Modern Instrumentation PHYS 533/CHEM 620

1
Modern InstrumentationPHYS 533/CHEM 620

• Lecture 2
• AC Circuits
• Amin Jazaeri
• Fall 2007

2
AC circuits

• Unlike DC voltage sources, which provides a
  steady, constant emf, an AC source provides an
  emf, or voltage, which varies with time (sine
  function repeats when argument wt increases by
  2p)
• V Vmaxsin wt w
  2pf

• angular frequency
• T period 1/f 2p/w

The ac voltage in your house has a frequency f of
60 Hz 60 oscillations/sec.
3
AC Circuit Components

• Basic Linear Time Invariant (LTI) components
• Resistor, R, ? (Ohms)
• Inductor, L, H (Henrys)
• Capacitor, C, F (Farads)
• Frequency
• Repetition rate, f, Hz (Hertz)
• Angular, ? 2?f, 1/s (radians/sec)

4
AC Circuit Components Resistors

• The behaviour of a resistor in an AC circuit is
  the same behavior as a DC circuit.
• VI/R
• Where both V and I are time dependent
• VVmaxcos(wt)
• IImaxcos(wt)
• There is no phase relationship between the
  voltage and current.

5
AC Circuit Components Capacitors

• Charge in a capacitor produces an electric field
  E, and thus a proportional voltage,
• Q C V,
• Where C is the capacitance.
• The charge on the capacitor changes according to
• I (dQ/dt).
• The instantaneous current is therefore
• I C(dV/dt).
• This is the time domain behavior of a capacitor.

6
AC Circuit Components Capacitors

• If VV0cos(wt) then
• Where
• XC1/wC is known as reactance.
• There is a p/2 phase difference between V and I.
• The voltage lags behind current by p/2

7
Capacitor in an AC Circuit

• The voltage lags behind the current by 90.
  VQ/C When Igt0, capacitor is charging, When
  Ilt0, capacitor is dis-charging.
• The average power consumed by a capacitor in an
  ac circuit is zero.

8
f0 DC
XC ?
infinity
I0
fhigh
I high
XC ?
low
9
AC Components Inductors

• Current in an inductor generates a magnetic
  field,
• B K1 I
• Changes in the field induce an inductive voltage.
• V K2 (dB/dt)
• The instantaneous voltage is
• V L(dI/dt),
• where L K1K2.
• This is the time domain behavior of an inductor.

10
AC Components Inductors

• If VV0cos(wt) then
• Where
• XLwL is known as Inductive reactance.
• There is a p/2 phase difference between V and I
  (The voltage across L leads the current).

11
Inductor in an AC Circuit

• The average power consumed by an inductor in an
  ac circuit is zero.
• The voltage leads the current by 90.

12
Inductive reactance, XL
VLXL I

Dimensional analysis
R
L
XL has units of ohms
13
Inductive reactance, XL
VLXL I

An inductor has higher back emf when ?I/?t is
greater, i.e. at high frequency. Inductive
reactance higher at high frequency.
At low frequency, DC, the current is not blocked,
i.e. low reactance at low frequency.
14
f0
I high
XL ?
0
f high
XL ?
high
I low
15
AC circuits -- Impedance

• Impedance and Ohms Law for AC
• Impedance is Z R jX,
• where j ?-1, and X is the reactance in
  ?.
• Generalized Ohms Law V I Z
• Resistance R dissipates power as heat.
• Reactance X stores and returns power.
• Inductors have positive reactance Xl?L
• Capacitors have negative reactance Xc-1/?C

16
Impedance shortcuts

• The impedance of components connected in series
  is the complex sum of their impedances.

• The impedance of components connected in parallel
  is the reciprocal of the complex sum of their
  reciprocal impedances.

17
LRC Circuit
The effective resistance of the circuit is given
by the impedance Z
I0 V0 / Z II0e j(wtf)
SI unit of impedance ohm
18
LRC Series Circuit

• Resonance w021/(LC) XLXC
• The Impedance at particular frequency becomes a
  real number RZ

19
Resonance in a Series RLC Circuit
The current in a series RLC circuit is
The current will be maximum when XL XC at a
resonance frequency w of
20
LRC Series Circuit
We make a phasor diagram to figure out phase
angles or how current leads or lags voltage
21
LRC Series Circuit
To account for the AC input voltage, we allow
the diagram to rotate counter-clockwise at 2?f
radians/sec. We are still preserving the
relationships between the phase angles of the
resistor, inductor and capacitor.
22
LRC Series Circuit

• The projections of the voltage vectors on the
  x-axis represent the instantaneous voltage drops
  across the three circuit elements.
• To find out the overall voltage from the source,
  we have to make the sum of the voltage drops
  around the closed loop equal zero.
• This means a vector addition.

23
LRC Series Circuit
24
Impedance Matching

• We often have to connect one circuit to another
  circuit
• Measuring device connected to display unit such
  as oscilloscope or chart recorder, etc.
• Two cases are important
• First is to transfer maximum power
• Second is to disturb original circuit as little
  as possible

25
Bridge Circuits

• Bridge circuits are extensively used for
  measuring component values, such as resistance,
  inductance, or capacitance, and of other circuit
  parameters directly derived from component values
• Its accuracy can be very high.
• A bridge is just two voltage dividers in
  parallel. The output is the difference between
  the two dividers.

26
Generaized Bridge
Ratio Arms
a
I2
I1
R1
R2
c
d
Vbias
R3
R4
Unknown Arm
Standard Arm
I4
I3
b
27
Wheatstone Bridge
Ratio Arms
a
I1
I2
Ra
Rx
c
d
Vbias
Rb
R
Unknown Arm
Standard Arm
Ix
I3
b
28
Accuracy

• Theoretically the relative error of Rx is
  summation of others
• In general, it is about 0.2, in comparison to
  at least 4 by ohmmeter.

29
Wheatstone Bridge
Ratio Arms
a
I1
I2
Ra
Rx
c
d
Vbias
Rb
R
Unknown Arm
Standard Arm
Ix
I3
b
30
Thevenin Equivalent of Wheatstone Bridge
Use Thevenin equivalent circuit can calculate the
galvanometer current to check sensitive of the
meter.
Open-circuit
31
Thevenin Equivalent of Wheatstone Bridge
Replace battery by internal resistance
32
General AC Bridge
The general AC bridge can have unspecified
impedance on its four arms. Bridge balance is
obtained by using complex notation of impedance.
It requires equal magnitude and phase.
a
Z2
Z1
c
d
Z3
Z4
b
33
Maxwell Bridge

• Used to determine the L and R of an inductor
  having a large series resistance
• Lx R2R3C
• Rx R2R3/R1
• Appears to work best when ?L/Rlt10

34
Hay Bridge

• Used to measure the L and R of an inductor having
  a small series resistance
• Appears to work best when ?L/Rgt10

35
Schering Bridge

• Used to determine an unknown capacitance

36
Resonant Bridge
Used to measure L or C
If L and C are known the Balance condition can
be used to measure frequency
37
Wien Bridge
Measures frequency when the bridge is balanced