电子电工技术英文课件 (43).pdf

4.5 Circuit Theorems In the development of circuit technology,some circuit laws,which can be used to simplify circuit analysis,have been summarized,.These circuit laws are called circuit theorems,like superposition theorem,Thvenin theorem,Norton theorem,substitution theorem,reciprocity theorem,maximum power transfer theorem,and so on.In this section,superposition theorem,Thvenin theorem,and maximum power transfer theorem will be introduced.4.5.1 Superposition Theorem In order to introduce superposition theorem,we need to introduce linearity first.Linearity is a combination of both the additivity and the homogeneity.Here we only introduce the additivity.The additivity is that the response to a sum of inputs is equal to the sum of the responses to each input applied individually.Lets take a resistor as an example.For a resistor,viR=(1)From(1),11viR=(2)22viR=(3)123vviR+=(4)From(2)-(4),1212312vvvviiiRRR+=+=+(5)The expression(5)shows a resistor has the property of additivity.A resistor is a linear element.A linear circuit is one that only contains linear elements.Most of circuits are linear circuits.A linear circuit obeys superposition theorem,i.e.,the voltage(or current）response of any branch in the circuit is equal to the sum of individual responses associated with the individual sources,as if each response had been acting alone.Lets take a simple example of superposition theorem,as shown in Fig.1.11V2Vi Figure 1:Circuit for demonstrating superposition theorem.When each source operates separately,other source should be set zero.For voltage sources,they should be replaced with short circuits,as shown in Fig.2.11V12V(1)i(2)i Figure 2:Circuits with individual voltage source applying corresponding to the circuit in Fig.1 From Fig.2,it is obvious that (1)2),2A1Aii=(6)According to superposition theorem,(2)(1)3Aiii=+=(7)From Fig.1,we can also conclude 3Ai=,this result is in accordance with that result obtained by superposition theorem.4.5.2 Thvenin Theorem In 1883,a French engineer Leon Thvenin discovered a theorem:any linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTh in series with a resistor RTh,where VTh is the open-circuit voltage at the terminals and RTh is the equivalent resistance between the same terminals when all sources in the circuit are turned off.It is difficult to understand Thvenin theorem,so we graphically show Thvenin theorem in Fig.3.ThVThVOriginalLinearCircuitEquivalent CircuitThRCircuit Equivalence Figure 3:Circuit for demonstrating Thvenin theorem.From Fig.3,Thvenin theorem actually demonstrates a kind of circuit equivalence.Lets take a circuit as shown in Fig.4 to show how to get the Thvenin equivalent circuit.ThV3V21 Figure 4:A circuit to be equivalent to its Thvenin equivalent circuit.From Fig.4,the open-circuit voltage is Th232V12V=+(8)When the voltage source in Fig.4 is turned off(short circuit),the circuit is shown in Fig.5.21 Figure 5:Circuit when the voltage source is turned off.From Fig.5,the equivalent resistance is Th1 22123R=+(9)The Thvenin equivalent circuit of the circuit in Fig.4 is shown in Fig.6.ThV2V23 Figure 6:Thvenin equivalent circuit of the circuit in Fig.4.In comparison with the circuit in Fig.4,the Thvenin equivalent circuit in Fig.6 is much simpler.Any complex linear two-terminal circuit including DC circuits and AC circuits can be equivalent to a simple circuit consisting of a voltage source in series with a resistor using Thvenin theorem.Thvenin theorem is so amazing that it is very useful in circuit analysis.For example,Thvenin theorem can apply in solving maximum power transfer problem,which will be introduced in the following.4.5.3 Maximum Power Transfer Theorem Fig.7 shows a circuit with variable load resistance R.Sometimes we are concerned about how much is the maximum power that the load resistance R can acquire.OriginalLinearCircuitRI Figure 7:Circuit with variable load resistance.To solve the maximum power transfer problem,we can apply Thvenin theorem to simplify the circuit first,as shown in Fig.8.RThVTh venin Equivalent CircuitThRI Figure 8:Simplifying the circuit in Fig.7 by applying Thvenin theorem.From Fig.8,the load current is ThThVIRR=+(10)The power of the load resistance is ()222ThTh2ThThRVVPI RRRRRRR=+(11)From(11),we can get the following inequality:()Th2Th2Th2Th22Th22ThThThThTh2224RVPRRRVRRRR RVVRR RR RR=+=+=+(12)From(12),the maximum power of the load resistance is 2ThmaxThThwhen4RVPRRR=(13)From(13),maximum power transfer theorem states that maximum power transfer take place when the variable load resistance is equal to the Thvenin resistance and the maximum power that the variable load resistance can acquire is 2ThmaxTh4RVPR=.Lets take an example for maximum power transfer problem.Find the maximum power the variable load resistance shown in Fig.9 can acquire.3V21R Figure 9:Example for maximum power transfer problem.We can apply Thvenin theorem to simplify the circuit in Fig.9,as shown in Fig.10.2V23R Figure 10:Example for maximum power transfer problem.According to maximum power transfer theorem,maximum power transfer take place when 23R=.The maximum power the variable load resistance can acquire is 22ThmaxTh21.5W2443RVPR=(14)Maximum power transfer theorem introduced above is only applicable for DC circuits.For AC circuits,sometimes we are concerned about how much is the maximum average power that the load impedance Z shown in Fig.11 can acquire.LinearACCircuitZI Figure 11:An AC circuit with variable load impedance.To solve the maximum average power transfer problem in the AC circuit,we can also apply Thvenin theorem to simplify the circuit first,as shown in Fig.12.ZThVTh venin Equivalent CircuitThZI Figure 12:Simplifying the ac circuit in Fig.11 by applying Thvenin theorem.From Fig.12,the load current is ()()ThThThThThThThThRjXRjXRRj XX=+VVVIZZ(15)The average power of the load resistance is ()()22Th22ThTh12RPRRRRXX=+VI(16)From(11),maximum average power transfer take place when *ThTh0and,that isThXXRR+=ZZ(17)The maximum power that the variable load impedance can acquire is 2ThmaxTh8RPR=V(18)Lets summarize maximum average power transfer theorem for AC circuits:maximum average power transfer take place when the variable load impedance is equal to the conjugate of Thvenin impedance and the maximum average power that the variable load impedance can acquire is 2ThmaxTh4RPR=V.