ece140: add weird powers

docs/1b/ece140.md

@@ -454,3 +454,58 @@ Superposition must be summed at the end only, although individual components can
 4. Apply KCL/KVL
 
 When applying source transformations, different equivalent circuits for **each frequency** must be calculated individually — reducing it to one equivalent circuit is not possible.
+
+### Power
+
+The average power is the integral average of instantaneous power:
+
+$$P=\frac 1 T \int^T_0 p(t)dt$$
+
+!!! tip
+    The average of a sinusoid over its period is zero.
+
+Alternatively, power can be calculated with magnitudes:
+
+$$P=\frac 1 2\text{Re}[VI^*]=\frac 1 2 V_mI_m\cos(\theta_v-\theta_i)$$
+
+The same rules for maximum power transfer apply with resistance, but with $Z_L$ as the **complex conjugate** of $Z_{Th}$. The maximum power has a shortcut formula:
+
+$$P_{max}=\frac{|V_{Th}^2}{8R_{Th}}$$
+
+The **effective value** of a sinusoid is its DC equivalent. It is the root mean square.
+
+$$X_{rms}=\sqrt{\frac 1 T\int^T_0x^2dt}$$
+
+The **apparent power** $S$ is the seemingly true power.
+
+$$S=V_{rms}I_{rms}$$
+
+The **power factor (pf)** is the required factor to take the apparent power into real power.
+
+$$pf=\frac P S = \cos(\theta_v-\theta_i)$$
+
+The **power factor angle** $\theta_v-\theta_i$ is the angle of local impedance between voltage and current.
+
+$$Z=\frac{V_{rms}}{I_{rms}}\phase{\theta_v-\theta_i}=\frac{V_m}{I_m}\phase{\theta_v-\theta_i}$$
+
+- A **leading** power factor has current lead voltage (capacitive)
+- A **lagging** power factor has voltage lead current (inductive)
+- A **unity** power factor has no phase shift
+
+Complex power $\bold S$ stores more phase information where $\bold{V_{rms}}=V_{rms}\phase{\theta_v}$.
+
+$$\bold S=\frac 1 2\bold{VI}^*=\bold{V_{rms}I^*_{rms}}$$
+
+These have units volt-amperes (VA).
+
+$$\bold S=V_{rms}I_{rms}\phase{\theta_v-\theta_i}=V_{rms}I_{rms}\cos(\theta_v-\theta_i)+jV_{rms}I_{rms}\sin(\theta_v-\theta_i)$$
+
+The two components of complex power are actual power $P=I^2_{rms}R$ and reactive power $Q=I^2_{rms}X$, the latter with units VAR (volt-ampere reactive).
+
+$$\bold S=P+jQ$$
+
+Complex power still follows most DC laws:
+
+$$\bold S=I^2_{rms}\bold Z=\frac{V^2_{rms}}{\bold Z^*}=\bold{V_{rms}I^*_{rms}}$$
+
+All powers (instantaneous, real, reactive, and complex) are conserved, except for apparent power.