Capacitors: Review A capacitor is a device that stores electrical potential energy by storing separated + and – charges –2 conductors separated by insulating.

Capacitors: Review• A capacitor is a device that stores electrical potential

energy by storing separated + and – charges– 2 conductors separated by insulating medium– + charge put on one conductor, equal amount of – charge

put on the other conductor– A battery or power supply typically supplies

the work necessary to separate the charge

• Simplest form of capacitor is the parallel plate capacitor– 2 parallel plates, each with same area A,

separated by distance d– Charge +Q on one plate, –Q on the other– Looks like a sandwich on a circuit diagram

Capacitors: Review• The charge Q on and voltage V across a capacitor

are related through the capacitance C

– “Capacity” to hold charge for a given V – 1 F is very large unit: typical values of C are F, nF, or pF– Capacitance depends on the geometry of the plates and

the material (dielectric) between the plates– “Static” description of capacitors

• A “dynamic” description of capacitor behavior comes from taking the time derivative of the above:– Current passed by a capacitor

depends on rate of change of V

V

QC Units: C / V = Farad (F)

dt

dVCI

Capacitors: Review• Water–pipe analogy of a capacitor

– Capacitor can be regarded as an enlargement in a water pipe with a flexible membrane stretched across the enlargement (see figure below)

– No water actually passes completely through pipe, but a surge of water flows out of the right–hand pipe

– For capacitor, no DC current flows through, but AC current does

– A stiff (flexible) membrane corresponds to small (large) capacitance

(Introductory Electronics, Simpson, 2nd Ed.)

RC Circuits: Review• Consider a circuit with a resistor and an uncharged

capacitor in series with a battery:

– Voltage across capacitor increases with time according to:

• A = –Vi since V = 0 at t = 0 • Vi = maximum (battery) voltage (only reached at t = ,

but 99% of Vi reached in t = 5)

RCti eVV /1

R

VVI

dt

dVC i RCt

i AeVV /

V

Vi

0.63Vi

V

Vi – V

Vi

(Lab 2–1)

RC Circuits: Review• Consider a circuit with a charged capacitor, a

resistor, and a switch:

– Before switch is closed, V = Vi and Q = Qi = CVi

– After switch is closed, capacitor discharges and voltage across capacitor decreases exponentially with time:

RCti eVV /

R

VI

dt

dVC

= RC = time constantV

Vi

0.37Vi

(Lab 2–1)

RC Circuits: Differentiators• Now consider the series RC circuit as a voltage

divider, with the output corresponding to the voltage across the resistor:

– The voltage across C is Vin – V, so:

– If RC is small, then and

• Thus the output differentiates the input!– Simple rule of thumb: differentiator works well if

R

VIVV

dt

dC in

R

V

dt

dVC in )()( in tV

dt

dRCtV

dtdVdtdV // in

inout VV

(The Art of Electronics, Horowitz and Hill, 2nd Ed.)

(Lab 2–2)

RC Circuits: Differentiators

• Output waveform when driven by square pulse input:

• What would happen if = RC were too big? (See Fig. 1.38 in the textbook for an indication of what would happen)


RC Circuits: Integrators• Now flip the order of the resistor and capacitor, with

the output corresponding to the voltage across the capacitor:

– The voltage across R is Vin – V, so:

– If RC is large, then and

• Thus the output integrates the input!– Simple rule of thumb: integrator works well if

R

VV

dt

dVCI

in

R

V

dt

dVC in constant)(

1)( in dttV

RCtV

inVV

inout VV


(Lab 2–3)

RC Circuits: Integrators

• Output waveform when driven by square pulse input:

• What would happen if = RC were too small? (See Fig. 1.33 in the textbook)


(H&H)

Inductors: Review• Inductors act as current stabilizers

– The larger the inductance in a circuit, the larger the opposition to the rate of change of the current

– Remember that resistance is a measure of the opposition to current

• The rate of current change in an inductor depends on the voltage applied across it – Putting a voltage across an inductor causes

the current to rise as a ramp– Note the difference between inductors and capacitors

• For capacitors, supplying a constant current causes the voltage to rise as a ramp

• An inductor is typically a coil of wire (hence its appropriate circuit symbol)

dt

dILV

Voltage vs. Current in AC Circuits: Review

The RLC Series Circuit: Review

• The instantaneous current in the circuit is the same at all points in the circuit

• The net instantaneous voltage v supplied by the AC source equals the sum of the instantaneous voltages across the separate elements

Series circuit consisting of a resistor, an inductor, and a capacitor connected to an AC generator

ftIi 2sinmax

LCR vvvv

The RLC Series Circuit: Review• But voltages measured with an AC voltmeter (Vrms)

across each circuit element do not sum up to the measured source voltage– The voltages across each circuit

element all have different phases (see figure at right)

• We use the algebra of complex numbers to keep track of the magnitude and phases of voltages and currents

V(t) = Re(Ve jt) I(t) = Re(Ie

jt) where = 2f V, I are complex representations j = (–1)1/2 (see Appendix B)

(Phase relations for RLC circuit)

Impedance• With these conventions for representing voltages and

currents, Ohm’s law takes a simple form: V = IZ

– V = complex representation of voltage applied across a circuit = V0e

j

– I = complex representation of circuit current = I0e j

– Z = total complex impedance (effective resistance) of the circuit

• For a series circuit: Z1 + Z2 + Z3 + …

• For a parallel circuit: 1 / Z = 1 / Z1 + 1 / Z2 + 1 / Z3 + …

• The impedance of resistors, capacitors, and inductors are given by:– ZR = R (resistors)– ZC = XC = –j / C = 1 / jC (capacitors)– ZL = XL = jL (inductors)

Complex Representation Example• The presence of the complex number j simply takes

into account the phase of the current relative to the voltage

• Example: place an inductor across the 110 V (rms) 60 Hz power line– The phase of the voltage is arbitrary,

so let V = V0 V(t) = Re(Ve jt)

V(t) = Re(Vcost + j Vsint) = V0cost

– For an inductor, ZC = j L

– So the (complex) current is given by: I = V / Z = V0 / j L = –V0 j / L

– The actual current is then I(t) = Re(Ie jt)

= Re(Icost + j Isint) = (V0 / L)sint current lags the voltage by 90°

Phasor Diagrams• Can also use phasor diagrams to keep track of

magnitude and phases of voltages– x axis represents the “real” part of the circuit impedance

(resistance)– y axis represents the “imaginary” part of the circuit

impedance (capacitive or inductive reactance)– Draw vectors to represent the impedances (with their

signs); add the vectors to determine combined series impedance

– Axes also represent (complex) voltages in a series circuit since the current is the same everywhere, so voltage is proportional to the impedance

Phasor Diagrams• Example: series RC circuit

– Total (input) voltage is obtained from a vector sum– Note that the vectors indicate phase as well as amplitude– Remember the mnemonic “ELI the ICE man”

• In an inductive circuit (L), the voltage E leads the current I

• In a capacitive circuit (C), the current I leads the voltage E

(Student Manual for The Art of Electronics, Hayes and Horowitz, 2nd Ed.)

= phase angle between input voltage and voltage across resistor or between input voltage and current

AC Power• The instantaneous power delivered to any circuit

element is given by P(t) = V(t)I(t)• Usually, however, it is much more useful to consider

the average power: Pave = Re(VI*) = Re(V*I)– V and I are complex rms amplitudes

• Example: hook up an inductor to a 1 V (rms) sinusoidal power supply– V = 1– I = V / ZL = V / jL = –j V / L

– Pave = Re(VI*) = Re(j V / L) = 0 – Same result holds for a capacitor (this fun activity is free!)

• All power delivered to an AC circuit is dissipated by the resistors in the circuit: – In general: where cos = power factor

RVRIP R /22rmsave

cosrmsin,rmsave VIP

RC Circuits: High–Pass Filters• Let’s interpret the differentiator RC circuit

as a frequency-dependent voltage divider (“frequency domain”):

– ZC = –j / C = –j / 2f C As f increases (decreases), ZC decreases (increases)

– Thus Vout (= voltage across R) increases with increasing f and Vout / Vin 1

– Circuit passes high-frequency input voltage to output

R1

R2

Resistor–only divider:

in21

2out V

RR

RV

RC differentiator circuit:

R

C


(Lab 2–5)

RC Circuits: Low–Pass Filters• Now simply switch the order of the resistor

and capacitor in the series circuit (same order as the integrator circuit earlier):

– ZC = –j / C = –j / 2f C As f increases (decreases), ZC decreases (increases)

– Thus Vout (= voltage across C) increases with decreasing f and Vout / Vin 1

– Circuit passes low-frequency input voltage to output

R1

R2

Resistor–only divider:

in21

2out V

RR

RV

RC integrator circuit:


R

C

(Lab 2–4)

RC Filter Frequency Response• The point where the output “turns the corner” is

called the 3dB point– Output is attenuated by 3dB relative

to the input– Special because a signal reduced by 3dB delivers half its

original power

• A graph of Vout (or Vout / Vin) vs. f is called the frequency response of the RC filter:

RCf

2

1dB3

(for both types of filters)


Example Problem #1.25

Solution details given in class.

R

C

Use a phasor diagram to obtain the low-pass filter response formula (Vout vs. Vin) on p. 37 of Horowitz and Hill.

Example Problem: Additional Exercise #1.3

Solution details given in class.

Design a “rumble filter” for audio. It should pass frequencies greater than 20 Hz (set the –3dB point at 10 Hz). Assume zero source impedance (perfect voltage source) and 10k (minimum) load impedance (that’s important so that you can choose R and C such that the load doesn’t affect the filter operation significantly).

Capacitors: Review A capacitor is a device that stores electrical potential energy by storing separated + and – charges –2 conductors separated by insulating.

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