Electrical Resonance
In the most familiar form of mechanical oscillations, the pendulum, the total system energy constantly bounces back and forth between the kinetic and potential forms. In the absence of friction (i.e. energy dissipation), a pendulum would oscillate forever
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In the most familiar form of mechanical oscillations, the pendulum, the total system energy constantly bounces back and forth between the kinetic and potential forms. In the absence of friction (i.e. energy dissipation), a pendulum would oscillate forever. Similarly, after two ideal electrical elements capable of storing energy (a capacitor and an inductor) are connected in parallel then the total initial energy of the system bounces back and forth between the electric and magnetic energy forms. This process is observed as electrical oscillations and the parallel LC circuit is said to be “in resonance”. The phenomenon of electrical resonance is essential to wireless radio communications technology because without it, simply put, there would be no modern communications. In this chapter, we study behaviour and derive the main parameters of electrical resonant circuits.
10.1
The LC Circuit
The simplest electrical circuit that exhibits oscillatory behaviour consists of an inductor L and a capacitor C connected in parallel (see Fig. 10.1). Let us assume the initial condition where the capacitor contains q amount of charge, hence the initial voltage V across the LC parallel network is related to the charges as q = C VC = C v(max).
10.1.1 Qualitative Description of LC Resonance At time t = 0, the voltage across the charged capacitor is at its maximum v(max), its associated electric field and stored energy are also at maximum, and the network current is still at zero value. That is, at time t = 0, the inductor is still “seen” by the capacitor charge as an ideal wire. Naturally, due to the electric field, the capacitive charge is forced to move through the only available path, the inductive wire. However, as soon as the first electron leaves the capacitor plate, this movement qualifies as a change of current in time and, according to (2.40), this “ideal wire” starts to show strong inductive properties accompanied by an appropriate magnetic field. Hence, while this rising current flows through the inductor, it must obey Lenz’s law and create the magnetic field that opposes the change that produced it. Eventually, the current reaches its maximum value i(max) (at t = T/4) when the capacitor is fully discharged; the complete energy of the LC system is now stored in the inductor’s magnetic field. © Springer Nature Switzerland AG 2021 R. Sobot, Wireless Communication Electronics, https://doi.org/10.1007/978-3-030-48630-3_10
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10 Electrical Resonance
Fig. 10.1 Ideal LC resonance, the first cycle
+q
i = i(max)
i=0
v=0 -q
i = -i(max)
-q ---------
+++++++++
---------
-q
+q v = -v(max)
v,i amplitude
v = v(max)
T 4
i=0
v=0
+++++++++
---------
+q
+++++++++
T 2
v = v(max)
3T 4
T time
It is now up to the inductor to serve as the energy source in the circuit and to push charges inside the wire while gradually passing the magnetic energy into the capacitive electrostatic energy. The uninterrupted flow of the current continues to cause the charges to keep accumulating at the other capacitor plate an
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