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# Lecture 3 Circuit Theorems 电路定理

## motivation

which should be easy to solve according to the lecture 2.

## Linear property 线性定理

What if

• $I_0 = 12A$?
• $V_0=90V$?

Will  $I=aI_0+bV_0$？

Excitation(source)->Response (voltage/current)

$$x \to \fbox{system}\to f(x)$$

A linear circuit consists of only linear elements (resistors， capacitors and inductors), linear dependent sources, and independent sources.含有且仅有线性元件与电源

### Homogeneity property 齐次性

$$\alpha x \to \fbox{system}\to \alpha f(x)$$

### Example

Find f,g,where $I=f(I_0,V_0),P = g(I_0,V_0)$

solution:

Mesh analysis
$$-(I-I_0)R_1-IR_2-V_0=0 \\ -I(R_1+R_2) = V_0-I_0R_1 \\ I=\frac{R_1}{R_1+R_2}I_0+\frac{-1}{R_1+R_2}V_0$$

$$P = UI = V_{R_1}I = (I_0-I)R_1I\\ P = (\frac{R_1}{R_1+R_2}I_0+\frac{1}{R_1+R_2}V_0) R_1 (\frac{R_1}{R_1+R_2}I_0+\frac{-1}{R_1+R_2}V_0) \\= a'I_0^2+b'V_0I_+c'V_0^2$$

## Super position Theorem 叠加定理

What if

• $I_0 = 0A, I'=?$
• $V_0=0V,I''=?$

Will $I=I'+I''$？

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./用独立源计算（就是将其他电源完全关闭状态下测算

$$x_1+x_2\to\fbox{system}\to f(x)+f(x2)$$

### Proof

$$I' = \frac{R_1}{R_1+R_2}I_0 \\ I'' = \frac{-1}{R_1+R_2}V_0 \\ I = I'+I'' = \frac{R_1}{R_1+R_2}I_0+\frac{-1}{R_1+R_2}V_0 = aI_0+bV_0$$

### Note:

1. Linear Circuit
2. one independent source at a time
3. dependent source are left alone inside the circuit

1. simplify the analyzing process
2. Evaluate the sensitivity of response to a specific source

Super position Theorem is a inference of linear property

### Example

Find  $i$ using Super position Theorem

• $i$'s contribution is from the independent voltage source:

$$10v-2Ωi'-1Ωi'-2i'=0 \\ i' = 2A$$

• $i$'s contribution is from the independent current source:

$$-2Ωi''-V=0\\ V-(i''+5A)1Ω-2i''=0 i' = -1A$$

$$i= i‘+i'' =1A$$

## Thevenin’s theorem/戴维南定理 • Source transformation电源变换定理 • Norton’s theorem/诺顿定理

What if $R_2 = 1Ω,I=?$

What if $R_2 = 5Ω,I=?$

### Substitution Theorem

If we know the voltage across/ the current trough any branch of a network, the branch can be replaced by a voltage or current source that will make the same voltage and current through that branch.

#### proof

core: KCL,KVL 等物理关系在原电路中保持不变

### Thevenin’s theorem/戴维南定理

#### motivation

complex linear system (CLS) which contains resistors independent sources dependent sources only.

Equivalent circuit for CLS Handle variable loads.

#### proof:

$$U' = U_{oc} = U_{Th} \\ U'' = -R_{eq}i \\ U= U'+U'' =U_{Th}+R_{eq}i \\ U = U_{Th}-R_{Th}i$$

#### proof2:

In CLS assuming  M independent voltage sources

and N independent currunt sources
$$U= Ai+\sum^{M}{j=1}B_jV{S_j}+\sum^N_{k=1}C_kI_{S_k}\\ U= Ai+CONST \\$$
Find $CONST$ if $i=0$, $U=U_{oc} =U_{Th}$

Find $A$ if $V_{S_j},I_{S_k}=0,A=U/i=-R_{Th}$

$$U = U_{Th}-R_{Th}i$$

#### Note

1. $V_{Th}= U_{oc}$
2. to get $R_{Th}$
3. equivalent resistance with all indeoendent sources turned off (does not work for circuits with dependent sources)只适用于无非独立源
4. calculate shout circuit current (isc)-> $R_{Th} = \frac{V_{oc}}{i_{sc}}$
5. Deactivate all independent sources, add external source $V_{ex}$, slove current $i_{ex}$ $R_{Th} = \frac{V_{ex}}{i_{ex}}$
6. 任意取(u,i)算系数

### source transformation and norton Theorem

$$I_{sc} = I_{N}\\ R_N = R_{Th} \\ I_N = \frac{V_{Th}}{R_{Th}} \\$$

## Maximum Power Transfer Theorem

$$P_L = I^2R_L=(\frac{V}{R_s+R_L})^2R_L=\frac{V^2}{R_s^2/R_L+2R_s+R_L}\\ when \space R_L = \pm R_S, P_L \space gets \space its \space maximum \space power$$