Notes for course EE1.1 Circuit Analysis 2004-05 TOPIC 5 – THE CAPACITOR, INDUCTOR AND TRANSIENT ANALYSIS Objectives • Introduce the capacitor and the inductor • Consider parallel and series connections of capacitors and inductors • Consider power and energy for the capacitor and the inductor • Introduce the switch as a circuit element • Introduce transient analysis through a computer clock rate example • Generalise to transient analysis of any 1st order RC or RL circuit 1 THE CAPACITOR A capacitor is a circuit element that stores charge It can be made by sandwiching an insulator between two conducting plates in a structure called a parallel-plate capacitor: The two conducting plates have area Ac and are separated by a dielectric layer of thickness d As positive charge flows into the capacitor, it begins to accumulate on the plate connected to that terminal, since it cannot continue to flow through the insulating layer This accumulation of positive charge attracts negative charge to the opposite plate This negative charge has to come from somewhere; it can be supplied only by the wire connected to the opposite plate This flow of negative charge constitutes an electrical current oriented out of the opposing plate, which is equal in magnitude to the current flowing into the capacitor From a circuit point of view, it appears that a current is flowing directly through the capacitor Topic 5 – Capacitor, Inductor and Transient Analysis In order to analyse this capacitor, we define parameters as follows: q Charge on the plates v Voltage between plates Ac Area of the plates D Distance between plates ε Permittivity of the dielectric between the plates It can be shown that the charge stored on each of the capacitor plates depends on the voltage across the capacitor in the following way: εA q = c v d The relationship between these two quantities is linear; q gradient C 0 v We call the constant of proportionality the capacitance (symbol: C): q = Cv In general capacitance depends on the geometry and construction of the capacitor For the parallel-plate capacitor we have been considering, the capacitance is given by: εA C = c d ε is usually expressed as: ε = εRε0 –14 where εR is the dielectric constant and ε0 = 8.854 × 10 F/cm is the permittivity of free space Since q has units of coulombs, and v is measured in volts, the fundamental unit of capacitance is Coulomb per Volt (C/V) which is called a Farad (abbreviation: F) Any device capable of storing charge acts as a capacitance, including parasitic capacitance We can derive the terminal current-voltage characteristics of the device by using the definition of current: dq i = dt Differentiating the capacitor q expression results in: dq d dv dC = (Cv) = C + v dt dt dt dt 2 Topic 5 – Capacitor, Inductor and Transient Analysis For a time-invariant capacitor, C is a constant and therefore dC/dt = 0 Hence, the current-voltage relationship for a capacitor is given by: dq i = dt dv = C dt In this equation, i is the current flowing through the capacitor, C is the capacitance, and v is voltage across the capacitor This expression tells us that the current through a capacitor is proportional to the time derivative of the voltage across it As long as the voltage is changing, there is current flowing through the capacitor It follows that there is zero current through a capacitor if the voltage across it is constant, and vice versa The symbol for a capacitor is as follows: We have derived an equation for i in terms of v To obtain an expression for v in terms of i, we can integrate both sides of the equation: 1 v = i(t)dt + v(0) C ∫ We have replaced i by i(t) to emphasise that it is a function of time We have also included an arbitrary constant v(0) (v at time t = 0) which is necessary when integrating v(0) is the voltage stored on the capacitor before we apply the current that is integrated by the capacitor to produce a change in voltage Since t is the time at the end of the integration period, it is more correct to plot i as a function of a dummy variable x, ie i(x) against x, which has the same graph as i(t) against t Assuming that we start applying current at t = 0 and are interested in v at time t, we can more correctly say: t 1 v = i(x)dx + v(0) C ∫ 0 If we apply current at time t1 and are interested in the voltage at time t2, then we have: t 1 2 v = i(x)dx + v(1) C ∫ t1 2 THE INDUCTOR Another circuit element which stores energy is the inductor 3 Topic 5 – Capacitor, Inductor and Transient Analysis A common way of making an inductor is to wind a wire into a coil, as shown: Passing a current through a conductor results in a magnetic field encirc1ing the wire A change in the current results in a change in the magnetic field B If the current is time varying, the magnetic field also varies, in step with the current This causes the magnetic lines of force to cut across the conductor which generates a voltage It may be shown that flux linkage is given by: λ = NAB where A is the area of the coil and N is the number of turns For the coil shown, it may be shown that: Nµi B = l l is the axial length of the coil and µ is a constant known as the permeability Hence, N 2Aµ λ = i l Hence, the flux linkage for an inductor is proportional to the current The constant of proportionality is known as the inductance and is given the symbol L λ = Li For this particular form of inductor, we have: 4 Topic 5 – Capacitor, Inductor and Transient Analysis N 2Aµ L = l The unit of inductance is the Henry (abbreviation: H), which is equivalent to an ohm-second (Ω-s). The inductor equation λ = Li corresponds to the equation q = Cv for the capacitor. The voltage across the inductor terminals is equal to the rate of change of the flux linkage: dλ d di dL v = = (Li) = L + i dt dt dt dt Assuming that the inductance L is time-invariant, we have for the inductor: dt v = L dt This inductor equation corresponds to the equation i = Cdv/dt for the capacitor. Note that a constant current flowing through an inductor corresponds to a zero voltage drop Conversely, if there is zero volts across an inductor, the current through it is constant in time The circuit symbol for an inductor is as follows: The reference direction arrow for the current encounters the + sign of the voltage first We illustrated the phenomenon of inductance using the coil, but it is important to remember that inductance is an intrinsic property of all conductors, regardless of their shape and it exists whenever current flows; even a straight wire has an inductance, although it is very small. We have derived an equation for v in terms of i for the inductor To obtain an expression for i in terms of v, we can integrate both sides of the equation: t 1 i = v(x)dx + i(0) L ∫ 0 where x is a dummy variable. If we apply voltage starting at time t1 and are interested in the current at time t2, then we have: t 1 2 i = v(x)dx + i(t1) L ∫ t1 5 Topic 5 – Capacitor, Inductor and Transient Analysis We give a summary of the relationships between i and v for the capacitor and for the inductor: Expression for i Expression for v dv t Capacitor i = C 1 dt v = i(x)dx + v(0) C ∫ 0 t di Inductor 1 v = L i = v(x)dx + i(0) dt L ∫ 0 The fact that capacitors and inductors can differentiate and integrate voltages and currents makes them indispensable circuit components, especially for processing signals Apart from their use as wanted circuit components, it is the case that other types of components, including resistors and transistors, have unwanted parasitic capacitance and inductance which can be a limiting factor for performance of circuits which operate at high frequencies; hence it is very important to be able to analyze circuits containing capacitors and inductors 2.1 Mechanics analogy The equations describing electronic components can be regarded as direct mappings of the equations governing mechanics. This leads to an electrical–mechanical analogy: Electrical Mechanical Equations dq ds i = v = dt dt v = iR f = vk1 di dv v = L f = Ma = M dt dt 1 1 1 v = ∫ idt f = ∫ vdt = s C k2 k2 Corresponding parameters current i ≡ v velocity charge q ≡ s distance voltage v ≡ f force resistance R ≡ k1 damping constant inductance L ≡ M mass Capacitance C ≡ k2 inverse spring constant 3 THE SWITCH The operation of an electrical switch is familiar to all; it passes current when closed, and does not pass current when open A transistor can acts as a switch 6 Topic 5 – Capacitor, Inductor and Transient Analysis In fact it is this switching action of a transistor which is exploited in all digital circuits, including the computer The symbol for an ideal switch is as shown An ideal switch has two states, open and closed The "switching event" takes place at the time indicated; by convention the switch is drawn in the position it assumes before the switching event The switch shown has a switching event at t = t0 and is shown in the open position; thus, by convention, the switch is open for t < t0, and closed for t> t0 It is also possible to have switches which open at the switching event: "Open" and "closed" are defined as follows: • an open switch has i = 0, with no restriction on the voltage across it • a closed switch has v = 0, with no restriction on the current through it Upon closer inspection, we see that: • an open switch can be replaced with an open-circuit or current source of value zero • a closed switch is identical to a short-circuit or zero-valued voltage source Ideal switches neither draw nor supply power regardless of the position of the switch This can be seen from the power relationship