To understand the use of phasor diagrams in calculating the impedance and resonance conditions in a series L-R-C circuit. In this problem, you will consider a series L-R-C circuit, containing a resistor of resistance R, an inductor of inductance L, and a capacitor of capacitance C, all connected in series to an AC source providing an alternating voltage V(t)=V0cos(?t). (Figure 1) You may have solved a number of problems in which you had to find the effective resistance of a circuit containing multiple resistors. Finding the overall resistance of a circuit is often of practical interest. In this problem, we will start our analysis of this L-R-C circuit by finding its effective overall resistance, or impedance. The impedance Z is defined by Z=V0I0, where V0 and I0 are the amplitudes of the voltage across the entire circuit and the current, respectively. Part A Find the impedance Z of the circuit using the phasor diagram shown. Notice that in this series circuit, the current is same for all elements of the circuit: (Figure 2) You may find the following reminders helpful: In a series circuit, the overall voltage is the sum of the individual voltages. In an AC circuit, the voltage across a capacitor lags behind the current, whereas the voltage across an inductor leads the current. The reactance (effective resistance) of an inductor in an AC circuit is given by XL=?L. The reactance (effective resistance) of the capacitor in an AC circuit is given by XC=1?C. Express your answer in terms of R, L, C and ?. View Available Hint(s) Z = Part B Now find the tangent of the phase angle ? between the current and the overall voltage in this circuit. Express tan(?) in terms of R, L, C, and ?. tan(?) = Request Answer We may be interested in finding the resonance conditions for the circuit, in other words, the conditions corresponding to the maximum current amplitude produced by a voltage source of a given amplitude. Finding such conditions has an immediate practical interest. For instance, tuning a radio means, essentially, changing the parameters of the circuitry so that the signal of the desired frequency has the maximum possible amplitude. Part C Imagine that the parameters R, L, C, and the amplitude of the voltage V0 are fixed, but the frequency of the voltage source is changeable. If the frequency of the source is changed from a very low one to a very high one, the current amplitude I0 will also change. The frequency at which I0 is at a maximum is called resonance. Find the frequency ?0 at which the circuit reaches resonance. Express your answer in terms of any or all of R, L, and C. View Available Hint(s) ?0 = Part D What is the phase angle ? between the voltage and the current when resonance is reached? View Available Hint(s) ? = Part E Now imagine that the parameters R, L, ?, and the amplitude of the voltage V0 are fixed but that the value of C can be changed. This is one of the easiest parameters to change when "tuning" such (radio frequency) circuits in order to make them resonate. This is because the capacitance can be changed just by moving the capacitor plates closer or farther apart. Find the capacitance C0 at resonance. Express your answer in terms of L and ?. C0 = Request Answer Provide Feedback Figure1 of 2 The figure shows a circuit. The circuit consists of an alternate current source, an inductor with inductivity L, a capacitor with capacity C, and a resistor with resistance R connected in series.