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EGR 271 – Circuit Theory I Reading Assignment: Chapter 18 in Electric Circuits, 9th Ed. by Nilsson One-port networks Earlier in the course we saw that a network N consisting or resistors and dependent sources can be represented by an equivalent resistance, Req . This type of network is a one-port network since it has one pair of terminals. I I + Network N (dependent sources, resistors) V _ + R eq V _ 1 EGR 271 – Circuit Theory I Two-port networks Suppose that a network N has two ports as shown below. How could it be represented or modeled? A common way to represent such a network is to use one of 6 possible two-port networks. These networks are circuits that are based on one of 6 possible sets of two-port equations. These equations are simply different combinations of two equations that relate the variables V1 , V2 , I1 , and I2 to one another. The coefficients in these equations are referred to as two-port parameters. I I 1 2 + Input Port V 1 _ + Network N (dependent sources, resistors) V _2 Output Port (N is a 2-port network) 2 EGR 271 – Circuit Theory I Note that I1 , I2 , V1 , and V2 are labeled as shown by convention. Often there is a common negative terminal between the input and the output so the figure above could be redrawn as: I I 1 2 + Input Port + V 1 Network N (dependent sources, resistors) (N is a 2-port network) _ V 2 Output Port _ 3 EGR 271 – Circuit Theory I Two-port equations Two-port equations are sets of two equations relating the four variables labeled on the diagram of the two-port network above. There are 6 possible ways to form sets of two equations which express two of the variables in terms of the other two variables. The 6 possible sets of equations are shown below. z-parameters g-parameters z-parameter equations: g-parameter equations: V1 z11 I1 z12 I 2 I1 g11 V1 g12 I2 V2 z 21 I1 z 22 I 2 V2 g 21 V1 g 22 I2 y-parameters Chain (or T or ABCD) parameters y-parameter equations: Chain parameter equations: I1 y11 V1 y12 V2 V1 A V2 B (I 2 ) I 2 y 21 V1 y 22 V2 I1 C V2 D (I 2 ) h-parameters Inverse chain (or T-1 or A'B'C'D' parameters h-parameter equations: Inverse-chain parameter equations: V1 h 11 I1 h 12 V2 V2 A'V1 B'(I1 ) I 2 h 21 I1 h 22 V2 I2 C'V1 D'(I1 ) Applications: z- and y-parameters: circuit modeling y-parameters: modeling transistor capacitance at high frequencies h- parameters: electronics (modeling transistors) Chain- and inverse-chain parameters: system representation (relates input and output) 4 EGR 271 – Circuit Theory I Calculation of z-parameters Show that V z11 1 I1 I2 0 z - parameterequations: V1 z11 I1 z12 I 2 V2 z 21 I1 z 22 I 2 Similarly show expressions for the other z-parameters. 5 EGR 271 – Circuit Theory I Example: Determine the z-parameters using the z-parameter definitions for the network shown I I below. 1 2 + V 10 30 20 1 _ z - parameterequations: V1 z11 I1 z12 I 2 + V V2 z 21 I1 z 22 I 2 2 _ Circuit 2 6 EGR 271 – Circuit Theory I Calculation of y-parameters Show that I y11 1 V1 V2 0 y - paramet erequations: I1 y11 V1 y12 V2 I 2 y 21 V1 y 22 V2 Similarly show expressions for the other y-parameters. 7 EGR 271 – Circuit Theory I Example: Determine the y-parameters using the y-parameter definitions for the network shown below. I 1 + 10 1 0.3 I X _ I I1 y11 V1 y12 V2 2 I 2 y 21 V1 y 22 V2 + I V y - paramet erequations: 6 X V 2 _ Circuit 3 8 EGR 271 – Circuit Theory I Reading Assignment: Chapter 18 in Electric Circuits, 8th Ed. by Nilsson Example: Determine the h-parameters for the network shown below. I I 1 + V _ 200 1 100 h - paramet erequations: 2 V1 h11 I1 h12 V2 + 300 V I 2 h 21 I1 h 22 V2 2 _ 9 EGR 271 – Circuit Theory I Calculation of 2-port parameters using network equations Consider the z-parameter equations shown below. V1 z11 I1 z12 I 2 V2 z 21 I1 z 22 I 2 Note that V1 and V2 are functions of I1 and I2 . If general sources, I1 and I2 are added to a network and the voltages V1 and V2 are calculated, the result will be expressions for V1 and V2 that are functions of I1 and I2 . So the z-parameter equations are naturally generated. Similarly, y-parameters can be found by adding two general voltage sources V1 and V2 and solving for the currents I1 and I2 . 10 EGR 271 – Circuit Theory I Example: Determine the y-parameters using network equations for the network shown below. 1 I + V _ I 1 1 1 1 1 2 + V 2 _ 11 EGR 271 – Circuit Theory I Modeling Two-Port Networks We have seen how two-port parameters can be determined for a given network. Additionally, two-port parameters might be specified for a certain device by the manufacturer (such as h-parameter values for a transistor). How are these parameters used? They are used to form a circuit model for the device or circuit. A circuit model is developed using the two-port parameter equations. z-parameter model The z-parameter equations are: V1 z11 I1 z12 I 2 V2 z 21 I1 z 22 I 2 What sort of circuit model could be drawn that would satisfy these equations? I1 I2 + V1 _ + ? V2 _ 12 EGR 271 – Circuit Theory I Development of the z-parameter model: One possible circuit model could be developed by treating each of the two-port parameter equations as KVL equations (illustrate). This results in the following circuit. z - parameterequations: V1 z11 I1 z12 I 2 V2 z 21 I1 z 22 I 2 13 EGR 271 – Circuit Theory I Development of the y-parameter model: One possible circuit model could be developed by treating each of the two-port parameter equations as KCL equations (illustrate). This results in the following circuit. y - paramet erequations: I1 y11 V1 y12 V2 I 2 y 21 V1 y 22 V2 14 EGR 271 – Circuit Theory I Development of the h-parameter model: One possible circuit model could be developed by treating one of the two-port parameter equations as a KVL equation and the other as a KCL equation (illustrate). This results in the following circuit. h - paramet erequations: V1 h11 I1 h12 V2 I 2 h 21 I1 h 22 V2 15 EGR 271 – Circuit Theory I Summary: Note: This page will be provided on the final exam. I2 I1 + z11 V1 + z - parameterequations: V2 V1 z11 I1 z12 I 2 _ V2 z 21 I1 z 22 I 2 + y - paramet erequations: V2 I1 y11 V1 y12 V2 _ I 2 y 21 V1 y 22 V2 + h - paramet erequations: V2 V1 h11 I1 h12 V2 _ I 2 h 21 I1 h 22 V2 z22 z12 I 2 + _ + _ z21 I 1 _ z-parameter model I2 I1 + V1 y 11 y12 V2 y21 V1 y 22 _ y-parameter model I2 I1 + V1 h11 h12V 2 + _ h21I 1 _ h-parameter model h 22 16 EGR 271 – Circuit Theory I Example: Consider Network N shown below. I 2 10 V + _ I 1 + 2 + + Network N V _ 1 V _ 2 8 V L _ Two tests were run on the circuit above in order to determine the z-parameters. The results of the tests are shown below. Test #1 (output open-circuited) Test #2 (input open-circuited) V1 = 2V V2 = 4V V1 = 2V V2 = 4V I1 = 1A I2 = 0 I1 = 0 I2 = 2A A) Use the test results to determine the z-parameters. B) Redraw the circuit shown above using the z-parameter model and the z-parameter values determined in part A. C) Determine VL. 17 EGR 271 – Circuit Theory I Example: Consider Network N shown below. I 2k 50 V + _ I 1 + 2 + + Network N V _ 1 V 2 _ 10k V L _ Determine VL if the h-parameters for Network N are: h11 = 2.5k , h12 0.001, h21 = 125, h22 = 25 mho 18