Introduction To Resonant Circuits GITAM University EEE Department

Resonance In Electric Circuits Any passive electric circuit will resonate if it has an inductor and capacitor.   Resonance is characterized by the input voltage and current being in phase. The driving point impedance (or admittance) is completely real when this condition exists. In this presentation we will consider (a) series resonance, and (b) parallel resonance. 

Series Resonance Consider the series RLC circuit shown below. V = VM 0 R L + C V _ I The input impedance is given by: The magnitude of the circuit current is;

Series Resonance Resonance occurs when, At resonance we designate w as wo and write; This is an important equation to remember. It applies to both series And parallel resonant circuits.

Series Resonance The magnitude of the current response for the series resonance circuit is as shown below. |I| Half power point w1 wo w2 w Bandwidth: BW = wBW = w2 – w1

Series Resonance The peak power delivered to the circuit is; . The so-called half-power is given when . We find the frequencies, w1 and w2, at which this half-power occurs by using;

Series Resonance After some insightful algebra one will find two frequencies at which the previous equation is satisfied, they are: and The two half-power frequencies are related to the resonant frequency by

Series Resonance The bandwidth of the series resonant circuit is given by; We define the Q (quality factor) of the circuit as; Using Q, we can write the bandwidth as; These are all important relationships.

Series Resonance An Observation: If Q > 10, one can safely use the approximation; These are useful approximations.

Series Resonance An Observation: By using Q = woL/R in the equations for w1and w2 we have; and

BANDWIDTH OF SERIES RESONANCE The width of the response is measured by the BANDWIDTH. BANDWIDTH is the difference between the half-power frequencies. Resonance frequency can be obtained from the half-power frequencies. The SHARPNESS of the resonance is measured by the QUALITY FACTOR. QUALITY FACTOR is the ratio of the resonance frequency to the bandwidth. The higher the Q the smaller is the bandwidth.

QUALITY FACTOR OF SERIES RESONANCE

BANDWIDTH of SERIES RESONANCE Current versus frequency for the series resonant circuit. Half Power Frequencies Dissipated power is half of the maximum value. The half-power frequencies 1 and 2 can be obtained by setting,

Selectivity The frequencies corresponding to 0.707 of the maximum current are called the band frequencies, cutoff frequencies, or half-power frequencies (ƒ1, ƒ2). Half-power frequencies are those frequencies at which the power delivered is one-half that delivered at resonant frequency. The range of frequencies between the two are referred to as bandwidth (abbreviated BW) of the resonant circuit. Since the resonant circuit is adjusted to select a band of frequencies it is called a selectivity curve. The shape of the curve depends on each element of the series R-L-C circuit. If resistance is made smaller with a fixed inductance and capacitance, the bandwidth decreases and the selectivity increases. If the ratio L/C increases with fixed resistance, the bandwidth again decreases with an increase in selectivity.

PARALLEL RESONANCE Resonance is a condition in an RLC circuit in which the capacitive and inductive reactances are equal in magnitude, resulting in a purely resistive impedance. Parallel resonance circuit behaves similarly but in opposite fashion compared to series resonant circuit. The admitance is minimum at resonance or impedance is maximum. Parallel resonant circuit.

PARALLEL RESONANCE At Resonance frequency: 1) Admitance is purely resistive. 2) The voltage and current are in phase. 3) The transfer function H()= Y() is Minimum. 4) Inductor and capacitor currents can be much more than the source current.

PARALLEL RESONANCE Voltage versus frequency for the parallel resonant circuit. The half-power frequencies can be obtained as:

Summary of series and parallel resonance circuits: Characteristic Series circuit Parallel circuit ωo Q B ω1, ω2 Q ≥ 10, ω1, ω2

Parallel Resonance Parallel Resonance Series Resonance

Resonance Example 1: Determine the resonant frequency for the circuit below. At resonance, the phase angle of Z must be equal to zero.

Resonance Analysis For zero phase; This gives; or

Parallel Resonance Example 2: A series RLC resonant circuit has a resonant frequency admittance of 2x10-2 S(mohs). The Q of the circuit is 50, and the resonant frequency is 10,000 rad/sec. Calculate the values of R, L, and C. Find the half-power frequencies and the bandwidth. A parallel RLC resonant circuit has a resonant frequency admittance of 2x10-2 S(mohs). The Q of the circuit is 50, and the resonant frequency is 10,000 rad/sec. Calculate the values of R, L, and C. Find the half-power frequencies and the bandwidth. A parallel RLC resonant circuit has a resonant frequency admittance of 2x10-2 S(mohs). The Q of the circuit is 50, and the resonant frequency is 10,000 rad/sec. Calculate the values of R, L, and C. Find the half-power frequencies and the bandwidth. First, R = 1/G = 1/(0.02) = 50 ohms. Second, from , we solve for L, knowing Q, R, and wo to find L = 0.25 H. Third, we can use

Parallel Resonance Example 2: (continued) Fourth: We can use and Fifth: Use the approximations; w1 = wo - 0.5wBW = 10,000 – 100 = 9,900 rad/sec w2 = wo - 0.5wBW = 10,000 + 100 = 10,100 rad/sec