Download Part B: Input and Output Impedance and Impedance Matching

MCE603: Interfacing and Control of Mechatronic Systems
Chapter 1: Impedance Analysis for Electromechanical
Interfacing
Part B: Input and Output Impedance
Cleveland State University
Mechanical Engineering
Hanz Richter, PhD
MCE603 – p.1/14
Input and Output Impedances
The input impedance of a network is the phasor ratio of the applied voltage
to the resulting current at the terminals. A high input impedance does not
draw too much current (power) from the source and viceversa.
It is commonly said that the input impedance of a network is the impedance
“seen” by the signal connected at the input of the network. Accordingly, one
can determine the input impedance from direct measurement of amplitude
and phase of current and voltage at input terminals, over a frequency
range.
The output impedance of a network or source is the Thévenin impedance.
A low output impedance is desirable in power supplies and signal
sources, so that the current drawn by the load does not change the
voltage at the terminals.
In situations when input or output impedances are inappropriate, an
impedance matching device must be inserted between source and load.
More later.
MCE603 – p.2/14
Power in AC Circuits
Suppose a network has terminal voltage V hφ + θi, while the current
is Ihφi, that is, θ is the angle between voltage and current.
The instantaneous power flowing across the terminals is p(t) = V I.
The average value over a period is given by
P =
Z
T
V I sin(wt + φ + θ) sin(wt + φ)dt
0
Using the trig identity sin A sin B = 21 cos(A − B) − 21 cos(A + B) we
obtain
Z T
Z T
VI
cos(2wt + 2φ + θ)dt
cos θdt −
P =
2T
0
0
The second integral vanishes, while the second gives
P =
VI
cos θ
2
MCE603 – p.3/14
Power in AC Circuits...
Only a fraction of V I is power dissipated at the load. The term cos θ
is known as power factor.
The quantity V I is known as apparent power, measured in VA, while
VI
2 cos θ is the active power, measured in Watts.
Resistive networks draw only active power (θ = 0), while capacitors
and inductors introduce reactive power.
Reactive power is only exchanged between the source and reactive
elements in the load
Exercise: Prove that if Z = R + Xi, with tan θ = X/R then
P =
|V |2
R
2 R2 + X 2
(1)
MCE603 – p.4/14
Power in AC Circuits...
Resistive networks draw only active power (θ = 0), while capacitors and
inductors introduce reactive power.
Reactive power is only exchanged between the source and reactive
elements in the load.
Active power can be also expressed in terms of rms voltage and current:
s
s
Z T
Z T
1
1
I 2 dt, Vrms =
V 2 dt
Irms =
T 0
T 0
√
For sinusoids, Irms /I = Vrms /V = 1/ 2, giving
P = Vrms Irms cos θ
The rms values are also called effective, since the heat produced by a
resistor is Irms Vrms and not IV .
Is the 110 V at the electrical outlet the amplitude or the rms value?
Although the power required to drive a generator matches the one
MCE603 – p.5/14
consumed actively (assuming 100% efficiency), the utilities charge by the
Impedance Matching and Bridging
Two typical situations demand impedance adjustements:
1: A measurement instrument has a low input impedance and/or the
measured object has a high output impedance. In this case, the
instrument draws too much current, which creates a significant
voltage drop due to the high output impedance, altering the
measurement. In measurement, one wants high input impedance
for the instrument and low output impedance for the measurand.
Some transducers (piezoelectric accelerometers, for instance),
possess a very high ouput impedance, making voltmeter or
oscilloscope readings impossible without a charge amplifier,
providing impedance adjustment. Ideally, the amplifier must present
a high impedance to the transducer (comparable to the transducer’s
output impedance) and a very low impedance to the rest of the
circuit. This is called buffering, impedance bridging.
MCE603 – p.6/14
Impedance Matching...
2: A driven device has an input impedance that is different than the
driver’s output impedance. In this case, part of the power is reflected
back from the load to the source. For maximum power delivered
from a source to a load, impedances must be “matched” (more
next). Waves in a rope example.
MCE603 – p.7/14
Maximum Power Theorem - Variable Load
Consider a source represented by its electrical
Thévenin equivalent and connected to a variable load
Zb = Rb + Xb i. The power transferred to the load is
given by (show it by using Eq. 1)
1 |V |2 Rb
P =
2 |Zb + Zth |2
Consider this expression as a function of two real
variables P = P (Rb , Xb ), maximize and show that
∗
Zb = Zth
(∗ denotes complex conjugate) results in
maximum power. Find the percentage of total power
that is dissipated in each of Rth and Rb .
MCE603 – p.8/14
Maximum Power Theorem - Variable Internal Impedance
If Zb is fixed and Zth is adjustable, it’s easy to see
that maximum power occurs at Zth = 0
Edison’s intuition: See interesting article at
http://www.du.edu/ jcalvert/tech/jacobi.htm
This case makes sense for the design of generators and
power supplies
For sensor interfacing, we typically care about impedance
bridging
A simple way to adjust impedances is by using a transformer.
MCE603 – p.9/14
Characteristic Impedance of a Cable
A cable can be modeled as follows
SE
R
R
R
R
1
0
1
0
I
C
I
C
If the cable is connected to a DC supply on one end and it is
infinitely long (or very long with the opposite end open), a constant
current is theoretically calculated (or measured), even when the
leakage resistance is zero (or negligible). This is because energy
travels endlessly through the cable (no reflection). Viewed from the
battery, this is equivalent to a resistor dissipating the energy. This
resistance is the characteristic impedance of the cable.
MCE603 – p.10/14
Characteristic Impedance of a Cable
If a short cable is terminated with a resistance equal to its
characteristic impedance, the input impedance of the combination is
independent of length and equal to the characteristic impedance.
Typical coaxial cables are 50 or 75 Ω. When a signal generator is
connected to a receiving instrument with a cable, the impedances of
the three elements should be as matched as possible for maximum
power transfer, especially over long cables.
The characteristic impedance of a coaxial cable can be
approximated as
d1
138
Zo = √ log10
d2
ǫ
where d1 is the inner diameter of the outer conductor, d2 is the outer
diameter of the inner conductor and ǫ is the relative permittivity of
the insulation between the conductors (range from 2 to 4 from
common cable insulators like Nylon, Polyethylene, etc.).
MCE603 – p.11/14
Impedance Control in Tele-manipulation
MCE603 – p.12/14
Impedance Control in Tele-manipulation
MCE603 – p.13/14
Mandatory Reading
Applications of Micromechatronics in Minimally Invasive Surgery,
Tendick,F., Sastry, S., Fearing, R. and Cohn, M.,IEEE/ASME
Transactions on Mechatronics, v3 N.1, 1998, pp.34-42.
The article is available online through the CSU Library (use IEEE
Xplore). This material may be part of the course evaluation (any exam).
MCE603 – p.14/14