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Capacitor

Capacitor. A circuit element that stores electric energy and electric charges. A capacitor always consists of two separated metals, one stores +q, and the other stores –q. A common capacitor is made of two parallel metal plates. Capacitance is defined as: C=q/V (F); Farad=Colomb/volt.

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Capacitor

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  1. Capacitor A circuit element that stores electric energy and electric charges A capacitor always consists of two separated metals, one stores +q, and the other stores –q. A common capacitor is made of two parallel metal plates. Capacitance is defined as: C=q/V (F); Farad=Colomb/volt Once the geometry of a capacitor is determined, the capacitance (C) is fixed (constant) and is independent of voltage V. If the voltage is increased, the charge will increase to keep q/V constant Application: sensor (touch screen, key board), flasher, defibrillator, rectifier, random access memory RAM, etc.

  2. Capacitor: cont. • Because of insulating dielectric materials between the plates, i=0 in DC circuit, i.e. the braches with Cs can be replaced with open circuit. • However, there are charges on the plates, and thus voltage across the capacitor according to q=Cv. • i-v relationship: i = dq/dt = C dv/dt • Solving differential equation needs an initial condition • Energy stored in a capacitor: WC =1/2 CvC(t)2

  3. series parallel Capacitors in V=V1+V2+V3 q=q1=q2=q3 V=V1=V2=V3 q=q1+q2+q3

  4. Inductor i-v relationship: vL(t)= LdiL/dt L: inductance, henry (H) Energy stored in inductors WL = ½ LiL2(t) In DC circuit, can be replaced with short circuit

  5. Sinusoidal waves • Why sinusoids: fundamental waves, ex. A square can be constructed using sinusoids with different frequencies (Fourier transform). • x(t)=Acos(wt+f) • f=1/T cycles/s, 1/s, or Hz • w=2pf rad/s • f =2p (Dt / T) rad =360 (Dt / T) deg.

  6. Average and RMS quantities in AC Circuit It is convenient to use root-mean-square or rms quantities to indicate relative strength of ac signals rather than the magnitude of the ac signal.

  7. b a Complex number review Euler’s indentity

  8. Phasor How can an ac quantity be represented by a complex number? Acos(wt+q)=Re(Aej(wt+q))=Re(Aejwtejq ) Since Re and ejwtalways exist, for simplicity Acos(wt+q)  Aejq=Aq Phasor representation Any sinusoidal signal may be mathematically represented in one of two ways: a time-domain form v(t) = Acos(wt+q) and a frequency-domain (or phasor) form V(jw) = Aejq=Aq In text book, bold uppercase quantity indicate phasor voltage or currents Note the specific frequency w of the sinusoidal signal, since this is not explicit apparent in the phasor expression

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