Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Two-Port Networks Introduction In general, a network may have n ports. The current entering one terminal leaves through the other terminal so that the net current entering the port equals zero. Thus, a two-port network has two terminal pairs acting as access points. The current entering one terminal of a pair leaves the other terminal in the pair. Three-terminal devices such as transistors can be configured into two-port networks. Knowing the parameters of a two-port network enables us to treat it as a “black box” when embedded within a larger network. 1 Introduction To characterize a two-port network requires that we relate the terminal quantities V1, V2, I1, and I2, out of which two are independent. The various terms that relate these voltages and currents are called parameters. Our goal is to study six sets of these parameters. As with op amps, we are only interested in the terminal behavior of the circuits. Reminder: Only two of the four variables (V1, V2, I1, and I2) are independent. 2 Impedance Parameters V1 = z11 I1 + z12 I 2 V2 = z21 I1 + z22 I 2 The z terms are called the impedance parameters, or simply z parameters, and have units of ohms. The values of the parameters can be evaluated by setting I1 = 0 (input port open-circuited) or I2 = 0 (output port open-circuited). Therefore, the z parameters are also called the open-circuit impedance parameters. 3 Impedance Parameters Note: The two equations associated with a set of parameters determine how the individual parameters are obtained. V1 = z11 I1 + z12 I 2 V2 = z21 I1 + z22 I 2 V1 z11 = I1 V2 z21 = I1 I 2 =0 V1 z12 = I2 I 2 =0 V2 z22 = I2 [ ] [ ] [ ] I1 =0 [ ] I1 =0 Sometimes z11 and z22 are called driving-point impedances, while z21 and z12 are called transfer impedances. A driving-point impedance is the input impedance of a two-terminal (one-port) device. Thus, z11 is the input driving-point impedance with the output port open-circuited, while z22 is the output driving-point impedance with the input port open-circuited. 4 Impedance Parameters Determination of the z parameters When z11 = z22, the two-port network is said to be symmetric. This implies that the network has mirror-like symmetry about some center line; that is, a line can be found that divides the network into two similar halves. When the two-port network is linear and has no dependent sources, the transfer impedances are equal (z12 = z21), and the two-port is said to be reciprocal. This means that if the points of excitation and response are interchanged, the transfer impedances remain the same. 5 Impedance Parameters Any two-port that is made entirely of resistors, capacitors, and inductors must be reciprocal. A reciprocal network can be replaced by the T-equivalent circuit. If the network is not reciprocal, a more general equivalent network is obtained. Reciprocal network (z12 = z21) Non-reciprocal network Note that both equivalent circuits satisfy the z-parameter equations. V1 = z11 I1 + z12 I 2 , V2 = z21 I1 + z22 I 2 6 Impedance Parameters Some two-port networks parameters might not exist. As an example, consider the ideal transformer. It is impossible to express the voltages in terms of the currents. An ideal transformer has no z parameters. However, it does have hybrid parameters (later). 7 Example z-1 — Method 1 Determine the z parameters for the following circuit. V1 z11 = I1 V2 z21 = I1 V1 z12 = I2 V2 z22 = I2 = 12 I1 =0 = 4 + 12 = 16 I1 =0 = 1 + 12 = 13 I 2 =0 = 12 I 2 =0 é13 12 ù ú [ z ] = êê ú ë12 16 û Non-symmetric, reciprocal 8 Example z-1 — Method 2 z12 = 12 = z21 z11 - z12 = 1 z11 = 1 + z12 = 13 z22 - z12 = 4 z22 = 4 + z12 = 16 é13 12 ù ú [ z ] = êê ú ë12 16 û 9 Example z-2 Find I1 and I2 in the following circuit. V1 = 40 I1 + j 20I 2 = 100V0 V2 = j 30 I1 + 50I 2 = -10I 2 I1 = j 2I 2 100V0 = j80 I 2 + j 20I 2 100V0 I2 = = 1A- 90 j100 I1 = j 2I 2 = 2A0 10 Example z-3 Calculate I1 and I2 in the following two-port. z11 I1 + z12 I 2 = V1 = 2V30- 2I1 z21 I1 + z22 I 2 = V2 = 0 2V30 = 8I1 - j 4I 2 0 = - j 4I1 + 8I 2 I 2 = j 0.5 I1 2V30 = 8I1 + 2I1 Answer: I1 = 20030 mA I 2 = 100120 mA 11 Example z-4 Determine the s domain expressions for the z parameters. V1 z11 = I1 V2 z21 = I1 V1 z12 = I2 V2 z22 = I2 = 10 I1 =0 = (0.2 s + 10) I1 =0 I2 2 10 s + 2 = + 10 = s s =0 = 10 I 2 =0 é10 s + 2 ê [ z ]= ê s ê êë 10 ù 10 ú ú ú 0.2 s + 10úû Non-symmetric, reciprocal 12 Example z-5 Find the z parameters for the following circuit. z11 = V1 I1 = 20 (5 + 15) = 10 I 2 =0 When I 2 = 0, then V2 = V2 z21 = I1 z22 V2 = I2 V1 I2 0.75V1 = = 7.5 V1 10 = 15 (5 + 20) = 9.375 I1 =0 When I1 = 0, then V1 = z12 = I 2 =0 15 V1 = 0.75V1 15 + 5 = I1 =0 20 V2 = 0.8V2 20 + 5 0.8V2 = 7.5 V2 9.375 é 10 7.5 ù ú [ z ] = êê ú ë 7.5 9.375 û Non-symmetric, reciprocal 13 Admittance Parameters The y terms are known as the admittance parameters (or, simply, y parameters) and have units of Siemens. I1 = y11V1 + y12V2 I 2 = y21V1 + y22V2 Since the y parameters are obtained by short-circuiting the input or output port, they are also called the short-circuit admittance parameters. I1 I1 y11 = y12 = [S] [S] V1 V =0 V2 V =0 2 I2 y21 = [S] V1 V =0 2 1 I2 y22 = V2 [S] V1 =0 14 Admittance Parameters For a two-port network that is linear and has no dependent sources, the transfer admittances are equal (y12 = y21). A reciprocal network (y12 = y21) can be modeled by a equivalent circuit. If the network is not reciprocal, a more general equivalent network is obtained. Reciprocal network (y12 = y21) Non-reciprocal network Note: Impedance and admittance parameters are collectively referred to as immittance parameters. 15 Example y-1 — Method 1 Obtain the y parameters for the network. I1 1 1 = + = 0.25S y11 = V1 V =0 20 5 2 I2 1 == -0.2S y21 = 5 V1 V =0 2 I1 y12 = V2 V1 =0 I2 y22 = V2 1 1 = + = 0.267S 15 5 V =0 =- 1 1 = -0.2S 5 é 0.25S -0.2S ù ú [ y ] = êê ú ë-0.2S 0.267Sû Non-symmetric, reciprocal 16 Example y-1 — Method 2 1 = -0.2S = y21 y12 = 5 1 y11 + y12 = 20 1 y22 + y12 = 15 1 - y12 = 0.25S y11 = 20 1 - y12 = 0.267S y22 = 15 é 0.25S -0.2S ù ú [ y ] = êê ú ë-0.2S 0.267Sû 17 Example y-2 Obtain the y parameters for the following T network. I1 I1 1 = = 0.2273S y11 = V1 V =0 (2 + 4 6) 2 I2 V1 V2 I2 46 1 =⋅ = -0.0909S y21 = 4 6 + 2 6 V1 V =0 2 I1 y12 = V2 I2 y22 = V2 =V1 =0 24 1 ⋅ = -0.0909S 2 4 + 6 2 1 = = 0.1364S (6 + 2 4) V1 =0 é 0.2273S -0.0909Sù ú [ y ] = êê ú ë-0.0909S 0.1364S û 18 Example y-3 Obtain the s domain expressions of the y parameters. I1 1 1 + s 2 LC y11 = = sC + = V1 V =0 sL sL 2 I2 1 y21 = =V1 V =0 sL 2 I1 y12 = V2 1 =sL V =0 I2 y22 = V2 1 1 + s 2 LC = sC + = sL sL V =0 1 1 é1 + s 2 LC ê ê sL [ y]= ê ê ê -1 êë sL 1 ùú sL úú 1 + s 2 LC úú sL úû Symmetric, reciprocal 19 Example y-4 Determine the y parameters for the following two-port. Short circuit Port 2 V1 - Vo Vo Vo - 0V = 2I1 + + I1 = 8 2 4 V1 - Vo Vo +3 0= 8 4 0 = V1 - Vo + 6Vo V1 = -5 Vo 20 Example y-4 cont’d -5Vo - Vo = -0.75S ⋅ Vo 8 I1 -0.75S ⋅ Vo y11 = = = 0.15S V1 -5 Vo I1 = I2 Vo - 0V + 2I1 + I 2 = 0 4 -I 2 = 0.25 S ⋅ Vo -1.5S ⋅ Vo = -1.25S ⋅ Vo I 2 1.25S ⋅ Vo y21 = = = -0.25S V1 -5 Vo 21 Example y-4 cont’d Short circuit Port 1 Vo Vo - V2 0V - Vo I1 = = 2I1 + + 8 2 4 Vo Vo Vo - V2 0=+ + 8 2 4 0 = -Vo + 4Vo + 2Vo - 2V2 V2 = 2.5 Vo 22 Example y-4 cont’d - Vo 8 I1 y12 = = = -0.05S V2 2.5 Vo Vo - V2 + 2I1 + I 2 = 0 4 Vo 2.5Vo 2Vo -I 2 = = -0.625S ⋅ Vo 4 4 8 0.625S ⋅ Vo I2 = = 0.25S y22 = V2 2.5 Vo é 0.15S -0.05Sù ú [ y ] = êê ú ë-0.25S 0.25S û 23 Hybrid Parameters The z and y parameters of a two-port network do not always exist. So there is a need for developing other sets of parameters. This third set of parameters is based on making V1 and I2 the dependent variables. V = h I +h V 1 11 1 12 2 I 2 = h21 I1 + h22V2 The h terms are known as the hybrid parameters (or, simply, h parameters) because they are a hybrid combination of ratios. The values are: V1 h11 = [ ] I1 V =0 V1 h12 = V2 I2 h21 = I1 I2 h22 = V2 2 V2 =0 I1 =0 [S] I1 =0 24 Hybrid Parameters To be specific, h11 = short-circuit input impedance h12 = open-circuit reverse voltage gain h21 = short-circuit forward current gain h22 = open-circuit output admittance For reciprocal networks, h12 = −h21. V1 = h11 I1 + h12V2 I 2 = h21 I1 + h22V2 The h-parameter equivalent network of a two-port network 25 Example h-1: Ideal Transformer 1 V1 = V2 , I1 = -nI 2 n V1 = h11I1 + h12V2 I 2 = h21I1 + h22V2 é ê 0 ê [h] = ê ê 1 êêë n 1ù ú nú ú ú 0ú úû 26 Example h-2 Find the hybrid parameters for the following two-port network. V1 V1 h11 = h12 = [ ] I1 V =0 V2 2 I2 h21 = I1 I2 h22 = V2 V2 =0 I1 =0 [S] I1 =0 V1 h11 = = 2 + 3 6 = 4 I1 V =0 2 h21 = I2 I1 =V2 =0 6 2 =(3 + 6) 3 27 Example h-2 cont’d h12 = V1 V2 I2 h22 = V2 é ê4 ê [h] = ê ê 2 êêë 3 2ù ú 3ú ú 1 ú Sú 9 úû 6 2 = = (3 + 6) 3 I =0 1 1 1 = = S (3 + 6) 9 I =0 1 Note that h12 = −h21 since the network is reciprocal 28 Example h-3 Determine the h parameters for the following circuit. V1 = 20 5 = 4 h11 = I1 V =0 2 é 4 0.8 ù ú [ h ] = êê ú ë-0.8 1.067Sû h12 = −h21 I2 h21 = I1 20 == -0.8 (20 + 5) V =0 V1 h12 = V2 20 = = 0.8 (5 + 20) I =0 I2 h22 = V2 1 = = 1.067S 15 (5 + 20) I =0 2 1 1 29 Example h-4 Find the impedance at the input port of the following circuit. I 2 = V2 Z L V2 50k (1) V1 =h11 I1 h12V2 (2) I 2 =h21 I1 h22V2 V2 Z L h12 h21Z L (1) V1 =h11 I1 I1 1 h22 Z L h21Z L I1 V2 1 h22 Z L h12 h21Z L 1.667k Z in V1 I1 h11 1 h22 Z L 30 Inverse Hybrid Parameters A set of parameters closely related to the h parameters are the g parameters or inverse hybrid parameters. These are used to describe the terminal currents and voltages as I1 = g11V1 + g12 I 2 V2 = g 21V1 + g 22 I 2 The g-parameter model of a two-port network For reciprocal networks, g12 = −g21. 31 Inverse Hybrid Parameters The values of the g parameters are determined as I1 = g11V1 + g12 I 2 V2 = g 21V1 + g 22 I 2 I1 g11 = V1 V2 g 21 = V1 [S] I 2 =0 I 2 =0 I1 g12 = I2 V1 =0 V2 g 22 = I2 [ ] V1 =0 Thus, the inverse hybrid parameters are specifically called g11 = open-circuit input admittance g12 = short-circuit reverse current gain g21 = open-circuit forward voltage gain g22 = short-circuit output impedance 32 Example g-1 Find the g parameters as functions of s for the following circuit. I1 I2 V1 V2 I g11 = 1 V1 V2 g 21 = V1 I2 I1 = g11V1 + g12 I 2 V2 = g 21V1 + g 22 I 2 1 [S] = s +1 =0 I1 g12 = I2 1 =s +1 V =0 1 = s +1 =0 V2 g 22 = I2 1 1⋅ s s 2 + s +1 [] = + = s 1+ s s ( s + 1) V =0 I2 1 1 Note that g12 = −g21 since the network is reciprocal 33 Example g-2 Find the g parameters from measured data: measured (port 2 open): V1 50 mV, I1 5μA V2 200 mV BLACK BOX measured (port 1 short): V2 10 mV, I1 2μA I 2 0.5μA I1 g11 = V1 V2 g 21 = V1 = I 2 =0 I 2 =0 5μA = 0.1mS 50 mV 200 = =4 50 I1 g12 = I2 g 22 = V2 I2 V1 =0 2 == -4 0.5 = V1 =0 10 mV = 20 k 0.5μA g12 = −g21 34