diff --git a/docs/1b/ece140.md b/docs/1b/ece140.md index 2873021..958b982 100644 --- a/docs/1b/ece140.md +++ b/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.