Transcript Data Domains and Transduction

Circuit Theory 2
 Topics
• Impedance matching,
• Impedance transformation,
• Bandwidth and Noise.
 How you will use these topics
• Understanding the rangefinder circuits,
• Optimizing rangefinder performance,
• Understanding transducer models.
Impedance matching
 What is impedance matching?
• Any signal source has an associated source impedance.
• Any “load” circuit driven at a port exhibits an
associated impedance across the terminals of the port.
• Matching these impedances optimizes the transfer of
power from the source to the load.
 A source with a voltage divider:
Rsource 
R1 R2
R1  R2
R1
R2
 A composite load circuit:
Rload
R R  R2 R3  R3 R1
 1 2
R2  R3
R1
R2
R3
Thevenin’s and Norton’s Theorems
 Ideal sources
• An ideal voltage source has zero source impedance.
• An ideal current source has zero source admittance.*
 Thevenin’s theorem: any electrical signal source is
equivalent to an ideal voltage source in series with
a source impedance.
 Norton’s theorem: any electrical signal source is
equivalent to an ideal current source in parallel
with a source admittance.*
* (Admittance is the complex reciprocal of impedance)
Thevenin and Norton example
 Actual source
100
2V
 Thevenin equivalent
100
50
1V
 Norton equivalent
20
mA
50
DC matching: simple example and proof
 A battery with voltage Vg and
internal resistance Rg is
connected to a load resistor Rl.
 The power delivered to Rl is
P  Vl I l 
Vg Rl

Vg
Rg  Rl Rg  Rl

Rg
Vg
Il
Vl
V g2 Rl
( Rg  Rl ) 2
 The maximum deliverable power occurs for
2
2


V
R
V
( Rg  Rl )
dP
d 
l
g
g

0

, 
2
3
dRl
dRl  ( Rg  Rl )  ( Rg  Rl )
Rl  Rg
Rl
Impedance transformation
 Ideally is a lossless two-port circuit.
 Conservation of energy:
V
• avg( I1V1 )  avg( I 2V2 )  0
• Reciprocal: power can flow either way.
I1
I2
Port
1
1
I1
 Realizations:
• Wideband: transformer with n1, n2 turns,
Port
2
V2
I2
V1 n1
n2 V2
V2 / V1  n2 / n1  I1 / I 2 , Z2 / Z1  (n2 / n1 )2
• Resonant “Pi” network: at resonance,
Zin  R2 (C2 / C1 )2
L
Zin
C1
C2
R2
Magnetic transformers
 The magnetic field H  k1n1I1 is proportional to the
total primary current linking the magnetic path.
 Magnetic induction B  H , where is the
permeability of the magnetic path.
 The voltage induced per turn is V  k2dB / dt , so the
secondary voltage V2  kn1n2 dI1 / dt, where k  k1k2 .
 Similarly, V1  kn2dI1 / dt , so V2 / V1  n2 / n1.
 Conservation of energy requires I1 / I 2  V2 / V1, so
 Impedance is transformed as Z 2 / Z1  (n2 / n1 )2.
1
Pi net

C2 Ls 2+Ls+R2
Zin 
C1C2 LR2 s 3+C1Ls 2+(C1+C2 )R2 s+1
L
Zin
C1
C2
R2
 P lotof Z in ( f ) [] for
R2  100[k],
L  25.4 [mH],
C1  800[nF],
C2  400[pF],
f  40 60 [kHz].
(blue = real, olive = imaginary)
Bandwidth and Noise
 Note the Pi net bandwidth was about 3 [kHz].
 The transformer was wideband.
 Why is bandwidth important?
• Response time is inversely proportional to bandwidth.
This limits the range resolution of the rangefinder.
• Noise is directly proportional to bandwidth. This limits
the maximum operational distance for the rangefinder.
 Therefore, a design tradeoff exists between range
resolution and maximum range.
Impedance matching homework problem
Find the load impedance that
accepts the maximum deliverable
power from a 5 [V] sinusoidal
source Vg at 40 [kHz], having a
source impedance due to an
internal series resistance of 50
[Ohms] and inductance 500 [uH].
What components, values, and
topology will implement this load?
Rg
Vg
Lg
Sources homework problem
 Show the Thevenin and
Norton equivalents of
the circuit shown.
10
20
10
10
 Extra credit:
Suppose you were given a sealed box 0.05[m] x 0.05[m] x 0.02[m],
weighing 0.1[kg] with two electrical terminals. It is totally opaque,
even to x-rays and ultrasound and cannot be disassembled for
inspection. When open circuited, the voltage across the terminals is 1
[V]. When short circuited, the current between the terminals is 1 [A].
You are told this is either a Thevenin source or a Norton source. How
would you determine which? You may make any measurements you
like on the unit.