In a series \(L C R\) circuit with an applied voltage of constant amplitude and variable frequency, the maximum voltages across the \(L\), the \(C\) and the \(R\) occur at angular frequencies \(\omega_{L} \omega_{Q}\) and \(\omega_{k}\), respectively. Show that \(\omega_{0}^{2}=\omega_{0}^{2}\left(1-1 / \dot{Q}_{6}^{2}\right)\), \(\omega_{n}=\omega_{0}\) and \(\omega_{t}^{2}=\omega_{0}^{2} /\left(1-1 / 2 Q_{0}^{2}\right)\), where \(\omega_{0}=1 /(L C)^{1 / 2}\) and \(Q_{0}=\omega_{0} L / R\).

When beginning to understand LCR circuits, one must be well-versed in basics like Kirchhoff's Voltage Law (KVL). KVL is foundational as it allows you to write an equation that represents the conservation of energy around a loop. It states that the algebraic sum of all voltages in a closed loop must equal zero. For an LCR circuit, this involves the potential differences across the inductor (L), capacitor (C), and resistor (R), as well as the supply voltage. In essence, what goes in must come out - the energy supplied must be fully accounted for in the energy consumed by the components.

When you are solving for the voltages at resonance, KVL is the starting point as it sets the stage for the calculations to follow. By combining this law with calculus, as with the differentiation required to find maximum voltage conditions, you are incorporating the time-variable nature of AC circuits into a coherent framework of analysis.

Dealing with alternating currents (AC) can be complex, but that's where phasor representation shines—it simplifies the analysis of AC circuits. Phasors convert time-dependent sinusoidal signals into a frequency domain using complex numbers, representing magnitude and phase angle. This complex number representation makes calculations involving AC much easier, especially when dealing with components like inductors and capacitors that create phase shifts between voltage and current.

When you used phasor form to express the voltages and currents in this LCR circuit, it facilitated the identification of resonance conditions through manageable algebra, rather than dealing with intricate trigonometric functions. Phasors are pivotal in understanding how these components interact at different frequencies and how that affects the circuit's overall behavior.

Natural angular frequency, denoted by \(\omega_0\), is a key concept when discussing resonance in an LCR circuit. It corresponds to the frequency at which the system would oscillate if there were no damping forces—like the resistance in the circuit. For LCR circuits, \(\omega_0\) is determined by the inductance (L) and capacitance (C) with the formula \(\omega_0 = 1/\sqrt{LC}\).

This value sets an intrinsic resonance condition for the circuit where the reactive effects of the inductor and capacitor cancel each other out, leading to a purely resistive circuit at \(\omega_0\). In the exercise, identifying \(\omega_0\) was crucial to derive the maximum voltages at different frequencies and understand how energy oscillates within the LCR system.

The quality factor, or \(Q_0\), is a dimensionless parameter that quantifies the 'sharpness' of resonance in an LCR circuit. It’s defined by the ratio of the natural angular frequency \(\omega_0\) to the damping rate, the latter of which is associated with the resistance \((R)\) in the circuit. You calculate it using the formula \(Q_0 = \omega_0 L / R\). A higher \(Q_0\) indicates a narrower bandwidth and a sharper peak in the resonance curve.

The concept of \(Q_0\) is pivotal when extending the analysis beyond finding resonant frequencies, indicating energy loss in the system. The exercise explored the relationship between the quality factor and the resonant frequencies, revealing how different resistance values can shift the conditions for maximum voltage across each component.

The phenomenon of series resonance in LCR circuits is where everything aligns for maximum efficiency. At resonance, the inductive and capacitive reactances are equal in magnitude but opposite in sign, resulting in their cancellation. This leads to the circuit behaving like a pure resistive circuit at a particular frequency—known as the resonant frequency, which is determined by the inductor and capacitor values.

In the exercise, we manipulated the circuit conditions to identify the frequencies at which the voltage across each component is maximized. Series resonance is vital for applications such as radio receivers, where you need to select a specific frequency from a myriad of signals. Understanding how and why the voltages peak at their respective frequencies provides insight into designing and optimizing circuits for selectivity and performance.