To understand basic calculations involving l-r-c ac circuits. because the currents and voltages vary, ac circuits are more complex than dc circuits. consider a circuit consisting of a resistor of resistance r, an inductor of inductance l, and a capacitor of capacitance c connected in series to an ac power source. (figure 1) as with all circuit components connected in series, the same current flows through each of these elements. this is an ac circuit, so the current is changing with time. the current i at a time t can be found using the relationship: i=icos(ωt), where i is the maximum current and ω=2πf (f is the frequency of the current source). the relationship between the current and voltage in an ac circuit works according to ohm's law. consider just the resistor in the circuit. because the current changes in time, the voltage across the resistor vr also changes. however, both i and vr will be at a maximum at the same time. the maximum voltage across the resistor is given by vr=ir recall that an inductor is designed to oppose any change in current in the circuit. although an inductor has no resistance, there is a potential difference vl across the ends of the inductor. unlike the case for the resistor, i and vl do not reach maximum values at the same time. the voltage reaches a maximum before the current. the maximum potential difference across the inductor is vl=iωl. by defining the quantity ωl as the inductive reactance xl, vl can be rewritten as vl=ixl. this equation is similar to ohm's law. part a an l-r-c circuit, operating at 60 hz, has an inductor with an inductance of 1.53×10−3h, a capacitance of 1.67×10−2f, and a resistance of 0.329 ω. what is the inductive reactance of this circuit? a capacitor is designed to store energy by allowing charge to build up on its plates. although a capacitor has no resistance in an ac circuit, there is a potential difference vc across the plates of the capacitor. the maximum values of i and vc do not occur at the same time. the voltage reaches a maximum after the current. the maximum potential difference across the inductor is vc=iωc. by defining the quantity 1/ωc as the capacitive reactance xc, vc can be rewritten as vc=ixc. as with the case for the inductor, this equation is similar to ohm's law. part b what is the capacitive reactance of the circuit in part a? the capacitive reactance, the inductive reactance, and the resistance of the circuit can be combined to give the impedance z of the circuit. the impedance is a measure of the total reactance and resistance of the circuit and is similar to the equivalent resistance that can be found from various resistors in a dc circuit. because vl and vc do not reach maximum values at the same time as i, the impedance is not found by adding r, xl, and xc. instead, the impedance is found using z=r2+(xl−xc)2√. the analogy to ohm's law is then v=iz. part c what is the total impedance of the circuit in parts a and b? in ac circuits, i and v are not measured directly. instead, ac ammeters and voltmeters are designed to measure the root-mean-square values of i and v: irms=i2√andvrms=v2√, which can be used in the equation v=iz to yield vrms=irmsz part d if this circuit were connected to a standard 120 v ac outlet, what would the rms current in the circuit be? a frequent application of l-r-c ac circuits is the tuning mechanism in a radio. the l-r-c ac circuit will have a resonant frequency that depends on both the inductance and capacitance of the circuit according to the formula f0=12πlc√. this is the frequency at which the impedance is the smallest, which causes the largest current to appear in the circuit for a given vrms. the radio picks up this resonant frequency and suppresses signals at other frequencies. a variable capacitor in this circuit causes the resonant frequency of the circuit to change. when you tune the radio you are adjusting the value of the capacitance in the circuit and hence the resonant frequency. part e to see whether the l-r-c ac circuit from part a would be suitable for a tuner in a radio, find the resonant frequency of this circuit.