apacitors and Inductors Overview Defining equations Key concepts and important properties Series and parallel equivalents Integrator Differentiator Portland State University EE 221 apacitors and Inductors Ver. 1.46 1 apacitors: oncept v c i c d onducting plates each with area A Dielectric with permittivity e v c i c Physical oncept Symbol onceptually, a capacitor consists of two conducting plates harge builds up on each of the plates Does this violate K (conservation of charge)? apacitors and inductors are almost as common as resistors The resistance of physical capacitors is usually 100 MΩ and can be safely ignored Portland State University EE 221 apacitors and Inductors Ver. 1.46 3 Portland State University EE 221 apacitors and Inductors Ver. 1.46 4 apacitance usually represented by the letter 1 F = 1 /V The strength of capacitors is measured in Farads The charge stored is directly proportional to the voltage q = charge in oulombs () = capacitance in Farads (F) v = voltage in Volts (V) where q = v apacitors: Relationship of harge and Potential Portland State University EE 221 apacitors and Inductors Ver. 1.46 2 alled energy storage elements At different times they may either produce or absorb energy Theseelementscanstoreenergy We will now discuss elements that depend on time Up to this point we have assumed everything happens instantaneously Introduction: The Impact of Time
Portland State University EE 221 apacitors and Inductors Ver. 1.46 7 = 1 i(τ)dτ Alternatively, if we assume limt =0 = 1 t0 i(τ)dτ v(t0) t0 i(τ)dτ = dv(τ) i(τ)dτ = d v(t0) = d = d We can also solve for in terms of : apacitors: Voltage versus urrent Portland State University EE 221 apacitors and Inductors Ver. 1.46 5 = capacitance in Farads e = permittivity in 2 /N m 2 A = area in m 2 d = length in m = ea d,where Physical oncept Symbol v c v c d Dielectric with permittivity e i c i c onducting plates each with area A apacitors: Physical Dimensions Portland State University EE 221 apacitors and Inductors Ver. 1.46 8 apacitors are linear. = a1i1(t)a2i2(t). = d(a 1v1(t)a2v2(t)), = a1 dv 1(t) a2 dv 2(t), If i1(t) = dv 1(t) and i2(t) = dv 2(t),thenavoltage =a1v1(t) a2v2(t) would produce = a1i1(t) a2i2(t) i2(t) =f (v2(t)) =f (a1v1(t)a2v2(t)) i1(t) =f (v1(t)) =a1v1(t)a2v2(t) Adeviceislinear if and only if the relationship between the device voltage and current is linear: apacitors: inearity Portland State University EE 221 apacitors and Inductors Ver. 1.46 6 This equation defines the behavior of capacitors (just as Ohm s law is the defining equation for resistors). = d d q(t) = d q(t) = apacitors: Defining Equation
Portland State University EE 221 apacitors and Inductors Ver. 1.46 11 If the voltage is constant, =0for all t Thus, if a D voltage is applied, i =0 What other circuit element has this property? How quickly can the current i change? How quickly can the voltage v change? How much energy does a capacitor dissipate? = d apacitors: Key oncepts Portland State University EE 221 apacitors and Inductors Ver. 1.46 9 = 2 v(t o) 2 v(to) = v(τ)dv(τ) = v(τ)2 v(to) to to w(t) w(t0) = p(τ)dτ = v(τ) dv(τ) dτ dτ The energy stored by a capacitor from time to to time t is given by, The power absorbed by a capacitor is given by, ( p(t) = = d ) apacitors: Energy Stored Portland State University EE 221 apacitors and Inductors Ver. 1.46 12 Find an expression for and plot versus t. t (s) t (s) 0.1 mf 10 (V) (ma) Example 1: apacitor Defining Equations Portland State University EE 221 apacitors and Inductors Ver. 1.46 10 Thus, the energy stored in a capacitor is given by w(t) = 2 By convention, this is the 0reference At what v can we no longer extract energy from the capacitor? Similar to potential energy: w = mgh, whereish =0? We need a reference energy to answer this question The equation above only tells us how much energy was stored in the capacitor over a period of time w(t) w(t0) = 2 v(t o) 2 We want an absolute measure of energy so we can answer the question: How much energy is stored in the capacitor? From the previous slide, the energy stored by a capacitor from time to to time t is given by, apacitors: Energy Stored ontinued
Example 2: apacitor Defining Equations Example 3: apacitors in Parallel 1 2 eq i 1 i 2 Find i1(t) and i2(t). Portland State University EE 221 apacitors and Inductors Ver. 1.46 15 (ma) (V) 0.1 mf 1 t (s) t (s) Find an expression for and plot versus t. Thevoltageacross the capacitor at t =0is 5 V. Portland State University EE 221 apacitors and Inductors Ver. 1.46 13 apacitors: Parallel Equivalent i 1 i 2 i 3 1 2 3 eq We need to find the value of eq that makes the relationship between vs(t) and is(t) the same for both circuits. eft circuit: Right circuit, is(t) = i1(t)i2(t)i3(t) dvs(t) dvs(t) dvs(t) = 1 2 3 = (1 2 3) d(t) dvs(t) is(t) = eq eq = Portland State University EE 221 apacitors and Inductors Ver. 1.46 14 Example 3: Workspace Portland State University EE 221 apacitors and Inductors Ver. 1.46 16
apacitors: Series Equivalent 1 2 v 1 v 3 v 3 s eq Example 4: Workspace Portland State University EE 221 apacitors and Inductors Ver. 1.46 19 v 2 vs(t) = v1(t)v2(t)v3(t) vs(t) = eq = = 1 1 ( 1 = ( 1 1 eq is(τ)dτ 1 1 2 3 ) ) is(τ)dτ is(τ)dτ 1 3 is(τ)dτ is(τ)dτ Portland State University EE 221 apacitors and Inductors Ver. 1.46 17 Example 4: apacitors in Series 1 v 1 v 2 2 eq Find v1(t) and v2(t). Portland State University EE 221 apacitors and Inductors Ver. 1.46 18 apacitors: Series & Parallel Summary The equivalent capacitance of N capacitors connected in parallel is the sum of the individual capacitances eq = 1 2 N The equivalent capacitance of N capacitors connected in series is the reciprocal of the sum of reciprocals of the individual capacitances 1 eq = 1 1 1 N Two capacitors in series have an equivalent capacitance given by eq = 12 1 2 Will not discuss delta and wye configurations Portland State University EE 221 apacitors and Inductors Ver. 1.46 20
Inductors: oncept ore Material Number of Turns, N rosssecitonal area, A ength, l Example 5: apacitor Network Equivalent 60 µf 10 µf eq 30 µf 40 µf 50 µf 20 µf Find the equivalent capacitance eq. Portland State University EE 221 apacitors and Inductors Ver. 1.46 21 onceptually, an inductor can be thought of a coil of conducting wire wrapped around some magnetic material Energy is stored in the magnetic field created by current flowing through the wire Inductance (strength) measured in henrys (H) Portland State University EE 221 apacitors and Inductors Ver. 1.46 23 Inductor: Relationship Physical Dimensions ore Material Number of Turns, N rosssecitonal area, A ength, l Example 5: Workspace Portland State University EE 221 apacitors and Inductors Ver. 1.46 22 = N 2 µa,where l = inductance in Henrys (H) N = number of turns µ = magnetic permeability of the core A = crosssectional area (m 2 ) l = length (m) Portland State University EE 221 apacitors and Inductors Ver. 1.46 24
General Inductor Physics i v φ Faraday s aw: where v = N dφ φ = NPi v = N dφ di di = N 2 P di = di v = voltage in volts (V) N = number of turns φ = magnetic flux in webers (Wb) t = time in seconds (s) P = permeance of the flux space i = current in amperes (A) = inductance in henrys (H) Portland State University EE 221 apacitors and Inductors Ver. 1.46 25 Inductors: omments ike capacitors, inductors are linear devices (proof left as exercise) Physical (i.e. practical, nonideal) inductors Range is typically from a few µh to tens of henrys Resistance is usually not negligible (0.1 100 Ω) apacitors and inductors are duals Portland State University EE 221 apacitors and Inductors Ver. 1.46 27 Inductors: Defining Equations = d = 1 t0 w(t) w(t0) = t0 p(τ)dτ v(τ)dτ i(t0) = i2 (t) i2 (t0) p(t) = w(t) = i2 (t) We can not extract energy from the inductor when i =0,soweuse this as our reference for 0 energy. Portland State University EE 221 apacitors and Inductors Ver. 1.46 26 Inductors: Key oncepts = d If the current i is constant, v =0 Thus, if a D current is applied, v =0 What other circuit element has this property? How quickly can the voltage v change? How quickly can the current i change? How much energy does an inductor dissipate? Portland State University EE 221 apacitors and Inductors Ver. 1.46 28
1 v 1 Inductors: Series Equivalent 2 v 3 3 eq v 2 Example 6: Inductor Defining Equations We need to find the value of eq that makes the relationship between vs and is the same for both circuits. eft circuit: Right circuit, vs(t) = v1(t)v2(t)v3(t) dis(t) dis(t) dis(t) = 1 2 3 = (1 2 3) d(t) dis(t) vs(t) = eq eq = Portland State University EE 221 apacitors and Inductors Ver. 1.46 31 (A) (mv) 1 10 mh t (s) t (s) Find an expression for and plot versus t. Portland State University EE 221 apacitors and Inductors Ver. 1.46 29 Example 7: Inductor Defining Equations 10 10 mh t (s) t (s) Example 8: Inductors in Series 1 v 1 v 2 2 eq Find v1(t) and v2(t). Portland State University EE 221 apacitors and Inductors Ver. 1.46 32 (mv) (A) Find an expression for and plot versus t. The initial current in the inductor at t =0is 0.5 A. Portland State University EE 221 apacitors and Inductors Ver. 1.46 30
Example 8: Workspace Portland State University EE 221 apacitors and Inductors Ver. 1.46 33 Example 9: Inductors in Parallel 1 2 eq i 1 i 2 Find i1(t) and i2(t). Portland State University EE 221 apacitors and Inductors Ver. 1.46 35 Inductors: Parallel Equivalent i 1 i 2 i 3 1 2 3 eq eft: is(t) = i1(t)i2(t)i3(t) t = 1 vs(τ)dτ 1 vs(τ)dτ 1 1 2 3 ( 1 = 1 1 ) vs(τ)dτ 1 2 3 ( ) 1 t Right: is(t) = vs(τ)dτ eq eq = t t vs(τ)dτ Portland State University EE 221 apacitors and Inductors Ver. 1.46 34 Example 9: Workspace Portland State University EE 221 apacitors and Inductors Ver. 1.46 36
Inductors: Series & Parallel Summary The equivalent inductance of N inductors connected in series is the sum of the individual inductances eq = 1 2 N The equivalent inductance of N inductors connected in parallel is the reciprocal of the sum of reciprocals of the individual inductances 1 eq = 1 1 1 1 2 N Two inductors in parallel have an equivalent inductance given by Similar to resistors eq = 12 1 2 Portland State University EE 221 apacitors and Inductors Ver. 1.46 37 Example 10: Workspace Portland State University EE 221 apacitors and Inductors Ver. 1.46 39 Example 10: Inductor Network Equivalent 50 mh 20 mh eq 60 mh 50 mh 30 mh 70 mh 10 mh 40 mh Find the equivalent inductance eq. Portland State University EE 221 apacitors and Inductors Ver. 1.46 38 Example 11: apacitive Integrator v i R v o R Solve for vo(t). Portland State University EE 221 apacitors and Inductors Ver. 1.46 40
Example 12: Inductive Integrator R v i v o R Solve for vo(t). Portland State University EE 221 apacitors and Inductors Ver. 1.46 41 Example 14: Inductive Differentiator v i R v o R Solve for vo(t). Portland State University EE 221 apacitors and Inductors Ver. 1.46 43 Example 13: apacitive Differentiator R v i v o R Solve for vo(t). Portland State University EE 221 apacitors and Inductors Ver. 1.46 42 Applications: omments Recall: the op amp is useful for implementing a wide range of equations Integrators tend to saturate Often have a feedback resistor to eliminate this problem Will be a key building block in EE 222 Differentiators amplify noise Rarely used in practice Portland State University EE 221 apacitors and Inductors Ver. 1.46 44
Summary apacitors and inductors store energy They have many dual properties apacitors store charge, inductors store energy in a magnetic field Networks of both devices in parallel and series combinations have a single device equivalent Both have a defining equation in the form of a firstorder differential equation apacitors prevent instantaneous voltage changes, inductors prevent instantaneous current changes This makes time an important factor in circuit analysis Portland State University EE 221 apacitors and Inductors Ver. 1.46 45