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1、CHAPTER 3 CIRCUIT THEOREMS,3-1 叠加定理 SUPERPOSITION,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. 证明： n个节点，b条支路，l
2、=b-n+1个独立回路方程,：中第h列的余因子,US为第h个回路独立电压源的代数和,（独立电压源总计为q个）,Superposition is based on linearity Keep all dependent sources when turn off all independent sources except one source The reference direction Not applicable to the effect on power due to each source,NOTICE：,例：R1=1，R2=2，R3=3，R4=4，IS5=5A，US6=6V，用
3、叠加定理计算电源与R1上的功率。图(a),解： 1IS5单独作用时：US6置零，图(b)：,I6=I2 I1= -1A， U5=R1I1+R2I2=10V,2US6单独作用时，图(c)：,3IS5与US6共同作用，图(a)：,真实功率,线性定理 LINEARITY PROPERTY,A linear circuit is one whose output is linearly related to its input.,Gkj：conductance(S) ； kj： current gain, dimensionless,Rkj： resistance() ； kj： voltage ga
4、in, dimensionless,例：一线性网络，US1=0V，IS2=0A时，有U3=3V；US1=1V，IS2= -1A时，U3=2V；US1= -4V，IS2=1A时，U3=1V。求当US1=1V，IS2=2A时，U3=?,解： U3=aUS1+bIS2+c aUS1：US1单独激励产生的电压分量； bIS2：IS2单独作用产生的电压分量； c：由网络A内的独立源一起激励产生的电压分量；,3-2 戴维南定理 THEVENINS THEOREM,Thevenins theorem states that a linear two-terminal circuit can be repla
5、ced by an equivalent circuit consisting of a voltage source E0 in series with a resistor R0.,E0 is the open-circuit voltage at the terminals. R0 is the input or equivalent resistance at the terminals when the independent sources are turned off.,A original circuit and its Thevenin equivalent circuit,
6、NOTICE：,Linear circuit If the network has dependent sources, we turn off all independent source in finding the R0.,例：US1=1V，R2=2，R3=3，R4=4，R5=5，US5=5V，IS6=6A，R1可变。R1=?时I1= -1A，图(a)。,解： 开路电压E0，图(b)：,I5网孔：(R2+ R3+ R5)I5 -R3IS6=US5 I5=2.3A KVL：E0=US5-R5I5 - R4Is6= -30.5V,等效内阻R0，图(c)： R0= R4+R5/(R2+ R3)
7、=6.5,等效网络，图(d)：,例：R1=1，R3=3，R4=4，R5=5，US1=1V，US3=3V，US4=4V，US5=5V，IS2=2A，求I3 。图(a),解： 图(b)：,图(d)：,图(c)：,例：R1=1，R2=2，R3=3，r=1，US1=1V，求I3。图(a),解： # 开路电压，图(b)： I3=0， rI3=0；,# 等效内阻，图(c)：,# 等效内阻，图(d)：,或者，图(e)：令Is=1A，（I3= -1A） rI3= -1V,# 图(f)：,四、最大功率传输定理 MAXIMUM POWER TRANSFER,Maximum power is transferred
8、 to the load when the load resistance equals the Thevenin resistance as seen from the load (RL=RS).,The source and load are said to be matched when RL=RS. （匹配）,证明：,传输效率：负载吸收的功率PL与电源产生的功率PS之比：,电路匹配：,例：IS=4A，R1=1，R2=3，R可变，r=2。问R=?时吸收最大功率？图(a),解： # 开路电压，图(b)：,# 等效内阻，图(c)：,rI2=R2I2+R1I1，且I1+ IS=I2 I2=2A
9、 E0=R2I2=6V,I2=0，rI2= R1I1=0，I1=0，IS=I0 I0= 4A；,或者，图(d)：,或者，图(e)：,rI2=R2I2+R1I1，I1+IS=I2 I20.5A,# 由最大功率传输定理，当R=R0=1.5时，吸收最大功率：图(f),3-3 诺顿定理 NORTONS THEOREM,Nortons theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source IS in parallel
10、 with a resistor R0. IS is the short-circuit current through the terminals. R0 is the input or equivalent resistance at the terminals when the independent sources are turned off.,The Thevenin and Norton equivalent circuits are related by a source transformation.,To determine the Thevenin or Norton e
11、quivalent circuit requires : # The open-circuit voltage E0 across terminals a and b. # The short-circuit current IS at terminals a and b. # The equivalent or input resistance R0 at terminals a and b when all independent sources are turned off.,Check Your Understanding Find the Norton equivalent circ
12、uit of the circuit in Fig.(a),Solution：,Fig (b)：,Fig (c)：,3-4 替代定理,R1=1，R2=2，Us3V 图(a)：I1=I2=US/(R1+R2)=1A，U1=R1I1=1V，U2=R2I2=2V 图(b)：U22V， I21A 图(c)：U22V， I21A,在任意线性和非线性、定常和时变的网络中，如果某k支路的电压为Uk，电流为Ik，只要该支路和网络的其它支路之间无耦合，即k支路不是非独立电源支路，总可以用下列任一元件去替代该支路： 电压为UskUk，极性与Uk相同的独立电压源； 电流为IskIk，方向与Ik一样的独立电流源； 电
13、阻为RkUk/Ik的线性电阻器（假设Uk和Ik有关联参考方向）。 替代后整个网络中的电流和电压都保持不变。,替代定理：,证明： 设网络有b条支路 支路电流为I1，I2，.IK.Ib， 满足KCL 支路电压为U1，U2，.UK，.Ub，满足KVL 代替：ISK=IK ，方向与Ik相同 必定有： 各支路的电压和电流保持不变,例：N为含源线性电路，当IR=2A，I=1/3A。R增加10时，IR=1.5A，I=1/2A。当R减少10时，I=? 图(a),解： 用电流源替代IR，根据线性定理有： I=A1IR+A2， 即：1/3=2A1+A2，1/2=1.5A1+A2， A1= -1/3A，A2=1A。
14、 戴维南等效： 图(b)(c),当R= -10时，, I=A1IR+A2=0A,3-5 特勒根定理 TELLEGENS THEOREM,Let the graph have b branches, let us use associated reference directions. Let I be any set of branch currents satisfying KCL and let U be any set of branch voltages satisfying KVL, then：,U1=U-U ，U2=U-U ，U3=U-U ，U4=U ，U5=U ，U6=U P=U
15、1I1+U2I2+U3I3+U4I4+U5I5+U6I6 =(U-U)I1+(U-U)I2+(U-U)I3+ =U(I1+I2+I4)+U(-I2+I3+I5)+U(-I1-I3+I6) = 0,特勒根第二定理（特勒根似功率定理）：如电路N和N的拓扑图形完全相同，各有b条支路，n个节点，对应支路采用相同编号，支路电压和电流的参考方向取为一致，则有：,例：No为无源线性电阻网络，IS1=4A，U2=10V，IS2=2A，U1=?,解：设No内各支路电压、电流采用关联参考方向,图(a)中：,图(b)中：,3-6 互易定理 RECIPROCITY THEOREM,In any passive lin
16、ear bilateral network, if the single voltage source UX in branch x produces the current response IY in branch y, then the removal of the voltage source from branch x and its insertion in branch y will produce the current response IY in branch x.,In any passive linear bilateral network, if the single current source IX between nodes x and x
