we will use the subscript ``rms'' for some measurements and will use ``V'' or ``v'' or ``I'' or ``i'' without this subscript when referring to instantaneous values of the voltage or current. Thus, even if two rms voltages are equal, ( Vrms = vrms), V may not be equal to v since V and v may have different phases.
The impedance Z (in ohms) of any part of a circuit is the ratio of the rms voltage across that
part and the rms current though that part:
Because impedance is defined as a ratio of voltage/current, impedance is measured in ohms.
If
= 2
f and we measure L and C in henries and farads,
respectively, then it can be shown
that the impedance XR of an resistance R is R
and the impedance XL (in ohms) of an inductance L is
XL =
L = 2
fL.
and the impedance XC (in ohms) of a capacitance C is
XC =
=
.
It can also be shown that the impedance Z (in ohms) of
an RLC series circuit (shown in Fig. 1) is
The impedance Z of the RLC series circuit is a minimum for
XL = XC.
The frequency for which this occurs is the resonant frequency. At this
frequency, fr, the current thru R is maximum, but the voltage
Vrms across the LC series combination is a minimum and in fact would be zero
if the inductor had no resistance. Hence one can search for fr by
varying the frequency and looking either
1) for a maximum
Vrms R
signal, or 2) a minimum
Vrms LC signal.
A search for the minimum
has a practical advantage that near the resonant frequency, fr, one can
increase enormously the detection sensitivity by going to maximum signal
generator amplitude and also by going to higher scope gain.
SUGGESTIONED PROCEDURE:
Phase relations: Use scope channel 1 (plus a differential amplifier) to observe the total voltage Vrms RLC across the RLC combination. Use scope channel 2 (plus differential amplifier) to observe Vrms R. Fig. 2 shows the phasor relationships. |
Figure 2
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Find the resonant frequency by this method. Adjusting gain and position controls so the signals nearly overlap will help.
OPTIONAL:
INTRODUCTION:
Since iR is in phase with V, but iL lags V by 90o, and iC leads V by 90o, we can describe the situation by the rotating vectors in Fig. 3b where I is the vector sum of IR, IL, and IC. Hence from Fig. 3b
Since V is the same for all the parallel elements, the relevant phase
differences are between the currents.
To measure the total current Irms and
the phase between it and the voltage Vrms, we will insert a 22 k
sampling resistor RS in series with the signal generator. See Figure 4.
Measure the voltage Vrms S across the sampling resistor RS by connecting the resistor ends to scope channel 2 via a differential amplifier. Since the voltage and current are in phase across a resistor, this signal Vrms S is proportional to the total current Irms.
We use channel 1 and the other differential amplifier to view the common voltage V across all the parallel elements. At resonant frequency fr the currents from L and C will cancel since they are of equal magnitude but (always) 180o out of phase. Hence at fr the total current I will be just that thru R, i.e. I = IR, and V and VS (or I) will be in phase.
SUGGESTED PROCEDURE: