Resonant Electric Circuits ( Physics )

Alternating current (AC) Production of alternating p.d. The mechanical work of rotating a coil in a magnetic field produces electrical work i.e. moving charges. An alternating emf varies with time and is produced by an alternating voltage power supply. emf = e = Vmsin2ft = Vmsint An alternating emf will produce an alternating current (a.c.) The frequency of an alternating emf (and current) is the number of complete cycles an emf goes through each second. Frequency, f, is measured in hertz (Hz) and the angular frequency, , is measured in radians per second (rd s-1). Adding two p.d.’s with the same frequency Take two alternating potential differences with the same frequency e1 = Vm1 sin(ft + ) = Vm1 sin(t + ) e2 = Vm2 sin(ft + ) = Vm2 sin(t + ) Graphical solution The sum of p.d.'s is also a p.d. with the same frequency but differs in amplitude and initial phase. e3 = e1 + e2 = Vm1 sin(t + ) + Vm2 sin(t + ) = Vm3 sin(t + ) ,  and  are "phase angles". Phase amplitude diagram "Phasors" (not vectors) because the angles are not real in space but "phase angles". Resistors An alternating current in the ciruit with R only i = Iosin(t) and instantenous p.d. across R becomes vR = e = VmR sin(t) = IoRsin(t) Phase amplitude diagram for p.d. and current for the a.c. ciruit with resistor only. p.d. across resistor is in phase (or "in step") with its current. Power dissipated in a resistor Pinstantaneous = Ri2 = R Io2 (sint)2 Derivation is called the r.m.s. current because it is the square root of the mean of the square of the current. The r.m.s. value is effectively a d.c. current which gives the same heating effect as the a.c. current. Using r.m.s. current we may predict r.m.s. voltage. Example The r.m.s. 240 V, 50 Hz mains electricity supply is connected in series with a 1 kW heater. Find the rms and instantaneous values of the p.d. across the heater, and the current in the circuit, and the power dissipated in the heater. Solution Reactive Elements in AC circuits Coils and capacitors react against AC without dissipating power. Parrallel plate capacitor stores charge Alternating current flows in capacitive circuit, where direct current would not flow. The charge Q stored on the plates = CV, where V is the applied voltage Q = CVo sin t I = dQ/dt = d/dt (CVo sin t) = CVo cos t o = CVo   The reactance Xc = Vo/Io = 1/C Power dissipation in a capacitor Instantaneous power p(t) = i(t)v(t) Pav = total energy dissipated in the time interval t = 0 to t = T (period of oscillation) Coils in AC circuits Changing current in a coil produces changing magnetic field and e.m.f. is induced, which opposes the current change: self induction. In DC circuits coils retard current growth and prolong current decay. Magnetic field B in the coil is proportional to current I and B is proportional to I, define self-inductance L:  = LI self induced e.m.f: the unit of self-inductance is henry (H) Energy E stored in an inductor (coil) E = ½ LI2 Coil connected to AC power supply where  = 2f The peak voltage V0 = I0L The reactance, XL , is defined as L The p.d. across the coil is ahead of the current in the circuit by one quarter of a cycle: 90o The inductive reactance increases with frequency of the circuit current Similarly to average power in the capacitor the average power in the coil is zero. Resonant circuits LCR series circuit Define circuit impedance, Z, when XL = XC, Z = R and the circuit is in resonance Resonant frequency f0 (or 0 = 2f0) Resonant Circuit of alternating current-By: M. Abdul Mumeed. +966507433412 19 © Md. Abdul Mumeed, Senior Secondary Teacher in PHYSICS at I. I. School, Riyadh, K.S.A.