Post-Midterm - March 12 Flashcards
Introduction to systems
(This course focuses on linear systems)
System transforms a signal from one form to another
Ex. Telephone converts voice to electrical signals,
- Filter reduces noise in a signal
- Amp inc/dec signal strength
Types of systems
System can be linear/non-linear
Linear systems satisfy superposition (scaling + additivity), nonlin sys do not
Time-invariant or time-varying: TI sys behaviour does not change over time
TV sys behavior depends on time
Lumped or distributed: Lumped can be described with finite eqns, Distributed requires partial diff eqns
Mathematical representations of systems (Linear sys are described by)
Convolution eqns, differential eqn for CT, Difference eqn for DT, State-space eqn, and transfer functions
Black box model of a system
System modelled as a black box with inputs and outputs.
Input signal (excitation) -> Processed by system -> Output signal (Response)
Eg. An RC Circuit (in: voltage, out: capacitor voltage)
Eg. Mechanical system (in: force, output: displacement)
Types of black box systems
SISO (single in, single out): one input, one output
MIMO (multi-in, multi-out): Multiple inputs and outputs
CT System: Works with continuous signals
DT System: Works with discrete signals
Causality
System is causal if its output at time t only depends on past and present inputs, not future ones
y(t) depends on u(t) for t<= to
Eg. real time audio filter is casual because it cannot process future sounds
Time-invariance (TI)
System is time-invariant if shifting the input shifts the output by the same amount.
u(t) -> y(t) –» u(t-to) -> y(t-to): if a systems behavior changes over time, it is time-varying
Initial Relaxedness
System is initially relaxed if there is no output before an input is applied
if y(t) = 0 before t0, the system is initially relaxed
Memoryless systems
A system is memoryless if its output at time t only depends on the input at that same time
y(t0) = f(u(t0))
If a system is memoryless, it is automatically causal.
There are Time-varying memoryless systems, and Time-invariant Mem sys
Linearity in memoryless systems
A system is linear if it follows the superposition property:
1. Additivity: u1 + u2 -> y1+y2
2. Scaling: Bu1 -> Bu2
If these properties hold, the system is linear time-invariant
To check if a system is linear, check if all possible inputs and outputs satisfy the two properties.
To prove it is NOT linear, find one example where the property fails.
Introduction to Op-Amps
An op-amp is a key electrical comp in circuits
Has two input terminals: non-inverting terminal e+, and an inverting terminal e-
1 output terminal v0(t)
Input resistance is very large (>10^4 ohms, and output resistance is very small < 50 ohms.
Meaning op-amps amplify voltage with minimal current draw
Op-Amps as nonlinear memoryless systems
The relationship between input and output of an op-amp is v0(t) = f(e+(t) - e-(t))
This means the output depends on the difference between the two input voltages.
Opamps linear region vs saturation region
If input difference ed = e+ - e- stays within a certain range [-a, a], then:
Vout(t) = Aed(t)
where A (open-loop gain) is extremely large (10^5 - 10^10)
if |ed| > a, the output saturates at the maximum voltage +/- vs, making the system nonlinear.
Eg. if v1 > v2, output jumps to positive saturation +vs
Eg. if v1 < v2, output jumps to negative saturation -vs
This allows us to compare two voltages, determining which one is larger
op-amps as LTI memoryless systems
To make the op-amp linear, we introduce negative feedback
Voltage Follower (buffer) : connecting output directly to the inverting input creates a voltage follower.
Eqn becomes Vout(t) = A * vi(t) / 1 + A
This means voltage follower outputs exactly the same voltage as the input.
Why use this? : prevents loading effects when measuring signals, connecting a circuit can distort it.
Buffers signals: ensures accurate signal transmission without distortion
Limitations of op-amps in real circuits
Output voltage limits: Vout(t) is limited by the power supply (Vs)
If input signal too large, output will clip (flatten at max/min values)
Freq response and bandwidth: all op-amps have bandwidth limit– only works properly within a certain freq range [0, omega_b]
If input signal has freq beyond this range, distortion occurs
Feedback and stability issues
- Negative feedback (Stable system):
In a voltage follower, the output is fed back to the inverting input. This stabilizes the system and allows linear operation. - Positive feedback (unstable system): if output is fed back to non-inverting input, we get a positive-feedback system. Vout(t) = A * vi(t) / A - 1
Problem: if A is very large, the denominator A - 1 gets close to zero, making vout unstable.
What is finite memory?
System has memory if the output at time n depends on past inputs.
Finite memory depends on a fixed N of past inputs
Causal system with memory
current output y[n] depends on current input u[n], with a fixed number of past inputs N
Forced response
Sys with memory has 2 main types of responses
FR occurs when the initial conditions are zero, output is caused only by the input
Natural response
Occurs when input is 0, output is caused by initial conditions only.
Causal systems
Causal systems do not depend on future inputs, h[n] = 0 for n < 0 is required for causal systems
