Series RLC circuit

Since the unloaded QCM is an electromechanical transducer, it can be described by the Butterworth-Van Dyke (BVD) equivalent electrical circuit represented in Fig. 12.3 (box) which is formed by a series RLC circuit in parallel with a static capacitance C0. The electrical equivalence to the mechanical model (mass, elastic response and friction losses of the quartz crystal) are represented by the inductance L, the capacitance C and the resistance, R connected in series. The static capacitance in parallel with the series motional RLC arm represents the electrical capacitance of the parallel plate capacitor formed by both metal electrodes that sandwich the thin quartz crystal plus the stray capacitance due to the connectors. However, it is not related with the piezoelectric effect but it influences the QCM resonant frequency. 

Series RLC Circuit. Consider the series RLC circuit shown in Fig. 9.2. The three impedances add, so the overall impedance is analyzed by setting up an integrodifferential equation... 

In a series RLC circuit, the current z R(t) through the resistor R is in-phase with the voltage v(t) across it the current ic(t) through the capacitor C is n/2 radians ahead of the phase of the voltage across it the current iL(t)... 

The inhomogeneous integro-differential equation for this series RLC circuit now becomes... 

Therefore the instantaneous power in a series RLC circuit varies at twice the frequency of the applied voltage. At a very special frequency co0, called the resonant frequency ... 

The shape of the function depends on the details of the system discussed. For a series RLC circuit at resonance, the quality factor Q0 maybe defined as ... 

The high-frequency equivalent circuit for a piezoelectric sensor is complex because of its mechanical resonance. This can be modeled by adding a series RLC circuit in parallel with the sensor capacitance and leakage resistance. The high-frequency equivalent circuit and its frequency response are shown in figure 2.3. In some applications, the mechanical resonance is desirable for accurate frequency control, as in the case of crystal filters. 

Equation (1.67) represents equivalent resistance or AC impedance of the whole series RLC circuit. It indicates that when a circuit is operating at a resonant frequency (i.e., Xl = X ), the overall impedance is equal to a real resistance R. If the frequency is increased to exceed the resonance, the capacitive reactance will be reduced and the inductive reactance will be increased. The situation is likewise reversed at frequencies below resonance. 

Initial analysis of this model reviews the real part of impedance as a function of frequency in Figure 6.12b for a hypothetical series RLC circuit in parallel with a leakage resistance. A division of the real impedance plot into four distinct regions provides a means of quantifying separate resistance elements detailed as follows. 

The simple series RLC electrical circuit of Fig. 9.2 consists of a direct-current (DC) power source (here a 3-V battery), a relay, and three loads in series a resistor of resistance R, a capacitor of capacitance C, and an inductor of inductance L. Assume first a DC potential E = E0, in series with R, C, and L the capacitance stores charge, the inductance stores current, and the resistance dissipates some of the current into Joule13 heating. The arrow shows the direction of the current (which, thanks to Franklin s unfortunate assignment, is the direction of motion of positive holes—that is, the opposite of the flow of negative electrons) the relay across L avoids conceptual difficulties about an initial current through the inductor. The current is usually denoted by I (from the French word "intensite"). These three components (R, C, and L) will be explored in sequence. 

Here, L is the dynamic inductance, a measure of the oscillating mass of die quartz, and Cs is the dynamic capacitance or series capacitance, a measure of the elasticity of the oscillating body [9], The resistance of the RLC circuit is related to the quality factor Q (the width of the resonance), the dissipation D [10], and the full width at half maximum f of the resonance by the relationship ... 

The Fig. 4.2.1 shows that when a ground fault occurs, at the end of one of the feeders there is an RLC circuit formed by the resistor (Rll and R12), inductance (Lll and LI 2) and capacitance (C12) of the feeders. Under fault conditions the current flows through these elements, this stream may contain a component of a frequency (1 to 10 kHz) that could generate a series resonant circuit. 

The Bode magnitude response of the simple RC circuit will always have the shape shown in Fig. 1.5 and can never exhibit resonance. On the other hand, the capacitor voltage in the series or parallel RLC circuit shown in Fig. 1.6 has the sharp Bode magnitude response shown in Fig. 1.7 when R = 10 L = 1 H, and C = 1 F (frequency scaling can be... 

FIGURE 1.8 Pole-zero pattern of a resonant series-parallel RLC circuit. 

The voltage across the series RLC output circuit alternates polarity between -F Vfir) and — Vdd so that Vac(6>) is of the same form as Eq. (7.89). When L and C are tuned to fundamental resonance... 

There are different RLC circuits with various R, L, and C combinations. Other RLC circuits include parallel RLC (R/L/C), parallel LC series with R ((L/O-R), series CL parallel with R (C-L)/R), parallel RL series with C (R/L)-C), series RL parallel with C ((RL)-C), parallel RC series with L (R/C)-L), and series RC parallel with L ((R-O/L) [3]. The expressions of AC impedance for these RLC circuits can be obtained and summarized as follows ... 

The series resonant frequency of an RLC resonant circuit is given by ... 

Finally, let us consider a simple circuit containing an RLC connection in series. The impedance of such a circuit is... 

Series-Parallel RLC Resonant Filter The Pole Zero Pattern Description of Resonance Time-Domain Description of Resonance Resonance and Energy Storage in Inductors and Capacitors Physical Hazards with Resonant Circuits... 

Electric circuit schematic of resistor, inductor, and capacitor (RLC) connected in series. 

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