# eet 202

EET 202

LAB EXPERIMENT INSTRUCTIONS

AC circuit analysis

Lab # 5: Resonance

Experiment: Parallel Resonance.

Objective:

After performing this experiment, you will be able to:

1. Compute the resonant frequency, Q and the bandwidth of a parallel resonant circuit.
2. Measure the frequency response of a parallel resonant circuit.
3. Use the frequency response to determine the bandwidth of a parallel resonant circuit.

Materials Required:

Inductor: 100 mH – 1 Piece.

Resistor: 1 KΩ – 1 Piece.

Capacitor: 0.047 µ F – 1 Piece.

Summary of theory:

In an RLC parallel circuit, the current in each branch is determined by applied voltage and the impedance of that branch. For an “ideal” inductor (no resistance), the branch impedance is XL and for a capacitor the branch impedance is XC. Since XL and XC are functions of frequency, it is apparent that the currents in each branch are also dependent on frequency. For any given L and C, there is a frequency at which the currents in each are equal and of opposite phase. This frequency is the resonant frequency and is found using the same equation as was used for series resonance.

The circuit and the phasor diagram for an ideal parallel RLC circuit at resonance as illustrated in Figure 5-2-1. Some interesting points to be observed are: The total source current at resonance is equal to the current in the resistor. The total current is actually less than the current in either the inductor or the capacitor. This is because of the opposite phase shift which occurs between inductors and capacitors, causing the addition of currents to cancel. Also, the impedance of the circuit is solely determined by R, as the inductor and capacitor appear to be open. In a two-branch circuit consisting of only L and C, the source current would be zero, causing the impedance to be finite, of course, this does not happen with actual components that do have resistance and other effects.

Figure 5-2-1

In a practical two-branch parallel circuit consisting of an inductor and a capacitor, the only significant resistance is the winding resistance of the inductor. Figure 5-2-2 (a) illustrates a practical parallel LC circuit containing winding resistance. By network theorems, the practical LC circuit can be converted to an equivalent parallel RLC circuit, as in the Figure 5-2-2 (b).  The equivalent circuit is easier to analyze. The phasor diagram for the ideal parallel RLC circuit can then be applied to the equivalent circuit as was illustrated in Figure 5-2-1. The equations to convert the inductance and its winding resistance to an equivalent parallel circuit are

Where R p (e q) represents the parallel equivalent resistance and RW represents the winding resistance of the inductor. The Q in the conversion equation is the Q for the inductor.

Q = XL / RW

Figure 5-2-2

The selectivity of series circuits was discussed in the previous part in series resonance. Similarly the parallel resonant circuits also respond to a group of frequencies. In parallel resonant circuits, the impedance as a function of frequency has the same shape as the current versus frequency curve for series resonant circuits. The bandwidth of a parallel resonant circuit is the frequency range at which the circuit impedance is 70.7% of maximum impedance. The sharpness of the response to the frequencies is again measured by circuit Q. The circuit Q will be different from Q of the inductor if there is additional resistance in the circuit. If there is no additional resistance in parallel with L and C, then the Q for a parallel resonant circuit is equal to the Q of the inductor.

Procedure:

1. Measure the value of a 100 mH inductor, 0.047 µF capacitor, and a 1.0 KΩ resistor. Enter the measured values in Table 5-2-1. If it is not possible to measure the inductor or capacitor, use listed values.

1. Measure the resistance of the inductor. Enter the measured inductor resistance RW in Table 5-2-1.

 Listed Value Measured Value L1 100 mH C1 0.047 µF RS1 1.0 KΩ RW (L1 resistance)

Table 5-2-1

 Computed Measured fr Q BW fi = BW/4

Table 5-2-2

1. Construct the circuit in Figure 5-2-3.  The purpose of RS1 is to develop a voltage that can be used to sense the total current in the circuit. Compute the resonant frequency of the circuit using the equation

Enter the computed resonant frequency in Table 5-2-2. Set the generator to the fr at 1.0 VPP output as measured with your oscilloscope. Use peak-to-peak values for all voltage measurements in this experiment.

Figure 5-2-3

1. The Q of a parallel LC circuit with no resistance other than the inductor winding resistance is equal to the Q of the inductor. Compute the approximate Q of the

parallel LC circuit from;                 Enter the compounded Q in Table 5-2-2.

1. Compute the bandwidth from the equation

Enter this as the compounded BW in the Table 5-2-2.

1. Connect your oscilloscope across RS1 and tune for resonance by observing the voltage across the resistor, RS1. Resonance occurs when the voltage across RS1 is minimum, since the impedance of the parallel LC circuit is highest. Measure the resonant frequency (fr) and record the measured result in Table 5-2-2.

1. Compute a frequency increment (fr) by dividing the computed bandwidth by 4.

That is                               Enter the computed fr value in Table 5-2-2.

1. Use the measured resonant frequency (fr) and the frequency increment (fi) from the Table 5-2-2 to compute 11 frequencies according to the computed Frequency column of the Table 5-2-3. Enter the 11 frequencies in the column 1 of the Table 5-2-3.

 Computed Frequency VRS1 I Z fr – 5 fi = fr – 4 fi = fr – 3 fi = fr – 2 fi = fr – 1 fi = fr   = fr + 1 fi = fr  + 2 fi = fr + 3 fi = fr + 4 fi = fr  + 5 fi =

Table 5-2-3

1. Tune the generator to each of the computed frequencies listed in Table 5-2-3. At each frequency, check that the generator voltage is still at 1.0 VPP; then measure the peak-to-peak voltage across RS1. Record the voltage VRS1 in the column 2 of the Table 5-2-3.

1. Compute the total peak-to-peak current, I, at each frequency by applying Ohm’s Law to the sense resistor R (That is, I = VRS1/RS1). Record the current I in column 3 of Table 5-2-3.

1. Use Ohm’s law with the measured source voltage (1.0 VPP) and the source current at each frequency to compute the impedance at each frequency.

Complete Table 5-2-3 by listing the computed impedance Z.

1. On Plot 5-2-1, draw the impedance versus frequency curve. From your curve determine the bandwidth. Complete Table 5-2-2 with the measured bandwidth.

Plot 5-2-1