IMPEDANCE and NETWORKS. Transformers. Networks. A method of analysing complex networks. Y-parameters and S-parameters
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1 IMPEDANCE and NETWORKS Transformers Networks A method of analysing complex networks Y-parameters and S-parameters 1 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
2 Transformers Combining the effects of self and mutual inductance, the equations of a transformer are given by, V 21 = jωl 1 I i + jωmi o V 43 = jωmi i + jωl 2 I o 2 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
3 Simplified Transformer Circuit The following circuit is equivalent to the above. Note the definition of voltage sense. 3 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
4 Networks: Linearity Maxwell s equations are linear Thus if we have an arbitrary circuit, then the E.M.F seen at any two terminals must be a linear function of the current drawn. 4 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
5 Networks: Thevenin and Norton Equivalents Every linear circuit can be simplified to the following trivial forms. 5 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
6 Networks: Impedance and Admittance Impedance: V = ZI Impedance: Z = R + jx R = Resistance and X = Reactance Admittance: I = YV Admittance: Y = G + jb G = Conductance and B = Susceptance 6 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
7 Networks: Analysis of Linear Circuits One of the main differences between radiofrequency compared to low frequency circuits is the effect of parasitics. Even the simplest transistor or oscillator models give rise to relatively complex circuits. We need a circuit solver. I now introduce a simple MATLAB program to solve all linear circuits. 7 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
8 Networks: Analysis of Linear Circuits: SOLVE: The Circuit Solver SOLVE is a simple and short MATLAB solver for complex circuits Main advantage of SOLVE is that it is so simple that you can easily trace problems. You can of course use PSPICE or any other program of your choice. I recommend that you get to know PSPICE (or some of the more complex commercial products, ANSOFT, ADA, CADENCE) due to the built in support for RF components and the detailed analysis of non-linear behaviour that these packages can provide. 8 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
9 Networks: Analysis of Linear Circuits We consider the following circuit driven by a 1 Ampere current source: 9 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
10 Networks: Analysis of Linear Circuits Consider a typical node in the circuit. The current I is injected into the node. At the first node (on the current source)... N ( ) N 1 = y j1 V1 V j = j=1 y j1 j=1 V 1 N 1 j=1 y j1 V j where each voltage is referenced to ground. V N = 0 and y ii = ENGN4545/ENGN6545: Radiofrequency Engineering L#7
11 Networks: Analysis of Linear Circuits At the i th node where i 1, there is no current injected, 0 = N j=1 ( ) y ji Vi V j = N y ji j=1 V i N 1 j=1 y ji V j 11 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
12 Networks: Analysis of Linear Circuits We therefore have an (N 1) (N 1) matrix equation. where I = MV M ii = N k=1 y ki where the sum is such that k i and, M ij = y ji for i j and i = 1,...,(N 1). Notice that only the diagonal terms M ii require knowledge of admittance to ground, y kn. Solve for V. 12 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
13 Networks: SOLVE. Matlab Code 13 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
14 Networks: Analysis of Linear Circuits: Some Simple Rules to Remember Node 1 is at the current source. The N th node is ground. V N = 0 and is not used. A node might actually be a junction of many joined wires.. so be careful not to leave wires out accidently. V i represents the voltage from ground to the i th node. There is always only one admittance between each node. This means that we have to parallel up parallel admittances. By the way.. Parallel admittances add. Why? The current source injects 1 Ampere. Thus the input impedance is numerically equal to the voltage on node ENGN4545/ENGN6545: Radiofrequency Engineering L#7
15 Networks: Analysis of Linear Circuits: Simple Example A series LC circuit ENGN4545/ENGN6545: Radiofrequency Engineering L#7
16 Networks: Analysis of Linear Circuits: Simple Example Real(Z) at node Sove output for ADMITTANCE_PROGRAM == LC.m Imag(Z) at node ENGN4545/ENGN6545: Radiofrequency Engineering L#7
17 Networks: Analysis of Linear Circuits: More complex Example 17 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
18 Networks: Analysis of Linear Circuits: More complex Example 18 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
19 Networks: Analysis of Linear Circuits: More complex Example Real(Z) at node Sove output for ADMITTANCE_PROGRAM == LC.m Imag(Z) at node ENGN4545/ENGN6545: Radiofrequency Engineering L#7
20 Networks: Y-parameters Four port parameters describing a linear passive network I 1 = y i V 1 + y r V 2 I 2 = y f V 1 + y o V 2 where y i is the input admittance, y r the reverse-transfer admittance, y f the forward-transfer admittance and y o the output admittance. 20 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
21 Networks: S-parameters Four port parameters describing a linear passive network b 1 = S 11 a 1 + S 12 a 2 b 2 = S 21 a 1 + S 22 a 2 where S 11 is the input reflection coefficient or return loss, S 12 is the reverse transmission coefficient, S 21 the forward transmission coefficient, and S 22 is the output reflection coefficient. 21 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
22 Networks: S-parameters a 1 and b 2 are forward waves and a 2 and b 1 are backward waves. They may be voltages or currents. S 21 is also referred to as transfer function, insertion loss or gain. S 11 = (b 1 /a 1 ) a 2 = 0. To measure S 11 let Z L = Z 0 (the characteristic impedance) and measure the input reflection coefficient. S 21 = (b 2 /a 1 ) a 2 = 0. To measure S 21 let Z L = Z 0 and measure the ratio of the transmitted to the incoming signal. S 22 = (b 2 /a 2 ) a 1 = 0. To measure S 22 swap the source and Z L and let Z L = Z 0 22 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
23 Networks: Y-parameters vs S-parameters Y-parameters and S-parameters are equivalent. Y-parameters are best for calculations. S-parameters are best for measurements. It is easy to measure power in a terminated network. S-parameters are usually the parameters given in component datasheets. 23 ENGN4545/ENGN6545: Radiofrequency Engineering L#7
24 Networks: Y-parameters vs S-parameters Y-parameters and S-parameters are related: y i = (1 + S 22)(1 S 11 ) + S 12 S 21 Z 0 y r = 2S 12 Z0 y f = 2S 21 Z0 y o = (1 + S 11)(1 S 22 ) + S 12 S 21 Z 0 where = (1 + S 11 )(1 + S 22 ) S 21 S ENGN4545/ENGN6545: Radiofrequency Engineering L#7
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