advertisement

Lecture #12 ANNOUNCEMENTS • Graduate school workshop tomorrow (9/23) – 3-4 PM, Wozniak Lounge (Soda Hall) OUTLINE – – – – Capacitors in series and in parallel Practical capacitors The inductor Inductors in series and in parallel Reading Chapter 6.1-6.3 EECS40, Fall 2003 Lecture 12, Slide 1 Prof. King Correction to Lecture 9, Slide 3 • If there are no independent sources in a circuit, VTh = 0. – If there are dependent sources in the circuit, we need to apply an external voltage in order to determine RTh. Example: Circuit used in RTh Calculation Example #2, Lecture 8: ITEST = Vx VTEST − Vx VTEST − 40i∆ + 20 60 VTEST – + RTh = Applying KCL to node x : Vx − 40i∆ V −V − i∆ + x TEST = 0 16 20 − Vx Definition of i ∆ : i∆ = 80 EECS40, Fall 2003 Lecture 12, Slide 2 ⇒ Vx = VTEST 75 = Ω I TEST 4 8 VTEST 25 Prof. King 1 Example: Current, Power & Energy for a Capacitor t v (V) 1 0 1 2 i (µA) 0 3 i=C 1 2 3 EECS40, Fall 2003 4 5 dv dt 4 i(t) v(t) – + 1 v(t ) = ∫ i (τ )dτ + v(0) C0 t (µs) vc must be a continuous function of time; however, ic can be discontinuous. 5 t (µs) Note: In “steady state” (dc operation), time derivatives are zero Æ C is an open circuit Lecture 12, Slide 3 Prof. King p (W) i(t) 1 2 3 4 5 – + v(t) 0 10 µF 10 µF t (µs) p = vi w (J) 0 t 1 EECS40, Fall 2003 2 3 4 5 Lecture 12, Slide 4 t (µs) 1 w = ∫ pdτ = Cv 2 2 0 Prof. King 2 Capacitors in Parallel i(t) i1(t) i2(t) + C1 C2 v(t) dv dv + C2 dt dt dv dv i = (C1 + C2 ) = Ceq dt dt i = i1 + i2 = C1 – + Ceq = C1 + C2 v(t) Ceq i(t) – i = Ceq dv dt Equivalent capacitance of capacitors in parallel is the sum EECS40, Fall 2003 Lecture 12, Slide 5 Prof. King Capacitors in Series + v1(t) – + v2(t) – i(t) C1 + C2 i(t) Ceq v(t)=v1(t)+v2(t) – dv d (v1 + v2 ) = Ceq dt dt dv dv i = Ceq 1 + Ceq 2 dt dt 1 1 1 ⇒ = + Ceq C1 C2 dv1 dv = C2 2 dt dt dv i dv2 i ⇒ 1= and = dt C1 dt C2 i = C1 i = Ceq EECS40, Fall 2003 i i + Ceq C2 C1 i = Ceq Lecture 12, Slide 6 Prof. King 3 Capacitive Voltage Divider Q: Suppose the voltage applied across a series combination of capacitors is changed by ∆v. How will this affect the voltage across each individual capacitor? ∆v = ∆v1 + ∆v2 ∆Q1=C1∆v1 Q1+∆Q1 C1 v+∆v + – -Q1−∆Q1 Q2+∆Q2 C2 −Q2−∆Q2 ∆Q2=C2∆v2 EECS40, Fall 2003 + v1+∆v1 – Note that no net charge can can be introduced to this node. Therefore, −∆Q1+∆Q2=0 + v2(t)+∆v2 – ⇒ C1∆v1 = C2 ∆v2 ∆v2 = C1 ∆v C1 + C2 Note: Capacitors in series have the same charge. Lecture 12, Slide 7 Prof. King Application Example: MEMS Accelerometer • Capacitive position sensor used to measure acceleration (by measuring force on a proof mass) g1 g2 FIXED OUTER PLATES EECS40, Fall 2003 Lecture 12, Slide 8 Prof. King 4 Sensing the Differential Capacitance – Fixed electrodes are biased at +Vs and –Vs – Movable electrode (proof mass) is biased at Vo Circuit model Vs Vo = −Vs + C1 Vo C2 –Vs EECS40, Fall 2003 C1 C − C2 Vs ( 2Vs ) = 1 C1 + C2 C1 + C2 εA ε A − Vo g1 g 2 g 2 − g1 g 2 − g1 = = = Vs εA + εA g 2 + g1 const g1 g 2 Lecture 12, Slide 9 Prof. King Practical Capacitors • A capacitor can be constructed by interleaving the plates with two dielectric layers and rolling them up, to achieve a compact size. • To achieve a small volume, a very thin dielectric with a high dielectric constant is desirable. However, dielectric materials break down and become conductors when the electric field (units: V/cm) is too high. – Real capacitors have maximum voltage ratings – An engineering trade-off exists between compact size and high voltage rating EECS40, Fall 2003 Lecture 12, Slide 10 Prof. King 5 The Inductor • An inductor is constructed by coiling a wire around some type of form. + vL(t) iL _ • Current flowing through the coil creates a magnetic field or flux that links the coil: LiL • When the current changes, the magnetic flux changes Æ a voltage across the coil is induced: Note: In “steady state” (dc operation), time derivatives are zero Æ L is a short circuit EECS40, Fall 2003 vL (t ) = L Lecture 12, Slide 11 diL dt Prof. King Symbol: L Units: Henrys (Volts • second / Ampere) (typical range of values: µH to 10 H) Current in terms of voltage: 1 vL (t )dt L t 1 iL (t ) = ∫ vL (τ )dτ + i (t0 ) L t0 diL = iL + vL – Note: iL must be a continuous function of time EECS40, Fall 2003 Lecture 12, Slide 12 Prof. King 6 Stored Energy Consider an inductor having an initial current i(t0) = i0 di p (t ) = v(t )i (t ) = Li (t ) dt t i i dι w(t ) = ∫ p (τ )dτ = ∫ Lι dτ = ∫ Lιdι dτ t0 i0 i0 w(t ) = 1 2 1 2 Li − Li0 2 2 EECS40, Fall 2003 Lecture 12, Slide 13 Prof. King Inductors in Series v = Leq + v1(t) – + v2(t) – v(t) + – L1 i(t) L2 v(t) + – di dt i(t) Leq + v(t)=v1(t)+v2(t) – v = L1 di di di di + L2 = (L1 + L2 ) = Leq dt dt dt dt Leq = L1 + L2 Equivalent inductance of inductors in series is the sum EECS40, Fall 2003 Lecture 12, Slide 14 Prof. King 7 Inductors in Parallel + i1 i(t) + i2 i(t) v(t) L2 L1 Leq v(t) – – t t i = i1 + i2 = t 1 1 vdτ + i1 (t0 ) + ∫ vdτ + i2 (t0 ) ∫ L1 t0 L2 t0 i= 1 vdτ + i (t0 ) Leq t∫0 1 1 t i = + ∫ vdτ + [i1 (t0 ) + i2 (t0 )] L1 L2 t0 1 1 1 ⇒ = + with i (t0 ) = i1 (t0 ) + i2 (t0 ) Leq L1 L2 EECS40, Fall 2003 Lecture 12, Slide 15 Prof. King Summary Capacitor Inductor dv dt 1 w = Cv 2 2 i=C v=L w= v cannot change instantaneously i can change instantaneously Do not short-circuit a charged capacitor (-> infinite current!) n cap.’s in series: EECS40, Fall 2003 1 2 Li 2 i cannot change instantaneously v can change instantaneously Do not open-circuit an inductor with current (-> infinite voltage!) n 1 1 n ind.’s in series: =∑ Ceq i =1 Ci n cap.’s in parallel: Ceq = n ∑C i =1 i di dt n ind.’s in parallel: Lecture 12, Slide 16 n Leq = ∑ Li i =1 n 1 1 =∑ Leq i =1 Li Prof. King 8