ece140: add steady state analysis

docs/1b/ece140.md

@@ -297,3 +297,160 @@ $$i=\frac 1 L\int^t_{t_0}v(t)dt + i(t_0)$$
 Much like capacitors, inductors have energy now based on current.
 
 $$U=\frac 1 2 Li^2$$
+
+## First-order circuits
+
+!!! definition
+    - An **RC** circuit contains a resistor and a capacitor.
+    - An **RL** circuit contains a resistor and an inductor.
+    - **First-order circuits** contain derivatives.
+    - A **source-free circuit** assumes that energy already exists in the capacitor/inductor and no external energy enters the system.
+    - The **circuit response** is the behaviour of the circuit after excitation.
+    - The **natural response** is the behaviour of the circuit without external excitation.
+
+The **time constant** $\tau$ is the time requirement for the circuit to decay to $\frac 1 e$ of its initial value. For RC circuits:
+
+$$\tau=RC$$
+
+$$v(t)=v_0e^{-t/\tau}$$
+
+RL circuits have very similar formulae:
+
+$$\tau=\frac L R$$
+
+$$i(t)=i_0e^{-t/\tau}$$
+
+### Singularity functions
+
+The **unit step function** is a stair that is undefined at zero.
+
+$$
+u(t)=\begin{cases}
+0 & \text{if }t<0 \\
+1 & \text{if }t>0
+$$
+
+The **unit impulse/delta function** is the derivative of the unit step function.
+
+$$
+\delta(t)=\frac{d}{dt}u(t)=\begin{cases}
+0 & \text{if }t<0 \\
+\text{undefined} & \text{if }t=0 \\
+0 & \text{if }t>0
+$$
+
+The sudden spike at $t=0$ means that $\int^{0+}_{0-}\delta(t)dt=1$.
+
+This function is related to signal strength. For the function $a\delta(t+y)$, changing $y$ shifts the phase while shifting $a$ shifts amplitude.
+
+To obtain $f(t)$ at the impulse:
+
+$$\int^b_a\delta(t-t_0)dt=f(t_0$$
+
+
+The **unit ramp function** is the integral of the unit step function.
+
+
+\begin{align*}
+r(t)&=\int^1_{-\infty}u(\lambda)d\lambda=tu(t) \\
+&=\begin{cases}
+0 & \text{if }t\leq 0 \\
+t & \text{if }t\geq 0
+\end{cases}
+\end{align*}
+
+## Circuit responses
+
+The total response to a circuit $V$ can be expressed as various combinations of:
+
+- the natural response, $v_n=v_0e^{-t/\tau}$
+- the forced response (induced) $v_f=v_s(1-e^{-t\tau})$
+- the temporary response, $(v_0-v_s)e^{-1/t}$
+- the permanent/steady-state response, $v_s$
+
+$$
+v(t)=\begin{cases}
+v_0 & \text{if }t<0 \\
+v_s+(v_0-v_s)e^{-t/\tau} &\text{if }t>0
+\end{cases}
+$$
+
+In general, for current and voltage ($x$), where $x_\infty$ is the final value and $x_0$ is the initial value:
+
+$$\boxed{x(t)=x(\infty)+[x(0)-x(\infty)]^{-t/\tau}}$$
+
+A delayed response by $t_0$ shifts $t$ to $t-t_0$ and $x(0)$ to $x(t_0)$.
+
+## Alternating current
+
+Where $V_m$ is the amplitude of the voltage and $\omega$ is its angular frequency:
+
+$$v(t)=V_m\sin(\omega t)$$
+
+For a sinusoid's period $T$, a circuit is period if and only if, for all $n\in\mathbb Z$:
+
+$$v(t)=v(t+nT)$$
+
+### Phasors
+
+The **phasor** is the complex number vector version of the sinusoid in the time domain.
+
+$$v(t)=\text{Re}(\bold Ve^{j\omega t})$$
+
+Please see [MATH 115: Linear Algebra#Geometry](/1a/math115/#geometry) for more information.
+
+$$\bold V=V_m^{j\phi}$$
+
+To transform time domains to frequency domains:
+
+| Sinusoidal | Phasor |
+| --- | --- |
+| $V_m\cos(\omega t+\phi)$ | $V_m\angle\phi$ |
+| $V_m\sin(\omega t+\phi)$ | $V_m\angle\phi-90^\circ$ |
+
+The **derivative** of a phasor is itself multiplied by $j\omega$.
+
+$$\frac{d}{dt}\bold V=j\omega\bold V$$
+
+Adding sinusoids of the **same frequency** ($\omega$) is equivalent to adding their phasors.
+
+If $\bold V$ and $\bold I$ are phasors:
+
+- Inductors: $\bold V=j\omega L\bold I$ ($\bold I$ lags $\bold V$ by 90°)
+- Capacitors: $\bold V=\frac{I}{j\omega C}$ ($\bold V$ lags $\bold I$ by 90°)
+
+The **scalar** quantity of **impedance** represents the opposition to electron flow, measured in ohms.
+
+$$Z=\frac{1}{j\omega C}=j\omega L$$
+
+It is effectively generalised resistance. Where $X$ is a positive value representing **reactance** such that $+jX$ implies inductance while $-jX$ implies capacitance:
+
+$$Z=\frac{\bold V}{\bold I}=R\pm jX$$
+
+**Admittance** is the inverse of impedance with units Siemens/mhos with factors **conductance** and **susceptance**:
+
+$$Y=G+jB$$
+
+Arranging equations yields
+
+$$
+G=\frac{R}{R^2+X^2} \\
+B=-\frac{X}{R^2+X^2}
+$$
+
+### Steady state analysis
+
+**Kirchoff's laws** only hold for phasor forms.
+
+1. Convert to phasor forms
+2. Solve phasor forms
+3. Convert back to time domain
+
+Superposition must be summed at the end only, although individual components can first be solved.
+
+1. Convert to phasor forms
+2. Solve each individual current/voltage that make KCL/KVL
+3. Convert to time domain
+4. Apply KCL/KVL
+
+