#7: Series & Parallel Resonance – EEL 3123: Networks & Systems Lab Manual

Objectives

To study the behavior of series and parallel LC circuits at resonance.
To understand the resonance frequency, cut-off frequency, bandwidth and quality factor of a resonance circuit.
To determine if a circuit is inductive or capacitive.
To understand the circuit behavior at resonance.

Equipment

Breadboard
Function generator
Oscilloscope
Digital multimeter (DMM)

Background

Resonant circuits form the basis for filters that have better performance than first order (RL, RC) filters in passing desired signal or rejecting undesired signals that are relatively close in frequency. The resonance frequency is defined as the frequency at which the impedance of the circuit is purely real, that is, with zero reactance. For the reactance to be zero, impedance of the inductor must equal that of the capacitor. At resonance, the impedance of a branch with LC in series is equal to zero, which is equivalent to a short, and the admittance of a branch with LC in parallel is equal to zero, which is equivalent to an open. As the frequency increases, the magnitude of an inductive reactance increases, while the magnitude of a capacitive reactance decreases. A circuit is said to be inductive if the total reactance is positive, and a circuit is said to be capacitive if the total reactance is negative.

A bandpass, RLC circuit, will have two cut-off frequencies $\omega_{c1}$ and $\omega_{c2}$ where the amplitude is ${1 \over \sqrt{2}}$ of the maximum value. The cut-off frequency is also called the half-power frequency or 3-dB frequency in some cases.

The bandwidth BW (or passband bandwidth) is defined as the difference between the upper and lower cutoff frequencies. In case of a low-pass filter or baseband signal, the bandwidth is equal to its upper cutoff frequency.

Examples of the resonance frequency, cutoff frequencies and the bandwidth are shown in Figure 7 – 1 for a bandpass and a bandreject filter.

Figure 7 – 1 Resonance frequency, cutoff frequency, and bandwidth

The quality factor or Q factor of the frequency response is described quantitatively in terms of the ratio of the resonance frequency to the bandwidth,

$Q={\omega_0 \over \beta}$

with both $\omega_0$ and $\beta$ are in radians. This definition lends itself to laboratory measurement because it is possible to measure both the resonance frequency $\omega_0$ and the bandwidth $\beta$ . The Q factor is also defined as an energy ratio,

$Q=2 \pi {\text{maximum energy stored} \over \text{total energy lost per period}}$ .

The steady-state response of a circuit will in general have a maximum amplitude and phase angle that is different from that of the source. In some cases, the magnitude of the voltage response may exceed that of the voltage source.

Preparation

A. RLC circuit basic measurement

For circuits (a) through (d) in Figure 7 – 2, use C = 0.1 uF, L = 100 mH, R = 1 kΩ.

Derive the transfer function.
Find the resonance frequency, cutoff frequencies, bandwidth and Q factor for each circuit.
What is the phase relation between the total voltage and current, is it leading or lagging when the frequency is (i) below resonance, and (ii) above resonance? What is the nature of the circuit in those two regions, ie, is it capacitive or inductive?

B. RLC circuit at resonance

For circuit (e) in Figure 7 – 2, use C = 0.1 uF, L = 100 mH.

At resonance frequency, calculate V_C if:
1. R = 3 kΩ
2. R = 300 Ω
Find out whether the magnitude of V_C is larger or smaller than V_in. Explain your result.

Simulation

Build and simulate the circuits (a) through (d) in Figure 7 – 2 using Multisim. Find the resonance frequency, half-power frequencies bandwidth and Q factor for each circuit. Compare the result with that from PREPARATION.
Build and simulate the circuit (e) in Figure 7 – 2. Find the magnitude for V_C for the following cases.
1. R = 3 kΩ
2. R = 300 Ω

Compare the result with that from PREPARATION.

Experiment

On the breadboard, build circuit (a) in Figure 7 – 2. Connect Ch1 to input and Ch2 to output so that both the input and the output are displayed on the oscilloscope.
Set the input voltage to $\pm 5V$ and frequency to the value calculated in PREPARATION. For circuits (a) and (c), vary the frequency from the theoretical value to get a maximum output voltage. The frequency at this maximum voltage V_max is the resonance frequency. For circuits (b) and (d), vary the frequency from the theoretical value to get a minimum output voltage. The frequency at this minimum voltage V_min is the resonance frequency.
If $V_{cutoff}={V_{max} \over \sqrt{2}}$ , vary the frequency again until the output voltage equals $V_{cutoff}$ . This frequency is the cut-off frequency. There should be 2 cut-off frequencies for each case.
Calculate the bandwidth by subtracting the 2 cut-off frequencies.
Calculate the Q factor using the equation from PREPARATION.
Repeat step 1 through 5 for circuits (b) to (d) in Figure 7 – 2.
Compare your result with that from PREPARATION and SIMULATION.
Build circuit (e) in Figure 7 – 2. Connect Ch1 to input and Ch2 to output so that both the input and the output are displayed on the oscilloscope. Set the input voltage to $\pm 5V$ and frequency to the resonance frequency found in step 1 for circuit (b) in Figure 7 – 2.
Find the magnitude for V_C for the following cases.
1. R = 3 kΩ,
2. R = 300 Ω.

Compare the result with that from PREPARATION and SIMULATION. Explain what you observe.

Figure 7 – 2 Circuits

Report

Prepare your report as per the guidelines given in APPENDIX III.