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Circuits II EE221 Unit 8 Instructor: Kevin D. Donohue 2 Port Networks –Impedance/Admittance, Transmission, and Hybird Parameters 2-Port Circuits Network parameters characterize linear circuits that have both input and output terminals, in terms of linear equations that describe the voltage and current relationships at those terminals. This model provides critical information for understanding the effects of connecting circuits, loads, and sources together at the input and output terminals of a twoport circuit. A similar model was used when dealing with oneport circuits. Review example: Thévenin and Norton Equivalent Circuits: 100 10 i1 a i1 10 V 50 100 b Show that Voc=8 V, Isc = 0.08 A, and Rth = 100 2-Port Circuits: Now take away the source from the previous example: 100 10 ia ia 50 100 Why wouldn't it make sense to talk about a Thévenin or Norton equivalent circuit in this case? The Thévenin and Norton models must be extended to describe circuit behavior at two ports. Label the terminal voltage and currents as v1, i1, v2, and i2 and develop a mathematical relationship to show their dependencies. Inverse Transmission -Parameter Model: If the circuit is linear, then a general linear relationship between the terminal voltages and currents can be expressed as: v2 av1 bi1 V2 i2 cv1 di1 I 2 Geometrically each equation forms a planar surface over the v1-i1 plane, therefore, only three points on the surface are necessary to uniquely determine a, b, c, d, V2, and I2. So if the circuit response is known for three different values of the v1-i1 pairs, six equations with six unknowns can be generated and solved. This problem can be simplified by strategically setting v1 and i1 to zero in order to isolate unknown parameters and simplify the resulting equations. In general, if there are no independent sources in the circuit then V2, and I2 will be 0. This will be the case for the problems considered in this unit. Example Determine the inverse transmission parameter model for the given circuit. 100 10 ia i1 + v1 - ia 50 100 i2 + v2 - Show that a =18/5, b= 100, c = 7/250 Siemens, d= 1. Summary Formula for Inverse Transmission Parameters: If all independent sources are deactivated, set i1 = 0 to find: a v2 v1 c i1 0 i2 v1 i1 0 If all independent sources are deactivated, set v1 = 0 to find: b v2 i1 d v1 0 i2 i1 v1 0 Equivalent Circuit for Inverse Transmission Parameter Model: If inverse transmission parameters are known, then the following circuit can be used as an equivalent circuit: i1 + - + v1 - i2 1 i2 c + - d c + - av1 bi1 + v2 - This circuit is helpful when implementing in SPICE without knowledge or details of circuit from which parameters were derived. SPICE Solutions for Two-Port Parameters: As shown on previous slides, by strategically selecting the constraints on certain port variables, the two-port parameters are equal to ratios of other port variables. Therefore: Port variables can be constrained by attaching a zero-valued voltage or current source. The other port can be excited by a unity-valued source (or some other convenient value). Place meters at the remaining values need to evaluate the ratios. The two-port parameter can be found computing the ratio values from those found directly by SPICE. Example: Determine the SPICE commands to find the abcd parameters for the circuit below. 100 10 ia i1 + v1 - ia 50 100 i2 + v2 - SPICE Solutions for Two-Port Parameters: 1) Consider setting v1=0, then i v d 2 b 2 i1 i1 2) Excite the circuit with i2=1 then v 100 b 2 i1 1 d i2 1 1 i1 1 3) Use SPICE to compute v2 and i1 to solve for b and d. VAm1 H1 R2 -1000.00m 100 0 V1 R1 50 VAma 16.67n R3 100 1 IVm2 100.00 I2 SPICE Solutions for Two-Port Parameters: 4) Consider setting i1=0, then v2 a v1 5) i2 v1 Excite the circuit with v2=1, then a 6) c 1 1 3.6 v1 0.2778 c i2 .00778 28m v1 0.2778 Use SPICE compute v1 and i2 to solve for a and c. H1 R2 VAm2 7.78m 100 0 I1 IVm1 277.78m R1 50 VAma 5.56m R3 100 1 V2 Transmission -Parameter Model: Transmission parameters are related to the inverse transmission parameters by reversing the independent and dependent variables: Transmission Parameters v2 av1 bi1 i2 cv1 di1 v2 a b v1 i c d i 1 2 Inverse Transmission Parameters 1 a b v2 v1 c d i i 2 1 A B v2 v1 C D i i 2 1 v1 Av2 Bi 2 i1 Cv2 Di2 Impedance/Admittance-Parameter Model: Impedance Parameters v1 z11i1 z12i2 v2 z21i1 z22i2 v1 z11 v z 2 21 z12 i1 z22 i2 Admittance Parameters 1 z11 z12 v1 i1 z 21 z 22 v2 i2 y11 y12 v1 i1 y 21 y22 v2 i2 i1 y11v1 y12v2 i2 y21v1 y22v2 Hybrid (h)/Inverse Hybrid (g)-Parameter Model: Hybrid Parameters v1 h11i1 h12v2 i2 h21i1 h22v2 v1 h11 h12 i1 i h 2 21 h22 v2 Inverse Hybrid Parameters 1 h11 h12 v1 i1 h 21 h22 i2 v2 g11 g12 v1 i1 g 21 g 22 i2 v2 i1 g11v1 g12i2 v2 g 21v1 g 22i2 Relationship Between 2 Sets of Port Parameters: Since a single set of network parameters characterize the linear circuits completely at the input and output terminals, it is possible to derive other network parameters from this set. Example: Consider the z and y parameter characterization of a given circuit with no independent sources: v z v z 1 2 z i z i 11 12 21 i y i y 1 22 2 Show that: z z 11 21 z y z y 12 22 11 21 y y 1 12 1 11 2 21 y y y y y y y y y y 11 21 y z y z 11 12 22 11 21 12 22 1 22 2 22 21 12 21 22 z z 12 22 11 y y y v y v 1 22 21 12 z z z z z z z z z z 22 11 22 21 12 21 11 22 21 12 y y y y y y y y y y 12 11 22 21 12 21 12 11 11 22 z z z z z z z z z z 12 11 22 21 12 21 12 11 11 22 Relationship Between 2 Sets of Port Parameters: Example: Consider the abcd and h parameter characterization of a given circuit with no independent sources: v h i h v2 a b v1 i c d i 1 2 1 11 2 21 h i h v 12 1 22 2 Show that: h11 h 21 b h12 a h22 bc ad a 1 a c a 1 h11 a b h12 h 12 c d h22 h h22h11 21 h h 12 12 Solving for Terminal Currents and Voltages from Port Parameters: Once the port parameters are known, no other information from the circuit is required to determine the behavior of the currents and voltages at the terminals. Example: Given the z-parameter representation of a circuit, determine the resulting terminal voltages and currents when a practical source with internal resistance Rs and voltage Vs is connected to the input (terminal 1) and a load RL is connected to the output (terminal 2): v z v z 1 11 2 21 z i z i 12 22 Show that: z 11 L z s 12 RV v R R R 21 L s 1 L s z 22 L R L R R z R RV R R L 22 R R s R L z L s L 22 s 2 L s z R 22 s R v R R R z R z R 12 L L s s + v2 - + v1 - Vs 2 i2 i1 Rs 1 s z 22 R L z 11 - RL RV L s z R V z R 21 L 12 R R L s s s z R 22 s Combinations of Two-Port Networks: Consider circuits A and B described by their abcdparameters (assume independent sources zero). i1a - + v1a - A + v2a - i2a i1b - - i2b + v1b - B + v2b - - If A and B are connected in series, show that the abcd parameters for the new two-port (from v1a to v2b) is given by: v2b ab i c 2b b bb aa db ca ba v1a ab aa bb ca d a i1a cb aa db ca abba bb d a v1a cbba db d a i1a Combinations of Two-Port Networks: Consider circuits A and B described by their y-parameters (assume independent sources zero). i1a i1 + - v1 + v1a - A + v2a - - - - i1b - i2a + v1b - B + v2b - i2b i2 + v2- - If A and B are connected in parallel, show that the yparameters for the new two-port (from v1a to v2b) is given by: i1 y11b i y 2 21b y12b y11a y22b y21a y12a v1 y11a y11b y22a v2 y21a y21b y12a y12b v1 y22a y22b v2