The Current That Never Makes Up Its Mind

Every circuit studied so far carried direct current, flowing steadily in one direction, but the electricity actually delivered to homes and offices behaves completely differently: it oscillates smoothly, like a sine function, reversing direction on a steady, predictable schedule, called alternating voltage, and the current it drives is alternating current. The overwhelming majority of electrical devices in daily use run on ac, and the reason traces back to one specific practical advantage: ac voltage can be stepped up or down easily and efficiently using a transformer, a device with no equivalent for steady dc voltage. This single fact makes long-distance electrical transmission genuinely economical, since power can be sent out from a generating station at a very high voltage, keeping the current (and the resistive heating losses that scale with current squared) small, then stepped back down to a safe, usable voltage close to where it is actually consumed. Beyond this transmission advantage, ac circuits also exhibit genuinely distinctive behaviour that direct current circuits simply do not, behaviour exploited constantly in ordinary devices; tuning a radio to a favourite station, for instance, relies on exactly this kind of ac-specific property, one this chapter builds up to explaining in full.

Connect an ac source, v = vm sinwt, directly to a resistor, and Kirchhoff's loop rule gives the current immediately: i = im sinwt, where im = vm/R, exactly Ohm's law, working for ac exactly as it does for dc. Crucially, current and voltage here reach their zero, minimum and maximum values at exactly the same instants, they are perfectly IN PHASE, rising and falling together throughout every cycle. Since the current spends equal time positive and negative, its average value over a full cycle is exactly zero, but this does NOT mean zero power is dissipated: instantaneous power is p = i²R, and i² is always positive whether i itself is positive or negative, so genuine Joule heating occurs continuously. Averaging i²R properly over a full cycle (using <sin²wt> = 1/2) gives an average power of (1/2)im²R. To express this cleanly in the same familiar form as dc power, P = I²R, a special value called the root mean square (rms) current is defined, Irms = im/root-2, approximately 0.707im; an rms voltage is defined the same way, Vrms = vm/root-2. In these terms, average power becomes P = I²R = IV, identical in form to the dc case, which is exactly why rms values, not peak values, are what get printed on appliances and quoted for household supply: the familiar 220V mains rating is an rms value, corresponding to a genuine peak voltage of about 311V.

Current is in phase with voltage for a resistor, but this turns out NOT to be true for an inductor, a capacitor, or any combination of the three, making a clean way to track phase relationships genuinely necessary. A phasor is a vector of fixed length, equal to a quantity's own peak value, spinning steadily counterclockwise about the origin at angular speed w; at any instant, the quantity's actual value is simply the phasor's vertical projection. Two phasors, one for voltage and one for current, spinning together at the same rate w, capture everything about their relationship at a glance: their relative angle IS the phase difference between them, fixed and unchanging as both spin together, while their lengths show the two peak values, vm and im. For a resistor, the voltage and current phasors point in exactly the same direction at every instant, visually confirming zero phase difference. It is worth being precise about what a phasor actually is: voltage and current are genuinely scalar quantities, not vectors, but their amplitudes and phases happen to combine mathematically in exactly the way true rotating vectors would, which is the entire justification for borrowing vector-style diagrams and the familiar rules of vector addition to handle them.

Connect the same ac source to a pure inductor instead (assumed to have negligible resistance), and Kirchhoff's loop rule now reads v - L(di/dt) = 0, the L(di/dt) term being the inductor's own self-induced back emf, its negative sign a direct consequence of Lenz's law. Solving this, di/dt = (vm/L) sinwt, and integrating gives i = -(vm/wL) coswt; since the source oscillates symmetrically about zero, so must the current, ruling out any leftover constant term. Rewriting coswt as -sin(wt - p/2) gives the cleaner form i = im sin(wt - p/2), where im = vm/XL, and XL = wL is called the inductive reactance, playing exactly the role resistance plays for a resistor, limiting the current's amplitude, measured in the same unit, ohms, but growing directly with both inductance AND frequency, unlike ordinary resistance. Comparing this current to the source voltage shows the current reaches every peak, zero and minimum a full quarter-cycle LATER than the voltage does, the current LAGS the voltage by 90 degrees. Working out the instantaneous power delivered to the inductor, p = iv, and averaging it over a complete cycle, gives exactly zero: an inductor absorbs energy building up its magnetic field during part of each cycle, then returns every bit of it as the field collapses during the next part, with nothing net dissipated at all.

A capacitor in a dc circuit only ever passes current briefly, while charging, then blocks it completely once fully charged; connected instead to an ac source, it never gets the chance to fully charge before the voltage reverses, so it is repeatedly charged and discharged, letting current flow continuously, if not unimpeded. With charge q on the capacitor at time t, the voltage across it is v = q/C, and Kirchhoff's loop rule sets this equal to the source voltage, vm sinwt = q/C; differentiating q with respect to time to get current, i = dq/dt = wCvm coswt, which rewrites cleanly as i = im sin(wt + p/2), where im = wCvm = vm/XC, and XC = 1/wC is the capacitive reactance, again measured in ohms, but now shrinking as EITHER frequency OR capacitance increases, the reverse of an inductor's behaviour. Comparing this to the source voltage shows the current reaches its peak a full quarter-cycle EARLIER than the voltage does, the current LEADS the voltage by 90 degrees, exactly opposite to the inductor's lag. Exactly as with the inductor, the average power delivered to a capacitor over a complete cycle works out to precisely zero: energy is stored in the capacitor's electric field as it charges, then fully returned to the circuit as it discharges, with no net dissipation at all. A simple demonstration makes both reactances vivid: a lamp in series with a capacitor stays dark on dc (once charged, current stops) but glows on ac, more dimly if the capacitance is reduced, since a smaller capacitance means a larger XC and hence less current.

Wire a resistor, inductor and capacitor together in series, and the same current I flows through all three at once, but each element's own voltage sits at a different phase relative to that shared current: VR stays exactly in phase with I, VC lags 90 degrees behind I, and VL leads 90 degrees ahead of I. Since VL and VC point in exactly OPPOSITE directions on a phasor diagram (both perpendicular to VR, but one up and one down), they can be combined first into a single net phasor of magnitude |vCm - vLm|; the source phasor V is then the hypotenuse of a right triangle formed by VR and this combined (VC - VL) phasor, giving, by the Pythagorean theorem, vm² = (imR)² + (imXC - imXL)², which rearranges to im = vm/Z, where Z, the circuit's impedance, is Z = the square root of R² + (XC - XL)², the genuinely general AC counterpart to resistance, now built from all three circuit elements at once. The phase angle between the source voltage and the shared current follows the same triangle, tanf = (XC - XL)/R: if XC exceeds XL, f is positive and the circuit behaves predominantly capacitively, with current leading the voltage; if XL exceeds XC, f is negative and the circuit behaves predominantly inductively, with current lagging instead. A genuinely important, if initially strange-looking consequence: since VR and VC (or VL) are out of phase, their rms values do NOT simply add up to the source's rms voltage; adding VR and VC arithmetically can give a sum noticeably larger than V itself, and only combining them properly, using this same Pythagorean relation, recovers the correct total.

Resonance is a genuinely universal phenomenon among any system with its own natural oscillation frequency, a child on a swing, pushed at just the right rhythm, swings dramatically higher than the same push at any other rhythm, and a series LCR circuit behaves exactly the same way. Since XL = wL grows with frequency while XC = 1/wC shrinks with it, there exists exactly one special angular frequency, w0, at which the two become exactly equal, XL = XC; at that frequency, the impedance collapses to its absolute minimum, Z = R alone (the inductor and capacitor's opposing voltages cancelling completely), and the current reaches its true maximum, im = vm/R. Setting XL = XC and solving gives this resonant frequency directly: w0 = 1/root(LC), depending only on the inductance and capacitance present, not on the resistance at all; R's role is instead to control how sharp or broad the resonance peak is, a smaller R giving a taller, narrower spike in current amplitude as frequency is swept through w0, a larger R giving a shorter, broader one. Resonant circuits are put to genuinely practical use throughout everyday technology: a radio's tuning dial adjusts a variable capacitor to shift the circuit's own w0 until it matches an incoming broadcast signal's frequency, letting that one station's signal alone drive a strong current while all the others, at different frequencies, are left comparatively weak; airport metal detectors work the very same way, with a person's body (and any metal on it) walking through a resonant coil-and-capacitor circuit, subtly shifting its impedance enough to trigger an alarm. One condition is absolutely essential for any of this: resonance can only occur when BOTH an inductor and a capacitor are present together, since only their two, exactly opposite phase voltages can cancel out; a circuit with only R and L, or only R and C, has no way to reach this cancellation, and never resonates at all.

Instantaneous power delivered to a series LCR circuit, p = vi, expands into a term proportional to cos(f), constant over the whole cycle, plus a second, purely oscillating term whose average is exactly zero; averaging power over a full cycle therefore leaves only the constant piece, P = VI cosf, where V and I are rms values and cosf, called the power factor, is the single number that decides how much of the apparent product VI actually turns into real, dissipated power. Three cases are worth holding separately in mind. A purely resistive circuit has f = 0, so cosf = 1, and the full VI product is genuinely dissipated, maximum possible power. A purely inductive or purely capacitive circuit has f = 90 degrees, so cosf = 0, and NO power is dissipated at all, even though a perfectly real, measurable current is flowing continuously, a current sometimes called wattless current for exactly this reason. A general series LCR circuit sits somewhere in between, with power genuinely dissipated only in its resistive part, however the reactive elements shift the phase. This has a direct, costly consequence for real power transmission: delivering a fixed amount of power P at a fixed voltage V requires a current I = P/(V cosf), so a LOW power factor means a correspondingly LARGER current is needed, and since transmission losses scale as I²R, a poor power factor genuinely wastes power as unnecessary heat in the wires. Power factor can be improved, pushed back toward 1, by adding a capacitor of the right size in parallel with an inductive load (motors and transformers, which are commonly inductive, are the usual culprits): this supplies a leading wattless current that cancels the lagging wattless current already present, leaving only the genuinely power-carrying component of current behind, cutting transmission losses without changing the actual work being done.

A transformer changes an alternating voltage from one value to another using nothing more than mutual induction (Chapter 6): two separate coils, a primary of Np turns and a secondary of Ns turns, wound on a shared soft-iron core, insulated from each other but linked through the very same alternating magnetic flux. An alternating voltage applied to the primary drives a current that produces a changing flux f in the core; that same flux, threading through the secondary too, induces its own emf there, es = -Ns(df/dt), while inducing a back emf in the primary itself, ep = -Np(df/dt). For an ideal transformer, negligible primary resistance forces ep to equal the applied primary voltage vp exactly (otherwise an infinite current would flow), and, if the secondary draws only a modest current, es likewise closely matches the secondary's own terminal voltage vs; dividing these two induced-emf expressions, since both share the identical df/dt, gives a strikingly simple relationship: vs/vp = Ns/Np. Assuming, further, that the transformer loses no energy at all, input power must equal output power, ipvp = isvs, which combines with the voltage relation to give the current relation too: is/ip = Np/Ns. A secondary with MORE turns than the primary steps voltage UP but current correspondingly DOWN (a step-up transformer); a secondary with FEWER turns steps voltage DOWN and current UP (a step-down transformer); either way, the power delivered stays exactly the same, voltage and current simply trade off against each other. Real transformers, though often reaching upward of 95 percent efficiency, do lose some energy: flux leakage (not all the primary's flux actually threading the secondary), resistive heating in the windings themselves, eddy currents induced directly within the iron core (reduced by using a laminated, rather than solid, core), and hysteresis losses from repeatedly reversing the core's own magnetisation. It is exactly this ability to step voltage up and down so efficiently that makes long-distance power transmission genuinely practical: a generating station's output is stepped up to a very high voltage for transmission, cutting the current (and hence I²R losses) dramatically, then stepped back down, in stages, at substations and utility poles, until a safe voltage finally reaches an ordinary home.