March 20 - tbd Flashcards
Forced and natural responses
Sys response from two sources:
Forced (zero-state) response: due to external input u(t), assuming system starts from rest.
Natural (zero-input) : comes from initial energy in the system (like stored voltage/current) when u(t) = 0.
System is initially relaxed if: initial state is zero: x(0) = 0.
or u(t) = 0 for all t < 0.
Convolution for CT LTI Systems
When a system is LTI and initially relaxed, the output can be calculated using convolution.
h(t): impulse response of the system.
u(t): Input signal
This operation blends the input with how the system responds to impulses.
Modelling LTI Systems with circuits
Basic components
Resistor: v(t) = Ri(t)
Cap: i(t) = C dv(t)/dt
Inductor: v(t) = L di(t)/dt
All linear elements, meaning any circuit built with them is an LTI system
State- Modelling Spacing
To analyze circuits efficiently, esp in simulations, we use state space eqns
Steps:
1) Choose state variables: capacitor, voltages, and inductor currents
2) Use kirchhoff’s laws: KVL, KCL
3) write differential eqns that describe how each state var evolves
What are SS eqns
They provide an internal description of CT LTI systems (eg. RLC Circuit) by expressing its dynamics as a set of first order differential eqns.
How? : Instead of writing 1 high order diff eqn, we define state vars (eg. voltages across caps, or current through inductors) that capture the system’s stored energy.
Then write how these states evolve over time, and relate to the output.
Form of an SS
State eqn : xt = axt + but
Output eqn : yt = c*xt+dut
Convolution vs SS
SS provide an internal description that is more modular, easier to sim, and handles multi-var systems more naturally.
Conv : external description that it computationally heavy.
Block diagram elements
Multipliers / gain elements (eg. yt = sigma*ut
Adders: combine multiple signals
Integrators: compute the integral of a signal (realizing the differential eqn)
Laplace Transform
Definition : Laplace transform of a CT signal xt : int 0 to inf : xte^stdt
Why t>= 0?
Usually assume xt = 0 for t< 0, causal signal -> simplifies analysis and matches many physical systems
Laplace transform is linear.
Transfer functions in LTI Systems
Can use convolution representation.
Laplace domain, when you take the laplace transform of both sides of the convolution
you get YS= HS*US
HS is the transfer function. can take laplace of ht
Why use laplace?
What is transfer func
Laplace converts diff eqns into algebraic eqns in s.
Transf funcs tell you how the output of an LTI system responds to an input an input in the freq domain
Poles and zeroes of sys trans function -> why qualitative analysis?
Instead of computing exact response for every input, we want to know how sys behaves in general.
Stability: Will the system’s response eventually die out or grow unbound?
Transient response: How does the system react immediately after an input is applied?
Steady-State response: what does the output look like after transients dissapears?
Transfer function Hs gives us these insights by examining its poles and zeros.
Poles and zeros
Zeros: These are the values of s for which Hs = 0.
They cancel out certain frequency components.
Poles: These are the values of s where Hs becomes infinite.
They determine the natural response of the system (how it behaves without any input)
Time response and role of poles
Inverse laplace transform tells how system will respond in time domain.
Real Poles: simple pole at s = sigmoid yields time response e^sig*t
if sig < 0 -> decays
sig > 0 -> grows unbound
Complex conjugate poles: yields response of e^.. sin(Btang)
sigmoid controls decay or growth, B sets oscillation freq
Laplace transform pairs and ROC
ROC: Laplace transform only valid in region where integral converges
For causal systems, this usually means Re(s) must be larger than a certain value.
Inverse laplace transform
To find the time-domain response xt from Xs.
Use partial fraction expansion: breaks Xs into simpler terms with known inverse transforms.
Residue method: use complex integration to compute xt
Result is a sum of terms like …
Step response and its relation to poles and zeros
Step response is the output when the input is a step function
In Laplace domain, step response is Ys = Hs/s
When expanded into partial fractions, each pole contributes a term in the time response. Behaviour of decay, oscillation, growth depends on pole locations.
Pole location and Complex pole location and meanings
Poles in LHP = Stable decaying responses
Poles in RHP = unstable growing responses
Complex poles = oscillatory responses whose envelope is determined by the real part.
