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======= ELEN30009 - Electrical Network Analysis and Design =======
{{tag>notes unimelb electrical_engineering}}
====== Lumped circuit abstraction ======

Used to build up mathematical models of real world systems.
Abstraction is the process of generalising by reducing information content to retain only relevant information.
Circuit abstraction is systematically predicting behaviour of real world networks using models of interconnected elements.
Example is replacing a light bulb with a resistor.
===== Linear Time Invariant models =====

Good approximation for many real world circuits.
Suggests techniques useful for nonlinear systems.
LTI systems are defined by linearity and time invariance.
==== Linearity ====

- Response to input \(ax(t)=a\) times response to \(x(t)\) and \(a\) is a scalar.
  - Response to inputs \(x_1(t)+x_2(t)\) = response to \(x_1(t)\) + response to \(x_2(t)\)
==== Time Invariance ====

Input-output relationship does not change with time.
This is that if input \(x(t)\) produces output \(y(t)\), then delayed input \(x(t-\tau)\) produces \(y(t-\tau)\)
====== First order transients ======

Contains only one energy storage element (capacitor or inductor) and one loss element (resistor).
Described by first order differential equations.
Transient analysis looks at time varying currents and voltages.
===== Review of linear circuit elements =====

Linear relationship between current and time derivative of voltage.
==== Capacitor ====

\[i(t)=C\frac{dv(t)}{dt}\]
\[v(t)=v(t_0)+\frac{1}{C}\int_{{t_0}^t{i(t)}dt}=V_0+\int{i(t)}dt\]
Short circuit initially. Open circuit at steady state.
Stores charge.
==== Inductor ====

\[v(t)=L\frac{di(t)}{dt}\]
\[i(t)=I_0+\frac{1}{L}\int_{t_0}^t{v(t)dt}\]
Open circuit initially. Short circuit at steady state.
Stores flux.
===== Transient response =====

* Natural Response
    * Time varying currents and voltages arising from the capacitor or inductor being abruptly removed from a DC source
Stored energy is released into resistive network.
  * Step Response
    * Time varying currents and voltages arising from the sudden application of a DC source to a capacitor or inductor
Energy is acquired by capacitor or inductor.
==== Natural response of RC circuit ====

\[v(t)=V_0e^{-\frac{1}{RC}t}=V_0e^{-\frac{1}{\tau}t}, t\geq 0\]
The capacitor is at an initial voltage (\(V_0\)) and decays to \(0\).
\(\tau=RC\) is the time constant for the circuit.
From this, we can find other expressions for:

Current
\[i(t)=\frac{v(t)}{R}=\frac{V_0}{R}e^{-\frac{t}{\tau}}, t\geq0\]
Power
\[p(t)=v(t)i(t)=\frac{V_0^2}{R}e^{-\frac{2t}{\tau}}, t\geq0\]
Energy
\[w=\int_0^t{p}dt=\frac{CV_0^2}{2}\left(1-e^{-\frac{2t}{\tau}}\right), t\geq0\]
==== Step function ====

\[Ku(t)=K,t>0\]
\[Ku(t)=0,t<0\]
Not defined at \(t=0\).
Thought of as a switch immediately changing state.
==== Step response of RC circuit ====

\[v_c(t)=I_SR+(V_0-I_SR)e^{-\frac{t}{\tau}}\]
\[=I_SR(1-e^{-\frac{t}{\tau}}), t\geq0\]
Basically a non-homogeneous form of natural response.
Current can be found like above:
\[i(t)=\left(I_S-\frac{V_0}{R}\right)e^{-\frac{t}{\tau}}, t\geq0\]
==== Natural response RL circuit ====

\[i(t)=I_0e^{-\frac{R}{L}t}=I_0e^{-\frac{t}{\tau}}, t\geq0\]
\[v(t)=L\frac{di}{dt}=-\frac{I_0}{\tau}e^{-\frac{t}{\tau}}, t\geq0\]
\[p(t)=v_R(t)i(t)=RI_0^2e^{-\frac{2t}{\tau}}, t\geq0\]
\[w(t)=\int_0^t{p}dt=\frac{1}{2}LI_0^2\left(1-e^{-\frac{2t}{\tau}}\right), t\geq0\]
\(\tau=\frac{L}{R}\)
==== Step response ====

\[i(t)=\frac{V_S}{R}+\left(I_0-\frac{V_S}{R}\right)e^{-\frac{t}{\tau}}\]
\[=\frac{V_S}{R}\left(1-e^{-\frac{t}{\tau}}\right)\]
\[v(t)=(V_S-I_0R)e^{-\frac{t}{\tau}}\]
==== General solution ====

\[x(t)=K\tau+[x(t_0)-K\tau]e^{-\frac{t-t_0}{\tau}}\]
Need to identify the variable of interest (\(x(t)\)), the initial (\(x(t_0)\)), final/steady state values (\(X\tau\)) and the time of switching (\(t_0\)).
The first term identifies the steady state response and the final term identifies the transient response.
===== Square wave input (Impulse response) =====

==== Case b (\(\tau<<T\)) ====

Fast rise time during charge interval, reaching steady state before discharge period begins.
Same for discharge period.
Capacitor waveform closely resembles input except for rounding at corners.
==== Case c (\(\tau\leq T/10\)) ====

Transients at both intervals almost go to completion, reaching steady state.
Reaches steady state at switching time.
==== Case d (\(\tau>>T/10\)) ====

In both charge and discharge intervals, transients do not go to completion.
Capacitor waveform is almost triangular, i.e. integral of input wave.
===== Pulse response =====

Basically a single iteration of a square wave.
While charging, acts like a step response, discharging is a natural response.
If \(\tau << t_p\), can consider the circuit at steady state when the pulse is ending and the voltage to be at \(V_p\).

If the pulse is shorter, where \(\tau > t_p\), the circuit does not reach steady state, as a result the discharge voltage depends on the charge voltage.
\[v_c(t)=[V_p(1-e^{-t_p/\tau})e^{-(t-t_p)/\tau}]\]

If the pulse is make even shorter \(\tau >> t_p\), the charging wave can be approximated to a straight line.
\[v_c(t)=V_p(1-e^{-t/\tau})\approx V_p\frac{t}{\tau}\]
At the end of the pulse, \(v_c(t)\) has reached a maximum value of 
\[v_c(t_p)\approx\frac{V_pt_p}{\tau}=\frac{\text{area of pulse}}{\tau}\]
The capacitor's discharge equation is given by 
\[v_c(t)\approx\left(\frac{V_pt_p}{\tau}\right)e^{-(t-t_p)/\tau}\]
\[\approx\left(\frac{V_pt_p}{\tau}\right)e^{-(t)/\tau}, t\geq t_p\]
Where now the response is proportional to the pulse area rather than just \(V_p\).
===== Impulse response =====

If we use a small pulse of 0 time, we have a true impulse and the extreme case of the pulse response.
We can model this as
\[v_i(t)=K\delta(t)\]
where \(\delta\) is the delta function. Here \(v_i\) has an area of \(K\).

The impulse response is given by 
\[v_c(t)=\frac{K}{RC}e^{-t\tau}=\frac{K}{\tau}e^{-t\tau}\]

If an impulse (\(\delta(t)\)) is given to an LTI system, the impulse response is received (\(h(t)\)).
This can be used to model an arbitrary input, \(x(t)\) as \(y(t)=h(t)*x(t)\).

In an RL circuit, an impulse of voltage with strength K must appear over the inductor as it resists all changes in current, making \(v_L(0^+)=\frac{K}{L}\).
This establishes the initial current condition and stores energy in the inductor:
\[w(0^+)=\frac{1}{2}L\left(\frac{K}{L}\right)^2=\frac{K^2}{2L}\]
Following the impulse, the circuit decays in accordance to the natural response
\[i(t)=\frac{K}{L}e^{-t/\tau}, t\geq0\]
====== Second order transients ======

Circuits containing both inductors and capacitors.
For now we limit to parallel RLC and series RLC. For these we look at transient and step responses.
Second order circuits can oscillate, unlike first order.
===== General procedure =====

- Use KVL/KCL and derive DE, may be in terms of capacitor voltage or inductor current.
  - Calculate circuit parameters (\(\alpha,\omega_0,\omega_d\)) and determine whether circuit is under, over or critically damped.
  - Write in form for type of damping.
  - Determine initial conditions.
  - Solve for coefficients.
===== Parallel RLC transient analysis =====

From KCL for a natural response, we can find 
\[\underbrace{\frac{v}{R}}_{i_R}+\underbrace{(I_0+\frac{1}{L}\int_0^t{v}dt)}_{i_L}+\underbrace{C\frac{dv}{dt}}_{i_C}=0\]
Which becomes
\[\frac{d^2v}{dt^2}+\frac{1}{RC}\frac{dv}{dt}+\frac{v}{LC}=0\]
A second order homogeneous DE with constant coefficients.
These have general solutions in the form of 
\[v(t)=A_1e^{s_1t}+A_2e^{s_2t}\]

For a step response, we have
\[x(t)=\underbrace{X_f}_{\text{forced response}}+\underbrace{(A_1'e^{s_1t}+A_2'e^{s_2t})}_{\text{natural response}}\]

\(s_1\) and \(s_2\) are circuit parameters determined by R, L and C. They are known as the roots of the characteristic equation.
\(A_1\) and \(A_2\) are determined by the initial conditions.
==== Parallel natural response parameters ====

^ Parameter^ Terminology^ Value in natural response^
| |Characteristic equation|\(s^2+\frac{s}{RC}+\frac{1}{LC}=0\)|
|\(s_1\), \(s_2\)|Characteristic roots (rad/s)|\(s_{1/2}=-\alpha\pm\sqrt{\alpha^2-\omega_0^2}\)|
|\(\alpha\)|Neper frequency (rad/s)|\(\alpha=\frac{1}{2RC}\)|
|\(\omega_0\)|Resonant radian frequency (rad/s)|\(\omega_0=\frac{1}{\sqrt{LC}}\)|
Characteristic equation is found by setting all independent sources to 0 and replacing the derivative by \(s\) and second derivative by \(s^2\).
==== Types of natural response for parallel ====

* \(\omega_0^2<\alpha^2\)
\(s_1\) and \(s_2\) are real and distinct, \(v(t)\) is over-damped (no oscillation).
  * \(\omega_0^2=\alpha^2\)
\(s_1\) and \(s_2\) are real and equal, \(v(t)\) is critically damped (verge of oscillation).
  * \(\omega_0^2>\alpha^2\)
\(s_1\) and \(s_2\) are complex, \(v(t)\) is under-damped (voltage oscillates about final value with oscillation frequency \(\omega_d=\sqrt{\omega_0^2-\alpha^2}\)).

All of the following can be done in terms of current with minimal changes.
=== Over-damped response ===

\[v(t)=A_1e^{s_1t}+A_2e^{s_2t}\]
Can determine constants from initial conditions.
\[v(0^+)=A_1+A_2\]
Initial voltage on the capacitor.
\[\frac{dv(0^+)}{dt}=\frac{i_C(0^+)}{C}=s_1A_1+s_2A_2\]
Related to capacitor current (\(i_c\)).
From KCL, initial current can be found.
\[i_C(0^+)=-\frac{V_0}{R}-I_0\]
=== Under-damped response ===

Damped radian frequency can be expressed as \(\omega_d=\sqrt{\omega_0^2-\alpha^2}\).
This allows us to express the characteristic roots as
\[s_1,s_2=-\alpha\pm\sqrt{\alpha^2-\omega_0^2}=\alpha\pm j\omega_d\]
Using the general formula and Euler's identity, we get a general solution of:
\[v(t)=B_1e^{-\alpha t}\cos{\omega_dt}+B_2e^{-\alpha t}\sin{\omega_dt}\]
Where the constants \(B_1\) and \(B_2\) are both real.
Can determine their values from:
\[v(0^+)=B_1\]
\[\frac{dv(0^+)}{dt}=\frac{i_C(0^+)}{C}=-\alpha B_1+\omega_dB_2\]

As \(R\to\infty\), \(\alpha\to 0\) which tells us that \(\omega_d\to\omega_0\). 
When \(\alpha=0\), the amplitude remains constant and the oscillation is sustained.

Under-damped systems reach their final value faster than over-damped systems and should be used if this is required and the oscillations aren't a concern.
=== Critically damped response ===

When \(\omega_0^2=\alpha_0^2\),
\[s_1=s_2=-\alpha=-1/(RC)\]
The voltage response does not take the form of an under or over-damped system as it would then take the form of \(A_0e^{-\alpha t}\) which is not capable of satisfying initial conditions \(V_0\) and \(I_0\).
Instead it takes the form of:
\[v(t)=\underbrace{D_1t e^{-\alpha t}}_{\text{product of linear \& exponential terms}}+\underbrace{D_2e^{-\alpha t}}_{\text{exponential term}}\]
These systems are rarely encountered as they require very precise conditions.

The initial conditions can be found with
\[v(0^+)=D_2\]
\[\frac{dv(0^+)}{dt}=\frac{i_C(0^+)}{C}=D_1-\alpha D_2\]
===== Step response of parallel =====

This uses the KCL equation
\[i_C+i_R+i_L=I\]
\[\implies\frac{di_L^2}{dt^2}+\frac{1}{RC}\frac{di_L}{dt}+\frac{i_L}{LC}=\frac{I}{LC}\]
Being the non-homogeneous equivalent of the natural response.
Now the RHS is a constant forcing function causing
\[i_L=I_f+\{\text{natural response}\}\]
Where \(I_f\) is the value at steady state.
===== Impulse response of parallel =====

If a current impulse were supplied, all current would go to the capacitor as it acts like a short circuit at steady state.
After the impulse, the energy would be released as a natural response.
\[i_C=C\frac{dv}{dt}\implies v(0^+)=\frac{1}{C}\int_{0^-}^{0^+}{A_0\delta(t)}dt=\frac{A_0}{C}\]
We can then write 
\[i_L(0^+)=0\]
\[\frac{i_L(0^+)}{dt}=\frac{A_0}{LC}\]
\[i_R(0^+)=\frac{v(0^+)}{R}=\frac{A_0}{RC}\]
\[i_C(0^+)=-i_R(0^+)=-\frac{A_0}{RC}\]
\[\frac{dv(0^+)}{dt}=\frac{i_C(0^+)}{C}=-\frac{A_0}{RC^2}\]
The impulse is setting the initial conditions for a natural response.
===== Series transient analysis =====

Same general procedure are parallel analysis, except KVL is basis of analysis.
This results in \(\alpha\) and \(\omega_d\) having different formulas.
\[\underbrace{L\frac{di}{dt}}_{v_L(t)}+\underbrace{Ri+\int_0^t{i}dt}_{v_R(t)}+\underbrace{V_0}_{v_C(t)}=0\]
From this we obtain
\[\frac{d^2i}{dt^2}+\frac{R}{L}\frac{di}{dt}+\frac{i}{LC}=0\]
This gives
\[\alpha=\frac{R}{2L}\]
\[\omega_0=\frac{1}{\sqrt{LC}}\]
With the rest of the math being the same.
====== Convolution Integral ======

Integral relates output \(y(t)\) to input \(x(t)\) and circuits unit impulse response.
\[y(t)=h(t)*x(t)=\int_{-\infty}^\infty{h(\lambda)x(t-\lambda)}d\lambda=\int_{-\infty}^\infty{x(\lambda)h(t-\lambda)}d\lambda\]
===== The step function =====

\[u(t)=\begin{cases}1,&t>0\\0,&t<0\end{cases}\]
This is non-continuous, when we need a continuous function, we assume a linear transition between \(0^-\) and \(0^+\), with \(u(0)=0.5\).
This is the same if we shift the function to step at \(a\).
We can also define an inverse step function:
\[u(t)=\begin{cases}1,&t<0\\0,&t>0\end{cases}\]

Derivative of a step function is the delta impulse function.
We can call the area of the impulse function its strength, with the unit impulse having an area of 1.
The impulse function has a sifting property, i.e.
\[\int_{-\infty}^{\infty}{f(t)\delta(t-a)}dt=f(a)\]
This removes all values of \(f(t)\) except when \(t=a\).

We can characterise a LTI system by its impulse response and use it to compute the output for an arbitrary input using the convolution integral.
===== Approximating the input signal \(x(t)\) =====

Using an input \(x(t)=0,t<0\), we can approximate \(x(t)\) into a series of rectangular pulses with uniform width \(x(t)\approx x_0(t)+x_1(t)+x_2(t)+\ldots+x_i(t)+\ldots\), where \(x(\lambda_i)\) is uniform between \(\lambda_i\) and \(\lambda_{i+1}\).
This can be further expressed as a series of step functions
\[x_i(t)\approx x(\lambda_i)\{u(t-\lambda_i)-u(t-[\lambda_i+\Delta\lambda])\}\]
If we make \(\Delta\lambda\) small enough, we can approximate using impulse functions of strength \(x(\lambda_i)\Delta\lambda\)
\[x(t)\approx x_0(\lambda_0)\Delta\lambda\delta(t-\lambda_0)+x_1(\lambda_1)\Delta\lambda\delta(t-\lambda_1)+\ldots+x_i(\lambda_i)\Delta\lambda\delta(t-\lambda_i)+\ldots\]

We can then use this input approximation to approximate the output as
\[x(\lambda_i)\Delta\lambda\delta(t-\lambda_i)\xrightarrow{h(t)}x(\lambda_i)\Delta\lambda h(t-\lambda_i)\]
This means that we can approximate the whole output as 
\[y(t)\approx x_0(\lambda_0)\Delta\lambda h(t-\lambda_0)+x_1(\lambda_1)\Delta\lambda h(t-\lambda_1)+\ldots+x_i(\lambda_i)\Delta\lambda h(t-\lambda_i)+\ldots\]
\[=\sum_{i=0}^\infty{\underbrace{x(\lambda_i)\Delta\lambda}_{\text{scaling}} \underbrace{h(t-\lambda_i)}_{\text{delayed impulse response}}}\]
Meaning that the output is approximated by a series of uniformly delayed impulse responses with appropriate scaling.

As \(\Delta\lambda\to\infty\), 
\[\sum_{i=0}^\infty{x(\lambda_i)\Delta\lambda h(t-\lambda_i)}\to\int_0^\infty{x(\lambda)h(t-\lambda)}d\lambda=x(t)*h(t)\]
===== Basic convolution principle =====

Multiply \(h(\lambda)\) by \(x(t-\lambda)\).
Here \(t\) is a shift variable and \(\lambda\) is time.
We then need to integrate for all \(t\).
==== Limits ====

For physically realisable circuits, \(h(t)=0,t<0\).
  * No response before impulse applied
  * Causal system - output depends on past and current inputs
  * Lower limit goes from \(-\infty\) to \(0\)
Assuming the input \(x(t)\) starts at \(t=0\)
  * After flipping, nothing to the right of time \(\lambda=t\)
  * Upper limit goes from \(\infty\) to \(t\)
This all causes
\[y(t)=\int_{-\infty}^{\infty}h(\lambda)x(t-\lambda)d\lambda=\int_{0}^{t}h(\lambda)x(t-\lambda)d\lambda\]
===== Memory and weighing function =====

Flipping the input function allows viewing of past, present and future.
Past values lie to the right, future to the left and present at the vertical axis where \(\lambda=0\).
\(h(t)\) weights \(v_i(t)\) according to past and present values, as such the impulse response determines how much memory the circuit has.

Memory is the extent to which the circuit's response matches its input.
If \(h(t)\) is flat, equal weights are given to all present and past values. This circuit has perfect memory.
If \(h(t)\) is an impulse function, no weight is given to past values and the output matches the input. This circuit has no memory.
Therefore the more memory the circuit has, the more past values are weighted and the more distorted the output waveform.
====== Circuit Analysis in S-Domain ======

===== Laplace Transform =====

Transforms a function of time into a function of complex frequency.
In the time domain, we have \(y(t)=h(t)*x(t)\).
In frequency domain, we have \(Y(s)=H(s)X(s)\), which is a multiplication rather than a convolution integral.
By using a Laplace and inverse Laplace transformation, we can avoid doing a convolution integral.
In the frequency domain, \(H(s)\) is the transfer function and \(H(j\omega)\) is the frequency response.

The Laplace transform (\(\mathcal{L}\{f(t)\}\)) of \(f(t)\) is given by the expression:
\[\mathcal{L}\{f(t)\}=F(s)=\int_{0^-}^\infty{f(t)e^{-st}}dt\]
The Laplace transform is used to transform a set of constant coefficient differential equations from the time domain to a set of linear polynomial equations in the frequency domain.
After obtaining a frequency domain expression for the output, we transform it to the time domain.
===== Frequency domain =====

^ Type of signal^ Example^ Complex frequency^
|Constant, DC|\(x(t)=2\)|Real \(s=0\)|
|Exponential decay|\(x(t)=e^{-6t}\)|Real \(s=-6\)|
|Sinusoidal|\(x(t)=sin(25t)=\frac{1}{2}j(e^{j25t}-e^{j25t})\)|imag \(s=\pm 25j\)|
|Exp increasing sinusoid|\(x(t)=e^{2t}cos(50t)=\frac{1}{2}(e^{(2+50j)t}+e^{(2-50j)t})\)|\(s=2\pm 50j\)|
From this we can visualise the complex s-plane.
The larger the imaginary component, the faster the oscillations, the larger the real component, the faster the growth/decay.
===== Functional and operational transforms =====

* **Functional:** Transform of a specific function
\[\mathcal{L}\{\sin{\omega t}\}=\frac{\omega}{s^2+\omega^2}\]
  * **Operational:** Indicate how an operation is performed once converted to the opposite domain
\[\mathcal{L}\{Kf(t)\}=KF(s)\]
\[\mathcal{L}\{e^{-at}f(t)\}=F(s+a)\]
More of these can be found in tables.
===== Circuit components in s-domain =====

==== Resistors ====

From Ohm's law:
\[v(t)=Ri(t)\]
As R is constant, the Laplace transform is:
\[V(s)=RI(s)\]
An s-domain resistor R (ohms) carries current I (ampere-seconds) and has terminal voltage (volt-seconds).
==== Inductor ====

{{ indsdom.png?100}}
Terminal voltage to current relationship is given by:
\[v(t)=L\frac{di(t)}{dt}\]
Its Laplace transform is 
\[V(s)=L[sI(s)-i(0^-)]=sLI(s)-LI_0\]
The equivalent impedance therefore is an impedance of \(sL\) ohms in series with an independent voltage source of \(LI_0\) volt-seconds oriented with the flow of current.

{{ indsdom2.png?100}}
We can also derive an alternate equivalent circuit.
\[I(s)=\frac{V(s)+LI_0}{sL}=\frac{V(s)}{sL}+\frac{I_0}{s}\]
This has an impedance \(sL\) ohms in parallel with an independent current source of \(I_0/s\) ampere-seconds oriented with the flow of current.
==== Capacitor ====

{{ capsdom.png?100}}
Time domain relationship between terminal current and voltage:
\[i(t)=C\frac{dv(t)}{dt}\]
Laplace transform is:
\[I(s)=C[sV(s)-v(0^-)]=sCV(s)-CV_0\]
This is a capacitor with current \(sCV(s)\) in parallel with a current source \(CV_0\) oriented against the flow of current.

{{ capsdom2.png?100}}
The alternate equivalent circuit is:
\[V(s)=\frac{I(s)}{sC}+\frac{V_0}{s}\]
This represents an impedance \(1/sC\) ohms in series with an independent voltage source \(V_0/s\) volt-seconds oriented against the flow of current.
==== Ohm's Law ====

\[V(s)=ZI(s)\]
Here Z is the s-domain impedance of the element
  * \(R\) ohms for resistor
  * \(sL\) ohms for inductor
  * \(1/sC\) ohms for capacitor
or the s-domain admittance
  * \(1/R\) siemens for resistor
  * \(1/sL\) siemens for inductor
  * \(sC\) siemens for capacitor
===== s-domain analysis =====

\[\mathcal{L}\{f_1(t)+f_2(t)+...\}=F_1(s)+F_2(s)+...\]
We can also determine that Kirchhoff's Laws also apply to s-domain currents and voltages
\[\sum{I(s)}=0\]
\[\sum{V(s)}=0\]
Linear, lumped-parameter circuits with constant component values always have unknown voltages and currents that can be expressed as rational functions of s.
If the order of the denominator is greater than the numerator, the function is a proper rational function.
If not, the function is an improper rational function.

Proper rational functions can be expanded into a sum of partial fractions, for which the numerators can be found and the inverse Laplace transformation can be performed.
The roots of the denominator are referred to as poles of the function as they are the values for which it becomes infinitely large.
The roots of the numerator are referred to as the zeros of the function as they are the values where the function becomes zero.

Improper rational functions need to be expanded into a polynomial and a proper rational fraction with polynomial division.
==== Four useful transformation pairs ====

Most rational functions can be handled with these pairs
^ Nature of roots^ \(F(s)\)^ \(f(t)\)^
|Distinct Real|\(\frac{K}{s+a}\)|\(Ke^{-at}u(t)\)|
|Repeated Real|\(\frac{K}{(s+a)^2}\)|\(Kte^{-at}u(t)\)|
|Distinct Complex|\(\frac{\lvert{K}\rvert\angle{\theta}}{s+\alpha-j\beta}+\frac{\lvert{K}\rvert\angle{-\theta}}{s+\alpha+j\beta}\)|\(2\lvert{K}\rvert e^{-\alpha t}\cos(\beta t+\theta)u(t)\)|
|Repeated Complex|\(\frac{\lvert{K}\rvert\angle{\theta}}{(s+\alpha-j\beta)^2}+\frac{\lvert{K}\rvert\angle{-\theta}}{(s+\alpha+j\beta)^2}\)|\(2t\lvert{K}\rvert e^{-\alpha t}\cos(\beta t+\theta)u(t)\)|
==== Initial and final value theorems ====

Can be used to check solution in s-domain prior to transforming.
\[\lim_{f\to0^+}{f(t)}=\lim_{s\to\infty}{sF(s)}\]
\[\lim_{f\to\infty}{f(t)}=\lim_{s\to0}{sF(s)}\]
The first is the initial value theorem. The latter is the final value theorem and is only applicable if the poles (apart from the first order at the origin) lie in the left half of the s-plane.
===== Transfer function =====

A transfer function is a s-domain mathematical representation of the relation between the input and output of LTI systems.
\(H(s)\) is the s-domain ratio of the Laplace transform of the output (response) to the Laplace transform of the input (source).
\[H(s)=\frac{Y(s)}{X(s)}\]

For a circuit with a voltage source (\(V_g\)), resistor (\(R\)), inductor (\(sL\)) and capacitor (\(1/sC\)) in series, using KVL, we obtain:
\[V_g=IR+sLI+I/sC\]
If \(I\) is chosen as the output, then the corresponding transfer function is:
\[H(s)=\frac{I}{V_g}=\frac{1}{R+sL+1/sC}=\frac{sC}{s^2LC+RCs+1}\]
If the capacitor voltage is chosen as the output, then the corresponding transfer function is:
\[H(s)=\frac{V}{V_g}=\frac{1/sC}{R+sL+1/sC}=\frac{1}{s^2LC+RCs+1}\]
From this we can gather that a single circuit can have many transfer functions.

For linear lumped parameter circuits, \(H(s)\) is always a rational function of \(s\).
Complex poles and zeros always appear in conjugate pairs.
The poles of \(H(s)\) must lie in the left half of the \(H(s)\) plane if the circuit response to a bounded source is to be bounded (Bounded-Input Bounded-Output (BIBO) Stability).
The zeros of \(H(s)\) may lie in either side of the s plane.

The poles from \(H(s)\) produce the transient component of the total response and the poles of \(X(s)\) produce the steady-state component once the response have been inverse Laplace transformed.
Using the transfer function with a Laplace transform doesn't invalidate the time-invariant property of the LTI model.

For an input (sinusoidal source):
\[x(t)=A\cos{(\omega t+\phi)}\]
The inverse Laplace \(Y(s)\) ignoring terms generated from the poles of \(H(s)\):
\[y_{ss}=\underbrace{A\lvert H(j\omega)\rvert}_{\text{Amplitude of ss response}}\cos{[\omega t+\underbrace{\phi+\theta(\omega)}_{\text{phase angle of ss response}}]}\]
The amplitude is the product of the source amplitude and transfer function magnitude.
The phase angle is the sum of the phase angles of the source and transfer functions.
From this equation, we can easily find the steady state response if we know the transfer function.
  - Evaluate \(H(s)\) at \(H(j\omega)\), where \(\omega\) is the source frequency (sinusoidal input)
  - Rewrite \(H(j\omega)\) in polar form: \(\lvert H(j\omega)\rvert e^{j\theta(\omega)}=\lvert H(j\omega)\rvert \angle{\theta(\omega)}\)
  - Substitute values into \(y_{ss}(t)\) to find steady-state amplitude and phase angle
====== Frequency selective circuits ======

===== Introduction =====

Filters only allow certain frequencies to pass through.
Can be used to filter out noisy signals.
The **passband** is the frequencies that appear in the output.
The **stopband** is filtered out of the input.
===== Types =====

* **Lowpass:** Low frequencies pass through
  * **Highpass:** High frequencies pass through
  * **Bandpass:** Range of frequencies pass through
  * **Bandreject:** Range of frequencies rejected
Ideally there is a clear cut off frequency (\(\omega_c\)), which creates a vertical line in the transfer function.
Ideally, the phase angle phase plot varies linearly in the passband.
==== Passive vs Active Filters ====

Passive filters use only passive components (resistors, capacitors and inductors).
Active filters use operational amplifiers.
Active filters can provide signal gain, but require power to do so.
For passive filters, the largest \(\lvert H(j\omega)\rvert=1\). The exception to this is the series RLC bandpass, which can amplify its output.
===== Cutoff Frequency of real filters =====

Ideal filters use jump discontinuity, which is impossible with real devices.
In real filters \(\lvert H(j\omega_c)\rvert=\frac{1}{\sqrt{2}}H_{max}\).
This is the half-power frequency, since the power is associated with the square of the signal (\(V^2/R\) or \(I^2R\)).
===== Lowpass filters =====

==== Series RL circuit ====

{{ lowrl.png?100}}
A RL circuit where the output is taken over the resistor.
\[\mathbf{V}_o=\frac{R}{j\omega L+R}\mathbf{V}_i\]
The behaviour of the resistor is independent to \(\omega\). The behaviour of L will vary with \(\omega\).
At low frequencies (\(\omega L<<R\)), the inductor has very small impedance, acting like a short at \(\omega=0\).
As \(\omega\) increases, the inductor's impedance increases relative to R and the phase lag approaches \(90^\circ\).
At high frequencies (\(\omega L >> R\)), the inductor impedance is very large compared to the resistance.
At \(\omega=\infty\), the inductor acts like an open circuit, with a phase angle difference of \(90^\circ\).

The transfer function of a filter in s domain is given by
\[H(s)=\frac{V_0(s)}{V_i(s)}=\frac{R/L}{s+R/L}\]
Substituting \(s=j\omega\)
\[H(j\omega)=\frac{R/L}{j\omega+R/L}\]
\[\lvert H(j\omega)\rvert=\frac{R/L}{\sqrt{\omega^2+(R/L)^2}}\]
\[\theta(j\omega)=-\tan^{-1}\frac{\omega}{R/L}\]
From this, we can find the cutoff frequency is 
\[\omega_c=\frac{R}{L}\]
==== Series RC circuit ====

{{ lowrc.png?100}}
The output voltage (\(v_o\)) is taken across C.
\[H(s)=\frac{1/RC}{s+1/RC}\]
\[H(j\omega)=\frac{1/RC}{j\omega+RC}\]
\[\omega_c=\frac{1}{RC}\]
==== General form ====

The general form for these __simple__ low-pass filters.
\[H(s)=\frac{\omega_c}{s+\omega_c}\]
We can also note that as in a RL circuit, \(\tau=\frac{L}{R}\) and in a RC circuit, \(\tau=RC\), thus
\[\tau=\frac{1}{\omega_c}\]
==== Loaded filter ====

{{ lowrcload.png?100}}
For a RC filter, a resistor in parallel with the capacitor.
\[H(s)=\frac{(1/sC)\lvert\rvert R_L}{(1/sC)\lvert\rvert R_L+R}=\frac{1/RC}{s+\left(\frac{R+R_L}{RR_LC}\right)}\]
\[H(j\omega)=\frac{1/RC}{j\omega+\left(\frac{R+R_L}{RR_LC}\right)}\]
\(H_{max}=\frac{R_L}{R+R_L}\) at \(\omega=0\).
Here \(H_{max}<1\).
\[\omega_c=\frac{1}{RC}\left(\frac{R_L+R}{R_L}\right)\]
If we substitute this back into the equation for \(H(j\omega)\)
\[H(j\omega)=\frac{K\omega_c}{j\omega+\omega_c}, K=\frac{R_L}{R+R_L}\]
===== Highpass filter =====

==== Series RC circuit ====

{{ highrc.png?100}}
The output is taken across R.
At \(\omega=0\), the capacitor acts like an open circuit so \(\lvert H(j0)\rvert=0\) and \(\theta(j0)=+90^\circ=\), at \(\omega=\infty\) the capacitor acts like a short.
\[H(s)=\frac{V_o(s)}{V_i(s)}=\frac{s}{s+1/RC}\]
\[H(j\omega)=\frac{j\omega}{j\omega+1/RC}\]
Separating into magnitude and phase angle equations
\[\lvert H(j\omega)\rvert=\frac{\omega}{\sqrt{\omega^2+(1/RC)^2}}\]
\[\theta(j\omega)=\tan^{-1}\left(\frac{\omega/RC}{\omega^2}\right)=90^\circ-\tan^{-1}(\omega RC)\]
So the cutoff frequency is
\[\omega_c=\frac{1}{RC}\]
This is the same as for the lowpass configuration.
==== Series RL circuit ====

{{ highrl.png?100}}
The output is taken over the inductor.
\[H(j\omega)=\frac{j\omega}{j\omega+R/L}\]
\[\lvert H(j\omega)\rvert=\frac{\omega}{\sqrt{\omega^2+(R/L)^2}}\]
\[\omega_c=\frac{R}{L}\]
==== General form ====

\[H(s)=\frac{s}{s+\omega_c}\]
==== Loading RL filter ====

{{ highrlload.png?100}}
Putting a resistor in parallel changes the transfer function
\[H(s)=\frac{sL\vert\vert R_L}{sL\vert\vert R_L+R}\]
\[H(s)=\frac{Ks}{s+\omega'_c},K=\frac{R_L}{R+R_L},\omega'_c=\frac{KR}{L}\]
===== Bandpass filter =====

Pass voltages within band of \(\omega_{c1}\) and \(\omega_{c2}\).
Cutoff frequencies are defined by the half power frequencies.

The bandwidth is defined as
\[\beta=\omega_{c2}-\omega_{c1}\]
The centre (resonant) frequency - undamped natural oscillation frequency where transfer function is maximum
\[\omega_o=\sqrt{\omega_{c1}\omega_{c2}}\]
Quality (Q) factor - Bandwidth relative to centre frequency
\[Q=\frac{\omega_o}{\beta}\]
==== Series RLC circuit ====

{{ bpseries.png?100}}
\[H(s)=\frac{R}{sL+R+(1/sC)}\]
\[H(s)=\frac{(R/L)s}{s^2+(R/L)s+(1/LC)}\]
The output voltage is taken over the resistor.
At \(\omega=0\), the capacitor is an open circuit while the inductor is a short circuit.
At \(\omega=\infty\), the capacitor is a short circuit while the inductor is an open circuit.
The capacitor has a negative impedance (\(1/j\omega C=-j/\omega C\)) while the inductor has a positive impedance (\(j\omega L\)).
At the resonant frequency (\(\omega_o\)), these impedances cancel out. 
\[\lvert H(j\omega_)\rvert=1\]
\[\theta(j\omega_o)=0^\circ\]
At \(\omega<\omega_o\), \(\theta(\omega)>0\), \(\theta(j\omega)\) maximises at \(+90^\circ\).
At \(\omega>\omega_o\), \(\theta(\omega)<0\), \(\theta(j\omega)\) maximises at \(-90^\circ\).

The magnitude and phase are
\[\lvert H(j\omega)\rvert = \frac{\omega(R/L)}{\sqrt{[-\omega^2+(1/LC)^2]^2+[\omega(R/L)]^2}}\]
\[\theta(j\omega)=90^\circ-tan^{-1}\left[\frac{\omega(R/L)}{-\omega^2+(1/LC)}\right]\]
\[\omega_o=\sqrt{\frac{1}{LC}}\]

Finding the cutoff frequencies
\[\omega^2_c\pm\omega_c\frac{R}{L}-\frac{1}{LC}=0\]
This provides 4 frequencies, only two are positive so have any significance.
\[\omega_{c1}=-\frac{R}{2L}+\sqrt{\left(\frac{R}{2L}\right)^2+\frac{1}{LC}}\]
\[\omega_{c2}=\frac{R}{2L}+\sqrt{\left(\frac{R}{2L}\right)^2+\frac{1}{LC}}\]
The bandwidth is 
\[\beta=\omega_{c2}-\omega_{c1}=\frac{R}{L}\]
The quality factor is 
\[Q=\frac{\omega_o}{\beta}=\frac{\sqrt{1/LC}}{R/L}=\sqrt{\frac{L}{CR^2}}\]
Using this we can write the transfer function as 
\[H(s)=\frac{\beta s}{s^2+\beta s+\omega_o^2}\]
==== Parallel RLC circuit ====

{{ bpparallel.png?100}}
Output taken over the capacitor and inductor.
\[H(s)=\frac{s/RC}{s^2+s(1/RC)+(1/LC)}\]
\[\lvert H(j\omega)\rvert=\frac{\omega/RC}{\sqrt{[(1/LC)-\omega^2]^2+[\omega/RC]^2}}\]
\[\omega_o=\sqrt{\frac{1}{LC}}\]
\[\omega_{c1}=-\frac{1}{2RC}+\sqrt{\left(\frac{1}{2RC}\right)^2+\frac{1}{LC}}\]
\[\omega_{c2}=\frac{1}{2RC}+\sqrt{\left(\frac{1}{2RC}\right)^2+\frac{1}{LC}}\]
\[\beta=\omega_{c2}-\omega_{c1}=\frac{1}{RC}\]
\[Q=\frac{\omega_o}{\beta}=\sqrt{\frac{R^2C}{L}}\]
\[H(s)=\frac{\beta s}{s^2+\beta s+\omega_o^2}\]
==== General form ====

\[H(s)=\frac{\beta s}{s^2+\beta s+\omega_o^2}\]
\[\beta=2\alpha\]
\[Q=\frac{\omega_o}{2\alpha}\]
  * With critical dampening, \(Q=\frac{1}{2}\) (intermediate Q)
  * With over-dampening, \(Q<\frac{1}{2}\) (low Q)
  * With under-dampening, \(Q>\frac{1}{2}\) (high Q)
The higher the Q factor, the sharper the pass curve.
==== Non-ideal voltage source ====

Here the source introduces resistance \(R_i\).
\[H(s)=\frac{(R/L)s}{s^2+\left(\frac{R+R_i}{L}\right)s+\left(\frac{1}{LC}\right)}\]
\[\lvert H(j\omega)\rvert=\frac{\omega(R/L)}{\sqrt{[(1/LC)-\omega^2]^2+[\omega((R+R_i)/L/)]^2}}\]
\[\omega_o=\sqrt{\frac{1}{LC}}\]
The resonant frequency is the same as the ideal case.
\[\omega_{c1}=-\frac{R+R_i}{2L}+\sqrt{\left(\frac{R+R_i}{2L}\right)^2+\left(\frac{1}{LC}\right)}\]
\[\omega_{c2}=\frac{R+R_i}{2L}+\sqrt{\left(\frac{R+R_i}{2L}\right)^2+\left(\frac{1}{LC}\right)}\]
\[\beta=\omega_{c2}-\omega_{c1}=\frac{R+R_i}{L}\]
\[H(s)=\frac{K\beta s}{s^2+\beta s+\omega_o^2}, K=\frac{R}{R_{R_i}}\]
===== Bandreject filter =====

{{ brseries.png?100}}
Compliment to bandpass filters.
Series RLC with output taken over the inductor and capacitor.
\[\omega_o=\sqrt{\frac{1}{LC}}\]
Passbands below \(\omega_{c1}\) and above \(\omega_{c2}\).
Overall capacitive over \(0<\omega<\omega_o\), with \(\theta(j\omega)<0\).
Overall inductive over \(\omega_o<\omega<\infty\) with \(\theta(j\omega)>0\).
At \(\omega=\omega_o\), \(\lvert H(j\omega)\rvert=0\).
\[H(s)=\frac{s^2+(1/LC)}{s^2+(R/L)s+(1/LC)}\]
\[H(j\omega)=\frac{\omega^2+(1/LC)}{\omega^2+(R/L)j\omega+(1/LC)}\]
\[\lvert H(j\omega)\rvert=\frac{\lvert -\omega^2+(1/LC)\rvert}{\sqrt{[-\omega^2+(1/LC)]^2+[\omega(R/L)]^2}}\]
\[\theta(j\omega)=-\tan^{-1}\left[\frac{\omega(R/L)}{-\omega^2+(1/LC)}\right]\]
\[\omega_{c1}=-\frac{R}{2L}+\sqrt{\left(\frac{R}{2L}\right)+\frac{1}{LC}}\]
\[\omega_{c2}=\frac{R}{2L}+\sqrt{\left(\frac{R}{2L}\right)+\frac{1}{LC}}\]
These are the same cutoff as the series RLC bandpass filter.
\[\beta=\omega_{c2}-\omega_{c1}=\frac{R}{L}\]
\[Q=\frac{\omega_o}{\beta}=\sqrt{\frac{L}{CR^2}}\]
{{ brparallel.png?100}}
A parallel RLC circuit with output across R will produce a similar bandreject filter.
==== General Form ====

\[H(s)=\frac{s^2+\omega_o^2}{s^2+\beta s+\omega_o^2}\]
===== Bode Plots =====

{{ bodeeg.png?100}}
x axis (\(\omega\)) is on a log base 10 scale.
A decade is the range of frequencies for which the ratio of highest frequency to lowest is 10.
e.g. 50 to 5000 rad/s is two decades.
An octave is a two-to-one change in frequency.
e.g. 3 to 12 rad/s is 2 octaves.
The y axis is a decibel scale to represent magnitude: \(20\log_{10}\lvert H(j\omega)\rvert\) in units of decibels (dB).
Using decibels allows us to 
  * Simultaneously see small and large values
  * Reduce multiplication and division of factors associated with poles and zeros of \(H(j\omega)\) to addition and subtraction

The following only applies to real poles and zeros.
==== Standard form ====

\[H(j\omega)=\frac{K_0\left(1+j\frac{\omega}{z_1}\right)}{j\omega\left(1+j\frac{\omega}{p_1}\right)}\]
This can be achieved like so
\[H(j\omega)=\frac{K(j\omega+z_1)}{j\omega(j\omega+p_1)}*\frac{z_1/z_1}{p_1/p_1}=\frac{Kz_1\left(1+j\frac{\omega}{z_1}\right)}{p_1*j\omega\left(1+j\frac{\omega}{p_1}\right)}\]
and \(K_0=K\frac{z_1}{p_1}\).

We can express the standard form of \(H(j\omega)\) in polar form:
\[H(j\omega)=\frac{K_0\left\lvert 1+j\frac{\omega}{z_1}\right\rvert\angle{\phi_1}}{\omega\angle{90^\circ}\left\lvert 1+j\frac{\omega}{p_1}\right\rvert\angle{\beta_1}}=\frac{K_0\left\lvert 1+j\frac{\omega}{z_1}\right\rvert}{\omega\left\lvert 1+j\frac{\omega}{p_1}\right\rvert}\angle{\phi_1-90^\circ-\beta_1}\]
The magnitude and phase angle expressions are:
\[\lvert H(j\omega)\rvert=\frac{K_0\left\lvert 1+j\frac{\omega}{z_1}\right\rvert}{\omega\left\lvert 1+j\frac{\omega}{p_1}\right\rvert}\]
\[\theta(j\omega)=\phi_1-90^\circ-\beta_1\]
Bode Diagrams consist of plotting \(20\log_{10}\lvert H(j\omega)\rvert\) and \(\theta(j\omega)\) as functions of \(\omega\).
==== Magnitude ====

\[A_{dB}=20\log_{10}(\lvert H(j\omega)\rvert)=20\log_{10}K_0+20\log_{10}\left\lvert 1+j\frac{\omega}{z_1}\right\rvert-20\log_{10}\omega-20\log_{10}\left\lvert 1+j\frac{\omega}{p_1}\right\rvert\]
It is useful to plot each term independently and combine the plots graphically.

The first term, \(20\log_{10}K_0\) is a horizontal, straight line.

The second term is a first order zero.
  * For \(\omega << z_1: 20\log_{10}\lvert 1+j\omega/z_1\rvert\approx 0\)
  * For \(\omega >> z_1: 20\log_{10}\lvert 1+j\omega/z_1\rvert\approx 20\log_{10}(\omega/z_1)\)
Approximating by two straight lines who intersect at \(\omega=z_1\).
The first line is 0 dB up until the corner frequency and the second has a 20 dB/decade slope.

For the forth term, a first order pole.
  * For \(\omega << z_1: -20\log_{10}\lvert 1+j\omega/p_1\rvert\approx 0\)
  * For \(\omega >> z_1: -20\log_{10}\lvert 1+j\omega/p_1\rvert\approx -20\log_{10}(\omega/p_1)\)
Approximating by two straight lines who intersect at \(\omega=p_1\).
The first line is 0 dB up until the corner frequency and the second has a -20 dB/decade slope.

The third term, a first order pole through the origin.
At \(\omega=1\), the term equals 0.
As a result it is a line going through (1,0) and having a slope of -20 dB/decade.

The total plot is the graphical sum of these four plots.

In reality there are smooth curves, instead of these piece-wise curves.
The two differ a maximum of 3 dB at the corner frequency, \(\omega=z_1\).
At \(\omega=0.5z_1\) or \(2 z_1\), the difference shrinks to 1 dB.
Zeros have a positive difference, poles have a negative difference.
As such it serves as a decent approximation.
==== Angle ====

* The phase angle of a constant is zero.
  * The phase angle of a first order zero/pole at \(\omega=0\) is constant \(+90^\circ/-90^\circ\)
  * For zeros or poles not at \(\omega=0\), straight line approximations are as follows:
Let the corner frequency be \(\omega_{cf}=z_1\).
    * For \(\omega\leq\frac{\omega_{cf}{10}}\implies\theta(j\omega)=0\)
    * For \(\omega\geq 10\omega_{cf}\implies\theta(j\omega)=\begin{cases}+90^\circ&\text{zero}\\-90^\circ&\text{pole}\end{cases}\)
    * For \(\frac{\omega_{cf}}{10}<\omega<10\omega_{cf}\), phase plot straight line through \(+45^\circ\) (zero) or \(-45^\circ\) (pole) at corner frequency
==== Second order poles and zeros ====

=== Magnitude plot ===

* Approximate plot by two lines that intersect at corner frequency \(\omega=p_1\)
  * First line is constant 0 dB up until corner frequency
  * Second line is straight sloped at -40 dB/decade
=== Phase plot ===

* For \(\omega\leq\frac{p_1}{10}\implies\theta(j\omega)=0\)
  * For \(\omega\geq 10p_1\implies\theta(j\omega)=-180\)
  * For \(\frac{p_1}{10}<\omega<10p_1\), phase is straight line through \(-90^\circ\) at \(p_1\)

This is the opposite for zeros.
==== Nth order poles and zeros ====

* Magnitude plot: slope after a zero/pole is \(\pm 20n\) (dB/decade)
  * Phase plot:
    * Pole: \(\omega\geq 10p_1\implies\theta(j\omega)=-90n\)
    * Zero: \(\omega\geq 10z_1\implies\theta(j\omega)=+90n\)
    * Between 1/10 and 10 times corner frequency, connect a straight line
====== Operational Amplifiers ======

===== Why amplification? =====

Increases weak signal amplitudes.
Can let a comparator work as there is a larger difference between voltage levels.
Can also increase signal strength, in order to increase noise tolerance.
Amplifiers are two port networks, with a common ground.
===== Linear Signal amplifiers =====

The output signal is an exact, scaled replica of the input signal.
==== Linear Voltage Amplifier ====

Output voltage (\(v_o(t)\)) is a scaled replica of input voltage (\(v_i(t)\))
\[v_o(t)=A_vv_i(t))\]
The gain of the amplifier is
\[A_v=\frac{v_o(t)}{v_i(t)} (V/V)\]
If \(A_v\) is positive it is a non-inverting amplifier, if negative is an inverting amplifier.
==== Current gain and power gain ====

\[A_i=\frac{i_o}{i_i}(A/A)\]
If the power output exceeds the input power, there is power gain.
\[A_p=\frac{P_o}{P_i}=A_vA_i(W/W)\]
Gain is often measured in decibels, being a logarithmic scale. 
In this case cascaded amplifiers gain are added.
If gain is linear, then gain is multiplied.

If there is gain, the power delivered to the load (\(P_o\)) is greater than the power obtained from the source (\(P_i\)) and must come from an external DC supply.
The average power supplied by the amplifier is the product of the average current and supply voltage.
\[P_s=V_{AA}\overline{I_A}+V_{BB}\overline{I_B}\]
Power is also dissipated as heat in the internal circuits of the amplifier.
The power balance equation for the amplifier is:
\[P_s+P_i=P_o+P_{dissipated}\]
The efficiency can be found by
\[\eta=\frac{P_o}{P_s}\times 100\%\]
Because extra power needs to be supplied, there is a limit to the maximum amplification after which the signal can be amplified no further.
There limits are \(L_+\) for the maximum limit and \(L_-\) for the minimum limit.
This saturation causes a nonlinear distortion of the output waveform.
To avoid this, bias the circuit towards the middle of the transfer characteristic and within linear operating range.
===== Equivalent circuit model =====

{{ opeq.png?100}}
An equivalent circuit can be constructed as a input resistor between the input terminals and a dependent source and resistor between the output terminals.
This model provides a realistic input and output impedance.
This is useful as real amplifiers draw current from the source and have reduced real output voltage when connected to a load.
\[v_o=A_{voc}v_i\left(\frac{R_L}{R_L+R_o}\right)\]

When the whole circuit is connected, the overall gain is \(A_{vs}=v_o/v_s\).
Through some manipulation with voltage dividers, we can find
\[A_{vs}=\frac{v_o}{v_s}=\frac{v_o}{v_s}\frac{v_i}{v_s}=A_v\frac{R_i}{R_i+R_s}=A_{voc}\frac{R_L}{R_L+R_o}\frac{R_i}{R_i+R_s}\]
We can note from here that not all the source voltage appears over the input terminal and not all the voltage from the dependent source appears at the output.
These reductions are called loading effects and as a result \(A_v\) and \(A_{vs}\) are less than \(A_{voc}\).
The impedance can be important in a design, which involves the following principles:
  - \(R_s<<R_i\implies v_i\approx v_s\)
  - \(R_L>>R_o\implies v_o\approx A_{voc}v_i\implies A_v\approx A_{voc}\)
These cause \(A_{vs}\approx A_{voc}\).
An ideal voltage amplifier is one with \(R_i=\infty,R_o=0\), as this has no reduction issues from loading effects.
==== Amplifier models ====

^ Model^ Input signal^ Output signal^ Gain parameter^ Ideal \(R_i\)^ Ideal \(R_o\)^
|Voltage|Voltage|Voltage|\(A_{voc}\) Open-circuit voltage gain|\(\infty\)|0|
|Current|Current|Current|\(A_{is}\) Short-circuit current gain|0|\(\infty\)|
|Trans-conductance|Voltage|Current|\(G_m\) Short-circuit trans-conductance|\(\infty\)|\(\infty\)|
|Trans-resistance|Current|Voltage|\(R_m\) Open-circuit trans-resistance|0|0|
=== Voltage ===

Output resistor in series with voltage controlled voltage source.
Source dependent on voltage over input resistor.
\(A_{voc}\) is the open circuit voltage.
=== Current ===

Output resistor in parallel with current controlled current source.
Current dependent on current through input resistor.
\(A_{isc}\) is the short circuit current.
The open circuit voltage is \(A_{isc}i_iR_o\).
With this we can find the equivalent gain of the voltage model.
\[A_{voc}=A_{isc}\frac{R_o}{R_i}\]
=== Trans-conductance ===

Output resistor in parallel with voltage controlled current source.
Source dependent on voltage over input resistor.
Open circuit voltage is \(G_{msc}v_iR_o\), giving equivalent voltage gain of
\[A_{voc}=G_{msc}R_o\]
=== Trans-resistance ===

Output resistor in series with current controlled voltage source.
Source dependent on current through input resistor.
Open circuit voltage is \(R_{moc}i_i\), giving equivalent voltage gain of
\[A_{voc}=\frac{R_{moc}}{R_i}\]
===== Cascaded Amplifiers =====

Cascading amplifiers alters the gain of the overall system, and cascading can be used to increase the overall gain.
Cascading amplifiers multiplies the gain of the individual stages.
\[A_v=\prod{A_{vn}}\]
Cascading is also affected by the resistance of each stage.
===== Terminals =====

* 2 input terminals, non-inverting (+) and inverting (-)
  * 1 output terminal
  * 2 power supply terminals, \(V^+\) and \(V^-\)
  * Additional terminals, offset nulls and NC
The first five are shown on the circuit symbol.
Voltages are measured relative to a reference, commonly ground.
Positive supply voltage, \(V_{CC}\) is between \(V^+\) and the reference node.
Negative supply voltage, \(-V_{CC}\) if equal but opposite to the positive voltage or \(-V_{EE}\) if not, is connected between \(V^-\) and reference node.
Voltages between non-inverting and inverting input terminals and reference are \(v_p\) and \(v_n\) respectively.
Voltage between output and reference is \(v_o\).
All current references are into the op amp terminal.
  * **\(i_{c+}\):** current into positive supply terminal
  * **\(i_{c-}\):** current into negative supply terminal
  * **\(i_p\):** current into non-inverting input
  * **\(i_n\):** current into inverting input
===== Behaviour =====

Op amps have three distinct regions:
  * Two saturation regions with nonlinear behaviour
  * One linear region with linear behaviour
When operating linearly, output voltage is difference in input voltages times gain.
\[v_o=A(v_p-v_n)\]
Such amplifiers are called differential amplifiers.
Gain, \(A\), is open loop gain, usually very large.
In practice, saturation is within a volt of \(V_{CC}\).
The complete characteristic is:
\[v_o=\begin{cases}-V_{CC} &A(v_p-v_n)< -V_{CC}\\A(v_p-v_n) &-V_{CC}\leq A(v_p-v_n)\leq V_{CC}\\+V_{CC} &A(v_p-v_n)>+V_{CC}\end{cases}\]
If we want to constrain to the linear region, we need
\[\lvert v_p-v_n\rvert\leq\frac{V_{CC}}{A}\]
This is a very small difference, as \(A\) is often large.
When a op amp is constrained to its linear region and node voltages are larger than above, the constraint becomes \(v_p=v_n\), known as a virtual short circuit.
This characterises an ideal op-amp whose \(A=\infty\).
Real op amps have large input resistances \(\approx 1M\Omega\), whereas the ideal has infinite input resistance and zero output resistance.
A consequence of this is that the ideal op amp has \(i_p=i_n=0\).
Summing the currents into the op amp we find:
\[i_p+i_n+i_o+i_{c+}+i_{c-}=0\]
\[i_o=-(i_{c+}+i_{c-})\]
==== Ideal op amp ====

An ideal op amp has the following properties:
  * Gain \(A\) is infinite
  * Equivalent circuit impedance is infinite
  * Input currents are zero
  * Output impedance is zero
  * Infinite bandwidth
===== Closed loop configurations =====

Open loop gain is very high and is fixed.
Due to manufacturing, gain can vary from unit to unit.
To control the gain for practical applications, feedback is used.
Passive components are used to provide feedback.
==== Inverting amplifier ====

{{ invamp.png?100}}
A source, \(v_s\), is connected to a resistor, \(R_s\), which is connected to the inverting input of the op amp.
The non-inverting input is connected to ground.
A resistor, \(R_f\), is connected between the amp's output and the non-inverting input after \(R_s\), closing the loop.
Ideally \(v_p\) and \(v_n\) are the same (virtual short circuit), causing the inverting terminal to be a virtual ground.
The current through \(R_s\), \(i_s\) is 
\[i_s=\frac{v_s-v_n}{R_s}=\frac{v_s}{R_s}\]
The ideal op amp doesn't allow current into its terminals, so \(i_s\) will flow into \(R_f\) to the output terminal.
Using KVL around the loop, we find
\[-v_o-i_sR_f+v_n=0\]
\[v_o=-\frac{R_f}{R_s}v_s\]
This is an inverted, amplified replica of the input, \(v_s\).
The closed loop gain is
\[G=\frac{v_o}{v_s}=\frac{R_f}{R_s}\]
Compared to \(A\), this gain is smaller and more predictable, and determined only by external components \(R_s\) and \(R_f\), meaning there are no loading effects.
The input impedance to the amp from the source's point of view is \(R_s\), which we can control.

The upper limit of the gain can be found as follows:
This all assumes \(A\) is infinite. If \(A\) is finite instead, then
\[v_o=A(v_p-v_n)\implies v_n=-\frac{v_o}{A}\]
And the current through \(R_s\) is 
\[i_s=\frac{v_s-(-v_o/A)}{R_s}=\frac{v_s+v_o/A}{R_s}\]
So the KVL becomes
\[0=-v_0\left(1+\frac{1}{A}+\frac{R_f}{R_s}\right)-v_s\left(\frac{R_f}{R_s}\right)\]
So the closed loop gain is 
\[G=\frac{v_o}{v_s}=-\frac{R_f}{R_s}\frac{1}{1+\left(1+\frac{R_f}{R_s}\right)/A}\]
As \(A\) goes to infinity, \(G\) approaches the ideal value.
In order to minimise the effect of the finite loop gain, we choose
\[1+\frac{R_f}{R_s}<<A\]
=== Weighted Summer ===

{{ weisum.png?100}}
If we have two supplied connected to resistors feeding into the inverting terminal in parallel, we have an inverting summer.
\[i_f=i_a+i_b=\frac{v_a}{R_a}+\frac{v_b}{R_b}\]
From this we find the output voltage to be:
\[v_o=v_n-iR_f=-\left(\frac{R_f}{R_a}v_a+\frac{R_f}{R_b}v_b\right)\]
The output voltage is an inverted, weighted sum of the input signals, weighted by their resistors.
=== Variations ===

{{ invvar.png?100}}
We can replace \(R_f\) by three resistors in a T configuration with the vertical one being connected to ground.
This achieves a high gain magnitude with a small range of resistors.
\[v_o=-v_{in}\left(\frac{R_2}{R_1}+\frac{R_4}{R_1}+\frac{R_2R_4}{R_1R_3}\right)\]
\[G=-\left(\frac{R_2}{R_1}+\frac{R_4}{R_1}+\frac{R_2R_4}{R_1R_3}\right)\]
The input impedance is \(R_1\) and the output impedance is zero.
Having \(R_1=R_3=1\) and \(R_2=R_4=10\) yields a gain of \(-120V/V\).
Without this variation, a ratio of \(120:1\) would be required as opposed to \(10:1\).
==== Non-inverting amplifier ====

{{ noninvamp.png?100}}
The non-inverting terminal has the source connected through a resistor. 
The inverting terminal still has a resistor, \(R_s\) between it and ground and the loop is caused by \(R_f\) between the output and the inverting terminal.
We find the gain to be 
\[G=1+\frac{R_f}{R_s}\]
If the input changes, a large output voltage will be produces, quickly bringing \(v_n\) to equal \(v_p\), causing the difference to be zero.
This is known as negative feedback, because the feedback signal opposes the input signal.
=== Buffer amplifier ===

Setting \(R_f=0\) and \(R_s=\infty\) gives a unity gain non-inverting amplifier (\(G=1\)).
Here the output is the same as the input.
It is used in cascading amplifiers.
When cascading, 
\[V_{in2}=v_{o1}\frac{R_{in2}}{R_{o1}+R_{in2}}\]
If \(R_{o1}>>R_{in2}\), \(v_{in2}<<v_{o1}\).
Inserting a buffer amplifier removes these loading effects.
==== Upper limit on gain ====

Assuming equal supply voltages, if the output region is linear, then
\[\lvert v_o\rvert\leq V_{CC}\]
\[G\leq\left\lvert\frac{V_{CC}}{v_{in}}\right\rvert\]
\[1+\frac{R_f}{R_s}\leq\left\lvert\frac{V_{CC}}{v_{in}}\right\rvert\]
The upper limit is dependent on the supply voltages.
===== Finite open loop gain =====

\[v_o=\frac{v_{in}-(v_o/A)}{R_s}+\left(v_{in}-\frac{v_o}{A}\right)\]
\[G=1+\frac{R_f}{R_s}\frac{1}{1+\frac{1+(R_f/R_s)}{A}}\]
This approaches the ideal value as A goes to infinity.
===== Difference (Differential) amplifiers =====

{{ diffamp.png?100}}
We want to amplify the difference between the two input signals.
If we just plug the input signals into the op amp, we get a large, unpredictable output due to the large open loop gain.
Using feedback, we can provide a predicable and stable closed-loop gain.
The difference circuit has a voltage source (\(v_a\)) connected through a resistor (\(R_a\)) to the inverting input, and a resistor (\(R_b\)) connected through the output to the inverting terminal.
There is also a voltage source (\(v_b\)) connected to the non-inverting terminal through a resistor (\(R_c\)) with a resistor (\(R_d\)) connected to ground in parallel.
From voltage division we can find:
\[v_p=\frac{R_d}{R_c+R_d}v_b=v_n\]
Assuming the op amp is ideal, we can find the current through \(R_a\):
\[i_a=\frac{v_a-v_n}{R_a}\]
From this, we can use KVL to find the output voltage:
\[v_o=v_n-i_aR_b\]
\[v_n=v_o+i_aR_b\]
This gives us:
\[v_o=\frac{R_d(R_a+R_b)}{R_a(R_c+R_d)}v_b-\frac{R_b}{R_a}v_a\]
This shows that the output is a scaled (weighted) replica of \(v_b\) and \(v_a\) where the scaling is controlled by the resistors.
To make the scaling uniform, we can
\[\frac{R_a}{R_b}=\frac{R_c}{R_d}\implies v_o=\frac{R_b}{R_a}(v_b-v_a)\]

We can better understand the behaviour by redefining the inputs.
The differential mode input is:
\[v_{dm}=v_b-v_a\]
The common mode input is:
\[v_{cm}=\frac{v_b+v_a}{2}\]
From this we can find:
\[v_o=\frac{R_aR_d-R_bR_c}{R_a(R_c+R_d)}v_{cm}+\frac{R_d(R_a+R_b)+R_b(R_c+R_d)}{2R_a(R_c+R_d)}v_{dm}\]
The coefficient of \(v_{cm}\) is the common mode gain (\(A_{cm}\)) and \(v_{dm}\) is the differential mode gain (\(A_{dm}\)).
If we choose \(R_a=R_c\) and \(R_b=R_d\), then
\[v_o=(0)v_{cm}+\frac{R_b}{R_a}v_{dm}\]
Thus the ideal difference amplifier has zero common mode gain, amplifying only the differential mode gain.

A measure of how ideal a difference amplifier is known as the common mode rejection ratio (CMRR).
\[CMRR(dB)=20\log\left(\frac{\lvert A_{dm}\rvert}{\lvert A_{cm}\rvert}\right)\]
If the resistors are perfectly matched, \(A_{cm}=0\) and \(CMRR(dB)=\infty\).

Difference amplifiers should ideally also have infinite input resistance.
The input resistance can be found by
\[R_{id}=\frac{v_{dm}}{i_a}=2R_a\]
====== Two port circuit analysis ======

===== Overview =====

Current is directed into the positive terminal of a port and out of the negative terminal.
The currents are constrained so that the current into the upper terminal is equal to the current coming out of the lower one.
Generally, the voltage rises from the lower to the upper terminals.
The model needs to model the terminal variables (\(i_1,v_1,i_2,v_2\)) and how they relate to each other, there is no need to find voltages inside the circuit.

The circuit is simplified to a black box where:
  * No initial energy is stored inside
  * No independent sources are inside, dependent are allowed
  * Current into a port must equal current out of the same port, \(i_1=i_1'\)
  * All external connections are made to the input or output ports.
These assumptions carry when the circuit is transformed to s-domain.
===== Three terminal two port circuits =====

The most common type of two port circuit (op amps, transistors, etc).
One terminal is shared between the ports.
===== Terminal Equations =====

Two port circuits can be described by the following equation:
\[[\mathbf{V}]=[\mathbf{Z}][\mathbf{I}]\]
Here \([\mathbf{V}]\) is a vector of the port voltages and \([\mathbf{I}]\) is a vector of the port currents.
\([\mathbf{Z}]\) is a 2x2 matrix of the impedances, specifically the open circuit impedance.
Additionally we can use the following instead of the open circuit impedance, Z:
  * Short circuit admittance, Y
  * Transmission, A
  * Inverse transmission, B
  * Hybrid, H
  * Inverse hybrid, G
The following pairs are inverses, i.e. \(A^{-1}=B\): (Z,Y), (A,B), (H,G).
Z gives the input and output voltages as a function of input and output currents, Y does currents in terms of voltages.

Given a circuit, the parameters can be determined by shorting or opening a port.
The Z parameters can be found as follows:
Linear passive circuits are reciprocal. Circuits that contain dependent sources are usually not reciprocal.
A circuit is said to be reciprocal if when a current applied separately to each port yields the same voltage on opposite ports.
\[I_o\text{ applied to Port 1}\implies V_2=V_o\]
This manifests itself in the matrix where \(z_{12}=z_{21}\).

The Y-parameters have units of siemens and can be found by shorting ports:
We also know that since Z and Y and reciprocals

where \(\Delta z=z_{11}z_{22}-z_{12}z_{21}\).
==== Transmission matrices ====

The inverse transmission is the same swapping the \(a\)'s for \(b\)'s and the vectors for each other.
The units on each \(a\) are not the same, \(a_{11}\) and \(a_{22}\) have no units, \(a_{12}\) is in ohms and \(a_{21}\) is in mhos.
A and B are known as transmission parameters as they describe the voltage and current at one end of the network in terms of the other.
==== Hybrid matrices ====

===== Terminated two port circuits =====

A circuit is driven by source \(V_g\), has source impedance \(Z_g\) and load impedance \(Z_L\).
The characteristics that determine its behaviour:

All of these can be found from the matrix parameters above.
===== Interconnected two-port circuits =====

{{ int2port.png?100}}
Makes it easy to design complex systems.
There are five ways to connect two port circuits.
a) Cascade - A parameters
b) Series - Z parameters
c) Parallel - Y parameters
d) Series-parallel - H parameters
e) Parallel-series - G parameters
When cascading circuits, we multiply each circuit's A matrix to get an A matrix of the whole system.
This assumes \(I_2\) is flowing out of the circuit (all the elements of the matrix are positive), as a result we change the equation as follows:
We only use the minus sign when expressing the input-output relationship, not when terminating or cascading.

For series, parallel, series-parallel and parallel-series circuits, we sum the matrices rather than multiplying them.
====== Operational Amplifier Networks ======

===== Integrator =====

{{ opampint.png?100}}
Produces an output signal proportional to the running-time integral of the input voltages.
Resistor feeding into the inverting terminal of an op amp, with a capacitor in parallel. The non-inverting terminal is grounded.
\[v_0=-\frac{1}{R_sC_f}\int_{t_0}^tv_sdt+v_0(t_0)\]
The output is an inverted, scaled replica of the integral of the input voltage, plus the initial capacitor voltage.
The gain is set by \(R_s\) and \(C_f\). 
To produce positive gain, cascade with an inverting amplifier.
===== Differentiator =====

{{ opampdif.png?100}}
Produces an output proportional to the derivative of the input voltage.
Capacitor feeding into the inverting terminal of an op amp, with a resistor in parallel. The non-inverting terminal is grounded.
\[v_0(t)=-RC\frac{dv_{in}}{dt}\]
===== Active filters =====

Benefits over passive filters:
  * Over amplification (gain > 1)
  * No bulky and costly inductors
  * Loading does not affect cutoff frequency and passband magnitude
Active filters are generally preferred, especially for gain, load variation and size constraints.
==== First order low pass ====

{{ 1stlow.png?100}}
Resistor in parallel with a capacitor and the op amp's inverting terminal, with a resistor between input and elements.
Low frequencies, capacitor is open, turning the circuit into an inverting amplifier with gain \(-R_2/R_1\).
High frequencies cause the capacitor to short the rest.
Overall this is a low pass filter with a gain of \(-R_2/R_1\).
\[H(s)=-\frac{R_2||(1/sC)}{R_1}\]
\[=-\frac{R_2}{R_1}\frac{1/(R_2C)}{s+1/(R_2C)}=-K\frac{\omega_c}{s+\omega_c}\]
This is the same general form as a passive filter, except K can exceed 1.

The prototype filter has component values of \(R_1=R_2=1\Omega\) and \(C=1F\) and a cutoff frequency of 1 rad/s.
\[H(s)=-1\frac{1}{s+1}\]
This can be altered using magnitude and frequency scaling.
==== First order high pass ====

{{ 1sthigh.png?100}}
Resistor and capacitor in series feed into a resistor and op amp's inverting terminal in parallel.
\[H(s)=-\frac{R_2}{R_1+\frac{1}{sC}}\]
\[=-\frac{R_2}{R_1}\frac{s}{s+1/(R_1C)}=-K\frac{s}{s+\omega_c}\]
This is the same as the passive filter except K can exceed 1.

The prototype filter has \(R_1=R_2=1\Omega\) and \(C=1F\).
\[H(s)=-\frac{1}{s+1}\]
It produces a unity passband gain and a cut off frequency of 1 rad/s.
===== Scaling =====

==== Magnitude Scaling ====

Used to convert from the prototype to real values.
Uses a positive, real scaling factor \(k_m\).
\[R'=k_mR, L'=k_mL, C'=\frac{C}{k_m}\]
If we know the value of one final element, we can scale the rest appropriately.
This preserves the cutoff frequency.
==== Frequency Scaling ====

Shifts the frequency response to another frequency region without changing the shape of the response.
Uses a positive, real scaling factor \(k_f=\frac{\omega_c'}{\omega_c}\).
\[R'=R, L'=L/k_f, C'=C/k_f\]
==== Combined scaling ====

Simultaneous scaling can be done as follows:
\[R'=k_mR, L'=\frac{k_m}{k_f}L, C'=\frac{1}{k_mk_f}C\]
===== Multiple Op Amps =====

==== Cascaded Bandpass filter ====

{{ 1stbandpass.png?100}}
Consists of three components cascaded in series
  * Unity gain LPF with cutoff frequency \(\omega_{c2}\)
  * Unity gain HPF with cutoff frequency \(\omega_{c1}\)
  * Gain component to provide desired level of gain in the passband.
This approach requires \(\frac{\omega_{c2}}{\omega_{c1}}\geq 2\).
This provides a transfer function:
\[H(s)=\frac{-K\omega_{c2}s}{s^2+(\omega_{c1}+\omega_{c2})s+\omega_{c1}\omega_{c2}}\]
To conform to the standard form of a bandpass filter, \(\omega_{c1}+\omega_{c2}\approx\omega_{c2}\implies \omega_{c2}>>\omega_{c1}\).
This then causes the transfer function to become:
\[H(s)=\frac{-K\omega_{c2}s}{s^2+(\omega_{c2})s+\omega_{c1}\omega_{c2}}\]
\[\omega_{c2}=\frac{1}{R_LC_L}\]
\[\omega_{c1}=\frac{1}{R_HC_H}\]
To provide the desired level of passband gain, we compute the magnitude of the transfer function at the centre frequency, \(\omega_0\)
\[\lvert H(j\omega_0)\rvert=\lvert\frac{-K\omega_{c2}(j\omega_0)}{(j\omega_0)^2+(\omega_{c1}+\omega_{c2})(j\omega_0)+\omega_{c1}\omega_{c2}}\rvert=\frac{K\omega_{c2}}{\omega_{c2}}=K\]
Since \(K=\frac{R_f}{R_i}\), we can tailor it to satisfy any desired gain.
==== Parallel Bandreject filter ====

{{ 1stbandreject.png?100}}
Consists of three components
  * Unity gain LPF with cutoff frequency \(\omega_{c1}\)
  * Unity gain HPF with cutoff frequency \(\omega_{c2}\)
  * Gain component to provide desired level of gain in the passband.
The LPF and HPF are in parallel and added using a weighted summing amplifier.
Again, we need  \({\omega_{c2}}+\omega_{c1}\approx\omega_{c2}\) to conform to the standard form of a filter.
\[\omega_{c1}=\frac{1}{R_LC_L}\]
\[\omega_{c2}=\frac{1}{R_HC_H}\]
\[K=R_f/R_i\]
===== Higher order filters =====

First order filters have a 20db/decade roll-off in the transition region.
Adding filters in a cascade causes the transition to become sharper, e.g. 4 filters causes a slope of 80 db/dec.
Cascading also alters the cutoff frequency, so we need to adjust for that in the design.
For low pass filters
\[\omega_{cn}=\sqrt{\sqrt[n]{2}-1}\]
This gives the scaling factor as 
\[k_f=\omega_c/\omega_{cn}\]
===== Butterworth filters =====

Provides a steeper response and flatter passband gain, closer to an ideal filter.
The design procedure is as follows:
  - Determine order needed
  - Use normalised Butterworth polynomials to determine filter stages
  - Frequency and magnitude scale to actual cutoff frequency
  - Cascade with gain stage to achieve desired passband gain
==== Second order low pass ====

{{ 2ndbtrlow.png?100}}
\[V_0=\frac{V_i}{R^2C_1C_2s^2+2RC_2s+1}\]
\[H(s)=\frac{\frac{1}{C_1C_2}}{s^2+\frac{2}{C_1}+\frac{1}{C_1C_2}}\]
\[b=\frac{2}{C_1}\]
\[1=\frac{1}{C_1C_2}\]
==== Second order high pass ====

{{ 2ndbtrhigh.png?100}}
\[H(s)=\frac{s^2}{s^2+bs+1}\]
\[b=\frac{2}{R_2}\]
\[1=\frac{1}{R_1R_2}\]
==== Bandpass and bandreject ====

Cascade a nth order high pass with a nth order lowpass to get the nth order bandpass.
Add a gain stage at the end for gain.

A bandreject can be made from a nth order high pass with a nth order lowpass in parallel into a summing amplifier to get the nth order bandreject.
Here the summing amplifier provides gain.
===== Non-ideal properties of Op Amps =====

Op amps have;
  * Linear Imperfections
    * Finite open loop gain and bandwidth
    * Finite input and output impedances
  * Nonlinear Imperfections
    * Output current limits
    * Output voltage saturation
    * Slew rate limitation
  * DC Imperfections
    * Offset voltage
    * Input and offset bias currents
==== Frequency dependent open loop gain ====

High DC gain until a cutoff frequency, \(f_{3dB}\) where gain is constant. 
Above \(f_{3dB}\), the magnitude gain falls off at -20 dB/dec.
\[A(f)=\frac{A_0}{1+j(f/f_{3dB})}\]
\[\lvert A(f)\rvert = \frac{A_0}{\sqrt{1+(f/f_{3dB})^2}}\]
For \(f>>f_{3dB}\), we can approximate with
\[A(f)=\frac{A_0f_{3dB}}{j(f)}\]
\[\lvert A(f)\rvert=\frac{A_0f_{3dB}}{f}\]
If \(f_t\) is the frequency which \(\lvert A(f_t)\rvert=0dB\), then
\[f_t=A_0f_{3dB}\]
\[\lvert A(f)=\frac{A_0f_{3dB}}{f}=\frac{f_t}{f}\]
Hence we can determine \(\lvert A(f_t)\rvert\) at any frequency if \(f_t\) is known.

For closed loop inverting gain, we get
\[G(f)=\frac{-R_2/R_1}{1+\frac{j(f)}{f_t/(1+R_2/R_1)}}\]
\[\lvert G(f)\rvert=\frac{-R_2/R_1}{\sqrt{1+\left(\frac{j(f)}{f_t/(1+R_2/R_1)}\right)^2}}\]
Again we can see a lowpass response similar to the open loop gain.
\[GBP=G_0f_{3dB}=(R_2/R_1)\frac{f_t}{1+R_2/R_1}\]

For closed loop non-inverting gain, we get
\[G(f)=\frac{1+R_2/R_1}{1+\frac{j(f)}{f_t/(1+R_2/R_1)}}\]
\[\lvert G(f)\rvert=\frac{1+R_2/R_1}{\sqrt{1+\left(\frac{j(f)}{f_t/(1+R_2/R_1)}\right)^2}}\]
\[GBP=G_0f_{3dB}=\left(1+\frac{R_2}{R_1}\right)\frac{f_t}{1+R_2/R_1}\]

==== Voltage and current limits ====

Due to physical limits to the Op Amp, the output current is limited. 
This limit is usually on the data-sheet for the IC.

The output voltage limit is dependent on the input supplies.
==== Slew rate limitation ====

The slew rate is the output voltage's rate of change.
\[SR=\left.\frac{dv_o}{dt}\right\vert_{\max}(V/\mu s)\]
This can cause a saw-tooth waveform on the output as the output chases the input waveform at a slower rate.
==== Offset voltage ====

Small voltage offset due to internal op amp symmetry.
Can be modelled as a constant voltage source connected to one of the terminals.
If the op amp is not required to amplify low frequency signals, we can use capacitive coupling to prevent DC offset problems.
The capacitor filters out the DC component of the source signal.
The DC voltage can be trimmed to zero using a variable resistor to the offset nulling terminals.
This resistance creates a=n imbalance that counteracts the internal mismatch.
==== Input bias an offset currents ====

Can be modelled as two current sources connected to the input terminals.
Can be reduced by introducing a resistor to both terminals of the op amp.
The introduced resistor must be equal the other two resistors in parallel.
\[R_3=R_1||R_2\]
This sets the output dc offset voltage to zero.
==== Coupling ====

A DC coupled amplifier provides DC gain.
IC amplifiers are often DC coupled as capacitors cannot be fabricated in an integrated form.
Some applications require DC coupled amplifiers whereas others require AC coupling.
An AC coupled amplifier does not provide DC gain.
The use of coupling capacitors between cascading stages can be used to avoid amplifier saturation.

The gain of an amplifier drops off at high frequencies due to the capacitance and inductance inherent in the wires in the circuit.
The inductance causes a drop off at higher frequencies.

Amplifiers that are DC coupled, or have \(f_L\) that is a fraction of \(f_H\) are called **wideband** or **baseband** amplifiers.
Amplifiers that are deliberately limited to a small bandwidth compared to the centre frequency are called **narrowband** or **bandpass** amplifiers.

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