Inductor-Capacitor Circuits

As discussed earlier, the two-element series combinations such as RC and RL circuits showed the exponential fimctions for the growth or decay of charge, potential, and cmrent over time with a time scale measured by their respective time constants Tc and Tj. However, an inductor-capacitor (LC) circuit demonstrates different behavior from that of RC and RL circuits. There are two new parameters introduced to describe LC circuit behavior (1) a sinusoidal oscillation period T and (2) an angular frequency co. The LC circuit s capacitor is initially charged to a maximum potential V and the magnetic energy stored in the inductor is zero. 

The solution to this differential equation with a maximum charge Q (i.e., amplitude of oscillation) at time f = 0 can be written as  

The amplitude wQ is equivalent to the maximum current I passing through the circuit. Solving for w is done by substituting Equations (1.47) and (1.48) into Equation (1.46)  

This (0 is often called the natural angular frequency by which the EC circuit will oscillate devoid of any external power source. 

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Each NMR instrument has a depth gauge to allow us to position the NMR tube correctly with respect to the spinner. Figure 2.1 shows the correct spatial relationships between the tube, the spinner that holds it, and the region of the NMR tube that will occupy the probe s detector (a coil of wire that is an inductor in a resistor-inductor-capacitor circuit) when the spirmer-tube assembly is in the instrument. Note that not all of the sample volume occupies the detected region. 

In a parallel capacitor-inductor (CL) circuit (C/L), the overall AC impedance,... 

With the duality principle in mind, let us attempt to open the switch in the inductor circuit and try to predict the outcome. What happens No Unfortunately, things don t remain almost unchanged as they did for a capacitor. In fact, the behavior of the inductor during the off-phase is really no replica of the off-phase of the capacitor circuit. 

But coming to the inductor charging circuit (i.e. switch closed), we can t seem to connect this too readily to any of our immediate real-life experiences. Our basic question here is — why does the charging current in the inductor circuit actually increase with time. Or equivalently, what prevents the current from being high to start with We know there is no mutually repelling charge here, as in the case of the capacitor. So why ... 

Receiver coil. An inductor in a resistor-inductor-capacitor (RFC) circuit that is tuned to the Larmor frequency of the observed nuclide and is positioned in the probe so that it surrounds a portion of the sample. 

The projection of the Fr phasor results in the instantaneous potential difference i = fac (Fig. 5.24b). The voltage amplitude across an inductor is defined as Vc = IXc-In Fig. 5.24b, the voltage in a capacitor circuit lags the current by 90°. The projection of the phasor Fc onto the horizontal axis in Fig. 5.24b results in... 

An inductor/capacitor tank is driven by a DC-to-AC inverter, as illustrated in Fig. 10.144. The tank circuit acts to reconstruct the sine wave at the output of the system. Regulation is accomplished by varying the capacitance or the inductance to control partial resonance and/or power factor. Some systems of this... 

Traces, pads, and vias often act as inductors, capacitors, and coupling elements in the actual circuit. Their shapes may have a material effect on overall circuit performance. For example, the lead inductance and capacitance in a transistor collector circuit wire may act as the resonant components for an RF amplifier or it may degrade performance if it is unwanted. Figure 13.1 shows the impedance of traces as a function of their capacitance. 

Other examples are the voltage V in a resistor capacitor RC circuit without an applied input voltage, and the change of current i in a resistor-inductor RL circuit without an applied input current, as described by Equations 3.77 and 3.78 respectively. 

In maldug electrochemical impedance measurements, one vec tor is examined, using the others as the frame of reference. The voltage vector is divided by the current vec tor, as in Ohm s law. Electrochemical impedance measures the impedance of an electrochemical system and then mathematically models the response using simple circuit elements such as resistors, capacitors, and inductors. In some cases, the circuit elements are used to yield information about the kinetics of the corrosion process. 

Let us examine how one determines the values of the inductor and capacitor. Several assumptions have to be made at the beginning of the design process since several of the tank circuit s characteristics are variable within the application. The first is to assume a value for the Q of the tank circuit. In the application, the Q varies greatly with the amount of load placed on the output of the supply. So, a good value to start with is... 

It is desired that the resonanee frequency of the tank circuit be 1 MHz. In ZVS QR converters, the tank circuit is not responsible for storing and passing on energy as it is in ZCS QR converters. The tank circuit can be seen more as an off-time transition shaper similar to a snubber when used in PWM converters. Here a wide range of values for both the inductor and the capacitor will work as long as their combined resonant frequency is 1 MHz or... 

In the parallel configuration, the same potential difference occurs across each and every element with the total current being the algebraic sum of the current flowing through each individual circuit element. Table 2-35 summarizes the equivalent resistance, conductance, capacitance, and inductance of series-parallel configurations of resistors, capacitors, and inductors. 

Reactive circuit elements (e.g., capacitors and inductors) store, not dissipate, energy. While the energy stored is periodically returned to the rest of the circuit, reactive elements do require increased potential or current to flow in the circuit. The power that must be supplied for the reactive elements is termed reactive power, and it is calculated as... 

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