Input resistance of a single-port circuit without independent source and its equivalent circuit

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Input resistance of a single-port circuit without independent source and its equivalent circuit [Copy link]

For a single-port circuit without independent source, which has two lead-out terminals, has no independent power supply inside and the current on the two terminals is the same, it is called a single-port circuit without independent source, also called a one-port circuit without independent source, as shown in Figure 2-4-1 (a). We introduce the concept of input resistance (also called equivalent resistance), represented by R0. It is defined as the ratio of port voltage u to port current i, that is,　

R0=u/i.
The reciprocal of input resistance R0 is called the input conductance of the single-port circuit without independent source, represented by G0. That is,

G0=1/R0=i/u .
According to the above two formulas, the corresponding equivalent circuit can be made, as shown in Figure 2-4-1 (b). This circuit is called the equivalent circuit of the single-port circuit without independent source. But it should be noted that the "equivalent" here still refers to the equivalent circuit other than the single-port circuit without independent source, that is, when the original single-port circuit without independent source is replaced by the equivalent resistance R0, the relationship curve between the voltage u and the current i on the port is inconvenient.

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Figure 2-4-1 Single-port circuit without independent source and its equivalent circuit

The following two circuits are used to study the solution method of input resistance R0
. 1. Single-port circuit without any power supply
A single-port circuit without any power supply (independent source and controlled source) is usually represented by the letter P in a square box , as shown in Figure 2-4-1 (c) . Its input resistance R0 can be obtained by star-to-delta equivalent transformation and simplification of resistance series and parallel connection.
Example 2-4-1 Figure 2-4-2 (a) circuit. Find the equivalent resistance Rab and the current in each branch?

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; Figure 2-4-2 Circuit of Example 2-4-1

Solution: Transform the three 2 Ω star-connected resistors into three 6 Ω triangle-connected resistors, as shown in Figure 2-4-2 (b) . Then use the resistor series-parallel simplification principle to simplify the resistors in Figure (b) into the resistors shown in Figure (c) . According to Figure (c),
we get Rab = 4×3/4+3=12/7 Ω i 1=30/7 Ω uab =4 i 1=120/7 V Going back to the circuit in Figure (b) , we get i 2= uab /6=20/7 A i 3= i 4= i5=i6 =(10- i1-i2 )/2=10/7 A Going back to the circuit in Figure (a), we get i 2=10- i1-i5 = 30/7 A i3=i1+i6 -10=-30/7 A i4=i6-i5 =0

2. Single-port circuit with controlled source
The single-port resistance with controlled source can be obtained by equivalent transformation to obtain its equivalent resistance. When performing equivalent transformation, the controlled source is treated in the same way as the independent source, but the branch where the control variable is located must be kept unchanged. Then, based on the obtained simplified circuit, the method of applying voltage source u at the network port to obtain port current i , or applying current source i to obtain port voltage u , is applied to obtain its equivalent resistance according to the formula R0 = u/i . Example 2-4-2 Calculate the input resistance (equivalent resistance) R0 of the single-port network shown in
Figure 2=4=3 (a) ?

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　Figure 2-4-3 Single-port circuit with controlled source

　

Solution: First , the parallel combination of the controlled current source 2i1 and the 2Ω resistor is equivalently transformed into the series combination of the controlled voltage source 4i1 and the resistor 2Ω. The transformation principle is exactly the same as the mutual equivalent transformation principle of the independent voltage source and the independent current source, as shown in Figure 2-4-3 (b) . Then, the series combination of the controlled voltage source 4i1 and the resistor (2Ω+2Ω) is equivalently transformed into the parallel combination of the controlled current source i1 and the resistor 4Ω, as shown in Figure 2-4-3 (c) . It can be seen that the simplified circuit diagram (c) retains the control variable i1 branch unchanged. Therefore, according to the circuit in Figure (c) , a voltage u is applied from port ab to obtain the current i, that is:

; i=i1+(u- 3 i)/ 4 +i1
　

Again, i 1 =(u- 3 i)/1
　

The input resistance (equivalent resistance) is obtained by connecting the two solutions : R0=u/i=3 1/9 Ω.
Its equivalent circuit is shown in Figure 2-4-3 (d) . It can be seen that the single-port network containing a linear controlled source is equivalent to a linear resistance element, which indicates the resistance of the controlled source.

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