Part One: Analysis of Network (6-2, 6-3)
Review of Resistive Network
3) KVL and KCL
4) Equivalent Circuits
5) Nodal Analysis and Mesh Analysis
Solve for I1 and I2.
Characteristics of Dynamic Network
Dynamic Elements Ohm’s Law: ineffective
3) Example (Problem 5.9):
Why so simple? Algebraic operation!
Dynamic Relationships (not Ohm’s Law) Complicate the analysis
Define ‘Generalized Resistors’ (Impedances)
As simple as resistive network!
Solution proposed for dynamic network:
All the dynamic elements Þ Laplace Trans. Models.
Þ As Resistive Network
Important: We can handle these two ‘resistive network elements’!
3) Resistor V(s) = RI(s)
Mutual Inductance (Transformers)
(make sure both i1 and i2 either away
or toward the polarity marks to make
the mutual inductance M positive.)
Circuit (not transformer) form:
Benefits of transform
Let’s write the equations from this circuit form:
Just ‘sources’ and ‘generalized resistors’ (impedances)!
Circuit Analysis: Examples
Key: Remember very little, capable of doing a lot
How: follow your intuition, resistive network
‘Little’ to remember: models for inductor, capacitor and mutual inductance.
Example 6-4: Find Norton Equivalent circuit
*Review of Resistive Network
1) short-circuit current through the load:
2) Equivalent Impedance or Resistance or :
A: Remove all sources
B: Replace by an external source
C: Calculate the current generated by the external source ‘point a’
D: Voltage / Current Þ
I(s) be zero? We don’t know yet!)
(Will I(s) be zero? We don’t know yet!)
condition: 1 ohm = 3/s
or I(s) = 0
=>I(s) = 0 =>Zs ¥
Example 6-5: Loop Analysis (including initial condition)
Question: What are i0 and v0?
What is ?
Why this direction?
Important: Signs of the sources!
Simplified (Standard form)
Definition of a Transfer Function
System analysis: How the system processes the input to form the output, or
Input : variable used and to be adjusted
to change or influence the output.
Can you give some examples for input and output?
Quantitative Description of ‘ how the system processes
the input to form the output’: Transfer Function H(s)
The resultant output y(t) to d (t) input: unit impulse response
In this case: X(s) = L [d (t)] =
Y(s) = Laplace Transform of the unit impulse response
=> H(s) = Y(s)/X(s) = Y(s)
Therefore: What is the transfer function of a system?
Answer : It is the
of the system.
Facts on Transfer Functions
* Independent of input, a property of the system structure and parameters.
* Obtained with zero initial conditions.
(Can we obtain the complete response of a system based on its transfer
function and the input?)
* Rational Function of s (Linear, lumped, fixed)
* H(s): Transfer function
H( j2pf ) or H( jw : frequency response function of the system
(Replace s in H(s) by j2pf or jw
|H( j2pf )| or |H( jw |: amplitude response function
ÐH(j2pf) or ÐH( jw Phase response function
Properties of Transfer Function for Linear, Lumped stable systems
Rational Function of s
Lumped, fixed, linear system =>
Corresponding differential equations:
(2) all real! Why? Results from real system components.
Roots of N(s), D(s): real or complex conjugate pairs.
Poles of the transfer function: roots of D(s)
Zeros of the transfer function: roots of N(s)
(3) H(s) = N(s)/D(s) of bounded-input bounded-output (BIBO) stable
* Degree of N(s) £ Degree of D(s)
Why? If degree N(s) > Degree D(s)
where degree N (s) <degree D(s)
Under a bounded-input x(t) = u(t) => X(s) = 1/s
( not bounded!)
* Poles: must lie in the left half of the s-plan (l. h. p)
(Can we also include k=1 into this form? Yes!)
* Any restriction on zeros? No (for BIBO stable system)
Components of System Response
Because x(t) is input, we can assume
D(s): System parameters
C(s): Determined by the initial conditions (initial states)
Initial-State Response (ISR) or Zero-Input Response (ZIR):
Zero-State Response (ZSR) (due to input)
From another point of view:
Transient Response: Approaches zero as tà
Forced Response: Steady-State response if the forced
response is a constant
How to find (1) zero-input response or initial-state response? No problem!
(2) zero-state response? No prolbem!
How to find (1) transient response? All terms which go to 0 as tà¥
(2) forced response? All terms other than transient terms.
Initial capacitor voltage:
RC = second
Find total response
Find zero-input response and zero-state response
Find transient and forced response
Which terms go to zero as tà¥
What are the other terms:
Asymptotic and Marginal Stability
System: (1) Asymptotically stable if as tà¥ (no input) for all
possible initial conditions, y(0), y’(0), … y(n-1)(0)
è Internal stability, has nothing to do with external input/output
(2) Marginally stable
all t>0 and all initial conditions
grows without bound for at least some values
of the initial condition.
(4) Asymptotically stable (internally stable)
=>must be BIBO stable. (external stability)
6-5 Routh Array
System H(s) = N(s)/D(s) asymptotically stable all poles in l.h.p (not
include jw axis.
How to determine the stability?
Other method to determine (just) stability without factorization?
All (when is used)
ð any => system unstable!
Denote to esnure stability
When all Re(pj) > 0 , all coefficients must be greater than zero. If some coefficient is not greater than zero, there must
be at least a Re(pj) <= 0 (i.e., )
=> system unstable
Question: All implies system stable?
Judge the stability: Use Routh Array (necessary and sufficient)
2. Routh Array Criterion
Find how many poles in the right half of the s-plane
Formation of Routh Array
Number of sign changes in the first column of the array
=> number of poles in the r. h. p.
sign: Changed once =>one pole in the r.h.p
Sign: changed twice => two poles in r.h.p.
Modifications for zero entries in the array
Case 1: First element of a row is zero
ð replace 0 by ε (a small positive number)
Case 2: whole row is zero (must occur at odd power row)
construct an auxiliary polynomial and the perform differentiation
Example: best way.
Application: Can not be replaced by MatLab
Range of some system parameters.
to ensure system stable!
Frequency Response and Bode Plot
Real positive number: function of
Interest of this section
In particular, obtain
Bode plots of factors
Constant factor k:
Can we plot it?
: Can we plot for them?
step 1: Coordinate systems
step 2: corner frequency
step 3: Label 0.1wc wc wc
What is T : T = 0.2
What is wc wc = 1/T = 5
Example : 0.2s + 1
Example : (0.2s + 1)2, (0.2s + 1)-2
Example : (Ts + 1)±N
(4) (Complex --- Conjugate poles)
Step 3 : Before :
Right of :
point 2: ( , )
Actual and z (show Fig 6-20)
What’s resonant frequency: reach maximum:
Under what condition we have a resonant frequency:
: see fig 6-21
What about : ?
Bode plots: More than one factors
Can we sum two plots into one?
Can we sum two plots into one?
Show result in fig 6-24
6.7 Block Diagrams
What is a block diagram?
Concepts: Block, block transfer function,
Interconnection, signal flow, direction
System input, system output
Simplification, system transfer function
Assumption: Y(s) is determined
by input (X(s)) and block transfer
function (G(s)). Not affected by
Should be vary careful in
analysis of practical systems about the accuracy of this assumption.
5. Single-loop system
Let’s find Closed-loop transfer function
6. More Rules and Summary: Table 6-1
Example 6-14: Find Y(s)/X(s)
Example 6-15: Armature- Controlled dc servomotor
Input : Ea (armature voltage)
Output : (angular shift)
Can we obtain ?
Example 6-16 Design of control system
Design of K such that closed loop system stable.
System stable if k>0. If certain performance is required in addition to the stability, k must be further designed.
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