Experiment 2

1
Experiment 2

• Part A Intro to Transfer Functions and AC
  Sweeps
• Part B Phasors, Transfer Functions and Filters
• Part C Using Transfer Functions and RLC
  Circuits
• Part D Equivalent Impedance and DC Sweeps

2
In Class Solution
Question 1 From diagram A Which of the
following statements is true given the direction
of the current flow. c.) P2 gt P1Current flow
from high pressure to low pressure therefore P2
must be greater than P1.Question 2 Draw the
circuit equivalent of diagram A, label current
flow and voltage ( and on the voltage
source).Question 3 Draw an AC signal with the
following parameters Vp-p6V Vave0V Frequency
2KHzLabel the axis, label the amplitude and
period
Diagram A
3
In Class Solution
Amp 3V
Period 0.5ms
4
Circuit Analysis (Combination Method)
5
Part A

• Introduction to Transfer Functions and Phasors
• Complex Polar Coordinates
• Complex Impedance (Z)
• AC Sweeps

6
Transfer Functions

• The transfer function describes the behavior of a
  circuit at Vout for all possible Vin.

7
Simple Example
8
More Complicated Example
What is H now?

• H now depends upon the input frequency (w 2pf)
  because the capacitor and inductor make the
  voltages change with the change in current.

9
How do we model H?

• We want a way to combine the effect of the
  components in terms of their influence on the
  amplitude and the phase.
• We can only do this because the signals are
  sinusoids
• cycle in time
• derivatives and integrals are just phase shifts
  and amplitude changes

10
We will define Phasors

• A phasor is a function of the amplitude and phase
  of a sinusoidal signal
• Phasors allow us to manipulate sinusoids in terms
  of amplitude and phase changes.
• Phasors are based on complex polar coordinates.
• Using phasors and complex numbers we will be able
  to find transfer functions for circuits.

11
Review of Polar Coordinates
point P is at ( rpcosqp , rpsinqp )
12
Review of Complex Numbers

• zp is a single number represented by two numbers
• zp has a real part (xp) and an imaginary part
  (yp)

13
Complex Polar Coordinates

• z xjy where x is A cosf and y is A sinf
• wt cycles once around the origin once for each
  cycle of the sinusoidal wave (w2pf)

14
Now we can define Phasors

• The real part is our signal.
• The two parts allow us to determine the influence
  of the phase and amplitude changes
  mathematically.
• After we manipulate the numbers, we discard the
  imaginary part.

15
The VIR of Phasors

• The influence of each component is given by Z,
  its complex impedance
• Once we have Z, we can use phasors to analyze
  circuits in much the same way that we analyze
  resistive circuits except we will be using the
  complex polar representation.

16
Magnitude and Phase

• Phasors have a magnitude and a phase derived from
  polar coordinates rules.

17
Influence of Resistor on Circuit

• Resistor modifies the amplitude of the signal by
  R
• Resistor has no effect on the phase

18
Influence of Inductor on Circuit
Note cosqsin(qp/2)

• Inductor modifies the amplitude of the signal by
  wL
• Inductor shifts the phase by p/2

19
Influence of Capacitor on Circuit

• Capacitor modifies the amplitude of the signal by
  1/wC
• Capacitor shifts the phase by -p/2

20
Understanding the influence of Phase
21
Complex Impedance

• Z defines the influence of a component on the
  amplitude and phase of a circuit
• Resistors ZR R
• change the amplitude by R
• Capacitors ZC1/jwC
• change the amplitude by 1/wC
• shift the phase -90 (1/j-j)
• Inductors ZLjwL
• change the amplitude by wL
• shift the phase 90 (j)

22
AC Sweeps
AC Source sweeps from 1Hz to 10K Hz
Transient at 10 Hz Transient at
100 Hz Transient at 1k Hz
23
Notes on Logarithmic Scales
24
Capture/PSpice Notes

• Showing the real and imaginary part of the signal
• in Capture PSpice-gtMarkers-gtAdvanced
• -gtReal Part of Voltage
• -gtImaginary Part of Voltage
• in PSpice Add Trace
• real part R( )
• imaginary part IMG( )
• Showing the phase of the signal
• in Capture
• PSpice-gtMarkers-gtAdvanced-gtPhase of Voltage
• in PSPice Add Trace
• phase P( )

25
Part B

• Phasors
• Complex Transfer Functions
• Filters

26
Definition of a Phasor

• The real part is our signal.
• The two parts allow us to determine the influence
  of the phase and amplitude changes
  mathematically.
• After we manipulate the numbers, we discard the
  imaginary part.

27
Phasor References

• http//ccrma-www.stanford.edu/jos/filters/Phasor_
  Notation.html
• http//www.ligo.caltech.edu/vsanni/ph3/ExpACCircu
  its/ACCircuits.pdf
• http//ptolemy.eecs.berkeley.edu/eecs20/berkeley/p
  hasors/demo/phasors.html

28
Phasor Applet
29
Adding Phasors Other Applets
30
Magnitude and Phase

• Phasors have a magnitude and a phase derived from
  polar coordinates rules.

31
Eulers Formula
32
Manipulating Phasors (1)

• Note wt is eliminated by the ratio
• This gives the phase change between signal 1 and
  signal 2

33
Manipulating Phasors (2)
34
Complex Transfer Functions

• If we use phasors, we can define H for all
  circuits in this way.
• If we use complex impedances, we can combine all
  components the way we combine resistors.
• H and V are now functions of j and w

35
Complex Impedance

• Z defines the influence of a component on the
  amplitude and phase of a circuit
• Resistors ZR R
• Capacitors ZC1/jwC
• Inductors ZLjwL
• We can use the rules for resistors to analyze
  circuits with capacitors and inductors if we use
  phasors and complex impedance.

36
Simple Example
37
Simple Example (continued)
38
In Class Problems
Question 1 What is the equation for Rtotal?
(Combining R1, R2, R3, R4, and R5?) Question 2
What is the value for Rtotal? Question 3 What
is the transfer function for the above circuit?
39
High and Low Pass Filters
High Pass Filter H 0 at w 0 H 1 at w H
0.707 at wc
wc2pfc
fc
Low Pass Filter H 1 at w 0 H 0 at w H
0.707 at wc
wc2pfc
fc
40
Corner Frequency

• The corner frequency of an RC or RL circuit tells
  us where it transitions from low to high or visa
  versa.
• We define it as the place where
• For RC circuits
• For RL circuits

41
Corner Frequency of our example
42
H(jw), wc, and filters

• We can use the transfer function, H(jw), and the
  corner frequency, wc, to easily determine the
  characteristics of a filter.
• If we consider the behavior of the transfer
  function as w approaches 0 and infinity and look
  for when H nears 0 and 1, we can identify high
  and low pass filters.
• The corner frequency gives us the point where the
  filter changes

43
Taking limits

• At low frequencies, (ie. w10-3), lowest power of
  w dominates
• At high frequencies (ie. w 103), highest power
  of w dominates

44
Taking limits -- Example

• At low frequencies, (lowest power)
• At high frequencies, (highest power)

45
Our example at low frequencies
46
Our example at high frequencies
47
Our example is a low pass filter
What about the phase?
48
Our example has a phase shift
49
Part C

• Using Transfer Functions
• Capacitor Impedance Proof
• More Filters
• Transfer Functions of RLC Circuits

50
Using H to find Vout
51
Simple Example (with numbers)
52
Capacitor Impedance Proof
Prove
53
Band Filters
Band Pass Filter H 0 at w 0 H 0 at w H
1 at w02pf0
f0
Band Reject Filter H 1 at w 0 H 1 at w
H 0 at w0 2pf0
f0
54
Resonant Frequency

• The resonant frequency of an RLC circuit tells us
  where it reaches a maximum or minimum.
• This can define the center of the band (on a band
  filter) or the location of the transition (on a
  high or low pass filter).
• The equation for the resonant frequency of an RLC
  circuit is

55
Another Example
56
At Very Low Frequencies
At Very High Frequencies
57
At the Resonant Frequency
if L1mH, C0.1uF and R100W w0100k rad/sec
f016k Hz H01
58
Our example is a low pass filter
Phase f 0 at w 0 f -180 at w
-90
Magnitude H 1 at w 0 H 0 at w
1
f016k Hz
Actual circuit resonance is only at the
theoretical resonant frequency, f0, when there
is no resistance.
59
Part D

• Equivalent Impedance
• Transfer Functions of More Complex Circuits

60
Equivalent Impedance

• Even though this filter has parallel components,
  we can still handle it.
• We can combine complex impedances like resistors
  to find the equivalent impedance of the
  components combined.

61
Equivalent Impedance
62
Determine H
63
At Very Low Frequencies
At Very High Frequencies
64
At the Resonant Frequency
65
Our example is a band pass filter
Magnitude H 0 at w 0 H1 at w0 H 0 at w
Phase f 90 at w 0 f 0 at w0 f -90 at w

f0
66
In Class Problems
Question 1 What is the equation for Rtotal?
(Combining R1, R2, R3, R4, and
R5?) (R5R4)(R2R3/R2R3)/ (R5R4)(R2R3/R2R
3) R1 Question 2 What is the value for
Rtotal? 10 ohms Question 3 What is the
transfer function for the above circuit? (next
slides)
67
Find voltage at this point, then use voltage
divider (only use this in series)
68
VB0.6V
Now use the voltage divider to find Vout
Were not done thoughwe are looking for the
transfer function HVout/Vin so remember