 # Lec. (4) Chapter (2) AC- circuits Capacitors and transient current 1.

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Lec. (4) Chapter (2) AC- circuits Capacitors and transient current 1

2.1 – Introduction Capacitor displays its true characteristics only when a change in voltage is made in the network. A capacitor is constructed of two parallel conducting plates separated by an insulator. Capacitance is a measure of a capacitor’s ability to store charge on its plates. Transient voltages and currents result when circuit is switched 2

2.2 – Capacitance o Capacitance is a measure of a capacitor’s ability to store charge on its plates. o A capacitor has a capacitance of 1 farad (F) if 1 coulomb (C) of charge is deposited on the plates by a potential difference of 1 volt across its plates. o The farad is generally too large a measure of capacitance for most practical applications, so the microfarad (10  6 ) or picofarad (10  12 ) is more commonly used. 3

A charged parallel plate capacitor. Q = C V where C = e o A / d for a parallel plate capacitor, where e o is the permittivity of the insulating material (dielectric) between plates. Recall that we used Gauss's Law to calculate the electric field (E) between the plates of a charged capacitor: E = s / e o where there is a vacuum between the plates. V ab = E d, so E = V ab /d The unit of capacitance is called the Farad (F). 4

 Fringing – At the edge of the capacitor plates the flux lines extend outside the common surface area of the plates. 2.2 – Capacitance

1. Fixed: mica, ceramic, electrolytic, tantalum and polyester-film 2. Variable Capacitors: The capacitance is changed by turning the shaft at one end to vary the common area of the movable and fixed plates. The greater the common area the larger the capacitance. 2.3 Types of Capacitors: 6

The potential energy stored in the system of positive charges that are separated from the negative charges is like a stretched spring that has potential energy associated with it. Capacitors can store charge and ENERGY  U = q  V and the potential V increases as the charge is placed on the plates 7

(V = Q / C) Since the V changes as the Q is increased, we have to integrate over all the little charges “dq” being added to a plate:  U = q  V 8 U: is the energy stored in a capacitor

Energy density: This is an important result because it tells us that empty space contains energy if there is an electric field (E) in the "empty" space. If we can get an electric field to travel (or propagate) we can send or transmit energy and information through empty space!!! 9

The charges induced on the surface of the dielectric (insulator) reduce the electric field. 10

You slide a slab of dielectric between the plates of a parallel-plate capacitor. As you do this, the charges on the plates remain constant. What effect does adding the dielectric have on the energy stored in the capacitor? A. The stored energy increases. B. The stored energy remains the same. C. The stored energy decreases. D. not enough information given to decide Q1. 2 U α E 2 11

2.5 Capacitors are in Series: When capacitors are in series, the charge is the same on each capacitor.

2.5 Capacitors are in Parallel When capacitors are in parallel, the total charge is the sum of that on each capacitor.

Charging a capacitor that is discharged –When switch is closed, the current instantaneously jumps to E/R –Exponentially decays to zero When switching, the capacitor looks like a short circuit Voltage begins at zero and exponentially increases to E volts Capacitor voltage cannot change instantaneously

2.5 – Initial Conditions The voltage across a capacitor at the instant of the start of the charging phase is called the initial value. Once the voltage is applied the transient phase will commence until a leveling off occurs after five time constants called steady-state as shown in the figure.

R V C ++ -- a b RC-Circuit: Resistance R and capacitance C in series with a source of emf V. Start charging capacitor... Applying KVL R V C ++ -- a b i

RC Circuit: Charging Capacitor Rearrange terms to place in differential form: R V C ++ -- a b i Multiply by C dt :

RC Circuit: Charging Capacitor Instantaneous charge q on a charging capacitor:

RC Circuit: Charging Capacitor At time t = 0: q = CV(1 - 1); q = 0 At time t =  : q = CV(1 - 0); q max = CV The charge q rises from zero initially to its maximum value q max = CV

Example 2.1. What is the charge on a 4  F capacitor charged by 12V for a time t = RC? Time, t Q max q Rise in Charge Capacitor  0.63 Q The time  = RC is known as the time constant. R = 1400  V 4  F ++ -- a b i e = 2.718

Example 2.1 (Cont.) What is the time constant  ? Time, t Q max q Rise in Charge Capacitor  0.63 Q The time  = RC is known as the time constant. R = 1400  V 4  F ++ -- a b i In one time constant (5.60 ms in this example), the charge rises to 63% of its maximum value (CV).  = (1400  )(4  F)  = 5.60 ms

RC Circuit: Decay of Current As charge q rises, the current i will decay. Current decay as a capacitor is charged: when t = 0 when t =  i=0

Current Decay Time, t I i Current Decay Capacitor  0.37 I R V C ++ -- a b i when t = 0 when t =  i=0

24 The placement of charge on the plates of a capacitor does not occur instantaneously. Transient Period – A period of time where the voltage or current changes from one steady-state level to another. The current ( i c ) through a capacitive network is essentially zero after five time constants of the capacitor charging phase.

Steady State Conditions Circuit is at steady state When voltage and current reach their final values and stop changing Capacitor has voltage across it, but no current flows through the circuit Capacitor looks like an open circuit 25

Example 2.2. What is the current i after one time constant (  RC)? Given R=1400  and C=4  F. The time  = RC is known as the time constant. i.e t =  = RC

Charge and Current During the Charging of a Capacitor. Time, t Q max q Rise in Charge Capacitor  0.63 I Time, t I i Current Decay Capacitor  0.37 I In a time  of one time constant, the charge q rises to 63% of its maximum, while the current i decays to 37% of its maximum value.

28 Capacitor Discharging Assume capacitor has E volts across it when it begins to discharge Current will instantly jump to –E/R Both voltage and current will decay exponentially to zero

29 RC Circuit: Discharge R V C ++ -- a b After C is fully charged, we turn switch to b, allowing it to discharge. Discharging capacitor... loop rule gives: R V C ++ -- a b i Negative because of decreasing I.

Discharging From q 0 to q: Instantaneous charge q on discharging capacitor: R V C ++ -- a b i

Discharging Capacitor R V C ++ -- a b i Note q o = CV and the instantaneous current is: dq/dt. Current i for a discharging capacitor.

Prob. 2.3 How many time constants are needed for a capacitor to reach 99% of final charge? 4.61 time constants

Prob. 2.4. Find time constant, q max, and time to reach a charge of 16  C if V = 12 V and C = 4  F. R V 1.8  F ++ -- a b i 1.4 M  C 12 V

Prob. 2.4. continued Time to reach 16  C: t=3.4S

2.12 – Energy Stored by a Capacitor o The ideal capacitor does not dissipate any energy supplied to it. It stores the energy in the form of an electric field between the conducting surfaces. o The power curve can be obtained by finding the product of the voltage and current at selected instants of time and connecting the points obtained. o W C is the area under the curve.

2.14 – Applications Capacitors find applications in: o Electronic flash lamps for cameras o Line conditioners o Timing circuits o Electronic power supplies 36

37 An RC Timing Application RC circuits Used to create delays for alarm, motor control, and timing applications Alarm unit shown contains a threshold detector When input to this detector exceeds a preset value, the alarm is turned on

38 An RC Timing Application Pulses have a rise and fall time Because they do not rise and fall instantaneously Rise and fall times are measured between the 10% and 90% points

39 The Effect of Pulse Width Width of pulse relative to a circuit’s time constant Determines how it is affected by an RC circuit If pulse width >> 5  Capacitor charges and discharges fully With the output taken across the resistor, this is a differentiator circuit If pulse width = 5  Capacitor fully charges and discharges during each pulse If the pulse width << 5  Capacitor cannot fully charge and discharge This is an integrator circuit

40 Simple Wave-shaping Circuits Circuit (a) provides approximate integration if 5  >>T Circuit (b) provides approximate differentiation if T >> 5 

41 Capacitive Loading(stray capacitance): Stray Capacitance Occurs when conductors are separated by insulating material Leads to stray capacitance In high-speed circuits this can cause problems

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