Lecture 12 Caps & RC Circuits 10-­‐23-­‐11 Midterm Grades •  Mean = 80, Median = 83, Mode = 95 •  Standard DeviaBon = 14 DescripBon of a Capacitor •  A capacitor is a passive element designed to store energy in its electric field. UE =
1
2
r r
D
∫ • E dV
V
€
•  A capacitor consists of two conducting plates separated
by an insulator (or dielectric).
3 DefiniBon of Capacitance •  Capacitance C is the raBo of the charge q on one plate of a capacitor to the voltage difference v between the two plates, measured in farads (F). and
•  Where ε is the permittivity of the dielectric material between
the plates, A is the surface area of each plate, d is the distance
between the plates.
µF (10–6)
•  Unit: F, pF (10–12), nF (10–9), and
4 Some Typical Capacitors Voltage across a Capacitor •  If i is flowing into the +v terminal of C –  Charging => i is posiBve (> 0) –  Discharging => i is – (< 0) •  The current-voltage relationship of capacitor according to
above convention is
and
Energy Stored in a Capacitor •  The energy, w, stored in the capacitor is •  A capacitor is
–  an open circuit to dc (dv/dt = 0).
–  its voltage cannot change abruptly (depends on
integral of i).
7 Voltage across a Capacitor Example The voltage across a 5-­‐µF capacitor is v(t) = 10 Cos(6000 t) V. Calculate the current through it 8 Parallel Capacitors •  The equivalent capacitance of N parallel-­‐connected capacitors is the sum of the individual capacitances. 9 Series Capacitors •  The equivalent capacitance of N series-­‐connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances. 10 Series and Parallel Capacitors Example 3 Find the equivalent capacitance seen at the terminals of the circuit in the circuit shown below: Answer:
Ceq = 40µF
70µF
11 Source-­‐Free RC Circuit •  A first-­‐order circuit is characterized by a first-­‐order differenBal equaBon. By KCL
Ohms law
•
•
Capacitor law
Apply Kirchhoff’s laws to purely resistive circuit results in algebraic
equations.
Apply the laws to RC (and RL) circuits produces differential equations.
12 Source-­‐Free Response of an RC Circuit •  The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitaBon. Time constant
Decays more slowly
Decays faster
•
•
The time constant τ of a circuit is the time required for the response to decay
by a factor of 1/e or 36.8% of its initial value.
v decays faster for small τ and slower for large τ.
13 First Order DE SoluBon for RC Circuit v
dv
+C
=0
R
dt
•  Use t
lnv = −
+ ln A
RC
•  Manipulate equaBon and
•  Integrate v
t
€
ln( ) = −
•  Manipulate A
RC
and exponetiation both
•  ExponenBate −t
−t
•  Define Bme constant v(t) = Vo e RC = Vo e τ
where
τ = RC
€
sides
SoluBon of DE for Source-­‐Free RC Circuit The key to working with a source-­‐free RC circuit is finding: where
1.  The initial voltage v(0) = V0 across the capacitor.
2.  The time constant τ = RC.
15 Source-­‐Free RC Circuit Example Example 1 Refer to the circuit below, determine vC, vx, and io for t ≥ 0. Assume that vC(0) = 30 V. Answer: vC = 30e–0.25t V ; vx = 10e–0.25t ; io = –2.5e–0.25t A
16 Source-­‐Free RC Circuit Example Example 2 The switch in circuit below is opened at t = 0, find v(t) for t ≥ 0. •  Please refer to lecture or textbook for more detail elaboration.
Answer: V(t) = 8e–2t V
17 Unit-­‐Step FuncBon •  The unit step func<on u(t) is 0 for negaBve values of t and 1 for posiBve values of t. 18 Abrupt Change (Switch on or off) Represent an abrupt change for:
1.  voltage source. 2.  for current source: 19 DE for Step-­‐Response of a RC Circuit •  The step response of a circuit is its behavior when the excitaBon is the step funcBon, which may be a voltage or a current source. •  Initial condition:
v(0-) = v(0+) = V0
•  Applying KCL,
Node
or
•  Where u(t) is the unit-step function
20 Step-­‐Response of a RC Circuit •  IntegraBng both sides and considering the iniBal condiBons, the soluBon of the equaBon is: Final value at
t -> ∞
Complete Response =
=
Natural response
(stored energy)
V0e–t/τ
+
Initial value at
t=0
+
Source-free
Response
Forced Response
(independent source)
Vs(1–e–t/τ)
21 Finding the Step-­‐Response of a RC Circuit Three steps to find out the step response of an RC circuit: 1.  The initial capacitor voltage v(0).
2.  The final capacitor voltage v(∞) — DC voltage
across C.
3.  The time constant τ.
Note: The above method is a short-cut method. You may also determine the
solution by setting up the circuit formula directly using KCL, KVL , ohms law,
capacitor and inductor VI laws.
22