电路理论

线性电路

A linear circuit is one whose output is, linearly related (or directly proportional) to its input.

$$\begin{array}{rl}kiR& =kv\\ v& =(i_1+i_2)R=i_{1}R+i_{2}R=v_1+v_2\end{array}$$

叠加原理

The superposition principle states that the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone.

源变换

A source transformation is the process of replacing a voltage source v_s in series with a resistor R by a current source i_s in parallel with a resistor R, or vice versa.

$$v_s=i_{s}R\text{ or }{i}_s=\frac{v_s}{R}$$

戴维南定理

Thevenin's theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source V_th in series with a resistor R_th, where V_th is the open-circuit voltage at the terminals and R_th is the input or equivalent resistance at the terminals when the independent sources are turned off.

诺顿定理

Norton's theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source I_N in parallel with a resistor R_N, where I_N is the short-circuit current through the terminals and R_N is the input or equivalent resistance at the terminals when the independent sources are turned off.

$$\begin{array}{rl}{R}_N& =R_{Th}\\ I_N& =\frac{V_{Th}}{R_{Th}}\end{array}$$

最大功率传输

Maximum power is transferred to the load when the load resistance equals the Thevenin resistance as seen from the load (R_L = R_Th).

$$\begin{array}{rl}p& =i^2{R}_L={\left(\frac{V_{Th}}{R_{Th}+R_L}\right)}^2{R}_L\\ \frac{\mathrm{d}p}{\mathrm{d}{R}_L}& =V_{Th}^2\left[\frac{R_{Th}-R_L}{(R_{Th}+R_L{)}^3}\right]=0\\ \text{thus}{R}_L& =R_{Th}\\ p_{\text{max}}& =\frac{V_{Th}^2}{4R_{Th}}\end{array}$$