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# caps1 - ENG17 Capacitors and Inductors Prof Richard Spencer...

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1 ENG17 Capacitors and Inductors Prof. Richard Spencer Capacitors A capacitor is a component that stores electric charge, just like a reservoir stores water A simple capacitor can be made by putting any two conductive plates close to, but not touching, each other

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2 Uncharged Capacitor When a capacitor is uncharged, the plates are globally electrically neutral (i.e., they have equal amounts of positive and negative charge) negative charge (red) positive charge (blue) Charged Capacitor When an electrical potential is applied, mobile electrons are pulled from one plate and added to the other so that there is net positive charge on one plate and net negative charge on the other This global charge separation sets up an electric field between the plates Because the charges on the plates attract each other, the charge separation will persist if the capacitor is removed from the circuit
3 Capacitor Current Current cannot flow in one direction through a capacitor indefinitely - the cap will eventually charge up to the potential driving it and the current will stop charges never really flow “through” the capacitor, we speak of the displacement current (i.e., charge moving on one side displaces charge on the other due to the field) The capacitance is defined by C = Q / V and is measured in Farads (in honor of Faraday) The capacitor current is And the voltage is () dq t dCv t dv t it C dt dt dt == = 0 1 (0 ) ( ) t vt v i d C α =+ Energy Storage in a Capacitor The energy stored in a capacitor can be found by integrating the power delivered up to that time we have assumed that v (- ) = 0 and turns out to only be a function of the voltage on the capacitor at that time 2 1 () () ( ) 2 t c t t Ut p d dv vC d d Cv d v C αα −∞ −∞ −∞ = ⎛⎞ = ⎜⎟ ⎝⎠

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4 DC & Transients in a Capacitor Because the current is proportional to the time derivative of the voltage, if the voltage is constant (i.e., DC), the current is zero and a capacitor looks like an open circuit Physically, this simply means that the charge on the capacitor is not changing (see the water model next for good intuition for this!) Notice that because the charge must change in order to change the voltage, you cannot change the voltage on a capacitor instantly unless you supply an impulse of current, so it looks like a short circuit to step changes in current () dv t it C dt = Water Model of a Capacitor One water model for a capacitor allows a plunger to move left or right depending on the pressure difference across it Energy is stored in the spring, and current only flows while the plunger is moving, i.e., flow = C ( d / dt )pressure, or for a circuit, i = C ( dv / dt )
5 RC Circuit Example Consider the water circuit shown below Turn on the motor at t = 0 The current flow is largest at the start when the capacitor spring is relaxed The flow decreases as the spring is stretched and applies more back pressure (so the pressure across the resistor is lower)

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caps1 - ENG17 Capacitors and Inductors Prof Richard Spencer...

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