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2-PORT CIRCUITS Objectives Notes for course EE1.1 Circuit Analysis 2004-05 TOPIC 10 – 2-PORT CIRCUITS Objectives: . Introduction . Re-examination of 1-port sub-circuits . Admittance parameters for 2-port circuits . Gain and port impedance from 2-port admittance parameters . Impedance parameters for 2-port circuits . Hybrid parameters for 2-port circuits 1 INTRODUCTION Amplifier circuits are found in a large number of appliances, including radios, TV, video, audio, telephony (mobile and fixed), communications and instrumentation. The applications of amplifiers are practically unlimited. It is clear that an amplifier circuit has an input signal and an output signal. This configuration is represented by a device called a 2-port circuit, which has an input port for the input signal and an output port for the output signal. In this topic, we look at circuits from this point of view. To develop the idea of 2-port circuits, consider a general 1-port sub-circuit of the type we are very familiar with: The basic passive elements, resistor, inductor and capacitor, and the independent sources, are the simplest 1-port sub-circuits. A more general 1-port sub-circuit contains any number of interconnected resistors, capacitors, inductors, and sources and could have many nodes. Sometimes, perhaps when we have finished designing a circuit, we become less interested in the detail of the elements interconnected in the circuit and are happy to represent the 1-port circuit by how it behaves at its terminals. This is achieved by use of circuit analysis to determine the relationship between the voltage V1 across the terminals and the current I1 flowing through the terminals which leads to a Thevenin or Norton equivalent circuit, which can be used as a simpler replacement for the original complex circuit. Consider now the 2-port circuit. A 2-port is a network having two pairs of terminals: Some elements intrinsically have more than the two terminals; examples are the bipolar junction transistor or the MOSFET. Such circuits cannot be represented as a 1-port circuit and the 2-port is the simplest description available. 2-port circuits also have the same application as 1-port circuits, which is that they enable us to describe the input-output behaviour of a circuit without worrying about circuit details. As an example, consider the BJT amplifier: Topic 10 – 2-port Circuits This circuit has 6 nodes, 7 2-terminal elements and one 3-terminal element. Once the circuit has been designed we may be happy to describe its behaviour between the input port and the output port, perhaps using parameters like gain. A 2-port description of the circuit allows us to do this systematically. In practice, 2-port circuits often represent devices in which a source delivers energy to a load through the 2-port network. For example, stereo amplifiers take a low power audio signal and increase its power so that it will drive a speaker system. Determining and knowing ratios such as voltage gain, current gain, and power gain of a 2-port circuit is very important when dealing with a source that delivers power through a 2-port to a load. This topic deals with precisely this need. Before proceeding to look at 2-port circuits, we re-examine 1-port circuits, and in particular the derivation of Thevenin and Norton equivalent circuits, in a more systematic way than hitherto. 2 RE-EXAMINATION OF 1-PORT SUB-CIRCUITS In this section, we look again at deriving Thevenin and Norton equivalents for 1-port sub-circuits and introduce systematic ways of deriving the equivalents based on nodal analysis. Consider first the following example: Example: Evaluate the Thevenin and Norton equivalents for the following circuit when viewed from terminals 1 and 0: The Thevenin voltage is the open-circuit voltage at node 1; using current division and Ohm's law, we have: 1 1 1 6 8 V = × × 4 = × × 4 = V oc 3 1 1 3 6 3 2 11 1+ + + + 2 3 The Thevenin impedance is obtained by de-activating the current source and determining the equivalent impedance seen between terminals 1 and 0. 1 3 1 × 3 R = 3 2 = 2 = Ω eq 1 3 2 + 9 11 + 3 2 6 2 Topic 10 – 2-port Circuits Hence, we have the Thevenin equivalent circuit and, by calculating Isc = Voc/Req, the Norton equivalent circuit: Thevenin and Norton equivalent circuits are simplifications of a circuit; however large the number of nodes in the original circuit, the Norton equivalent has just two nodes (1 and 0). The simplification can be viewed as a process of eliminating nodes in the original circuit which are not port nodes (node 2 in the above example). This process can be viewed as a simplification of equations obtained by systematic nodal analysis of the circuit. We now apply this systematic method to the above circuit. We start by labelling the reference node as zero and the port node as 1; the remaining nodes are numbered sequentially from 2 onwards. At the port for which we require the Thevenin or Norton equivalent, i.e. port 1, we apply a test current source which we label I1: We now perform by-inspection nodal analysis: ⎡ 5 −2⎤ ⎡V1 ⎤ ⎡I1⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎣−2 3 ⎦ ⎣V2 ⎦ ⎣ 4 ⎦ Node 2 is not a port node and is therefore an internal node, whose voltage V2 must be eliminated; we use Gaussian elimination: −2 −2 −2 4 11 8 5 −2 V I ⎡ ( )( ) ⎤ V ⎡ ( ) ⎤ ⎡ ⎤ V ⎡ ⎤ ⎡ ⎤ ⎡ 1 ⎤ ⎡ 1⎤ ⎢5 − 0⎥ ⎡ 1⎤ ⎢I1 − ⎥ ⎢ 0⎥ ⎡ 1⎤ ⎢I1 + ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ≡ 3 ⎢ ⎥ = 3 ≡ 3 ⎢ ⎥ = 3 ⎣−2 3 ⎦ ⎣V2 ⎦ ⎣ 4 ⎦ ⎢ ⎥ ⎣ 0 ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ 0 ⎦ ⎢ ⎥ ⎣⎢ 0 0⎦⎥ ⎣⎢ 0 ⎦⎥ ⎣ 0 0⎦ ⎣ 0 ⎦ The resulting equation leads directly to the Norton and Thevenin equivalent circuits, derived above: 11 8 11 8 3 8 V = I + I = V − V = I + 3 1 1 3 1 3 1 3 1 11 1 11 If the original circuit has no independent sources, then the above process leads just to the equivalent Thevenin/Norton equivalent impedance. We consider a more complex example: Example: Determine the equivalent resistance between nodes 1 and 0 of the following circuit (note that I1 is a test current source): 3 Topic 10 – 2-port Circuits We first perform by-inspection nodal analysis: ⎡1 ⎤ ⎡ 3 ⎤ + 1 −1 0 −1 0 ⎢2 ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎡V1 ⎤ ⎡I1⎤ ⎢ ⎥ ⎡V1 ⎤ ⎡I1⎤ 1 5 ⎢ −1 1+ + 1 −1 ⎥ ⎢V ⎥ = ⎢ 0 ⎥ ⎢−1 −1⎥ ⎢V ⎥ = ⎢ 0 ⎥ ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢V ⎥ ⎢ 0 ⎥ ⎢V ⎥ ⎢ 0 ⎥ ⎢ 0 1 1 1⎥ ⎣ 3 ⎦ ⎣ ⎦ ⎢ 0 1 2 ⎥ ⎣ 3 ⎦ ⎣ ⎦ ⎢ − + ⎥ ⎢ − ⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ The partition lines show that although this circuit has 3 nodes, we are treating it as a 1-port circuit, i.e. we are only interested in the relationship between I1 and V1. Nodes 2 and 3 are not port nodes and therefore V2 and V3 must be eliminated; we use Gaussian elimination: ⎡ 3 ⎤ ⎡ 3 ⎤ −1 0 −1 0 ⎡ 3 ⎤ ⎢ 2 ⎥ ⎢ 2 ⎥ −1 0 ⎢ ⎥ ⎡V1 ⎤ ⎡I1⎤ ⎢ ⎥ ⎡V1 ⎤ ⎡I1⎤ ⎢ 2 ⎥ ⎡V1 ⎤ ⎡I1⎤ 5 5 (−1)(−1) ⎢ ⎥ ⎢−1 −1⎥ ⎢V ⎥ = ⎢ 0 ⎥ ⎢−1 − 0⎥ ⎢V ⎥ = ⎢ 0 ⎥ −1 2 0 ⎢V ⎥ = ⎢ 0 ⎥ ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ 2 2 ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢V ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 0 0 0⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ 0 −1 2 ⎣ 3 ⎦ ⎣ ⎦ 0 0 0 ⎣ ⎦ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎡ 3 ⎤ ⎡ 3 (−1)(−1) ⎤ ⎢ −1⎥ ⎡V1 ⎤ ⎡I1⎤ − 0 ⎡V1⎤ ⎡I1⎤ ⎡1 0⎤ ⎡V1⎤ ⎡I1⎤ 2 = ⎢2 2 ⎥ = = ⎢ ⎥ ⎢V ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢0 0⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ 1 2 ⎣ 2 ⎦ ⎣ ⎦ 0 0 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣⎢− ⎦⎥ ⎣⎢ ⎦⎥ The resulting equation leads directly to the relationship between V1 and I1: V1 = I1 Hence, the equivalent sub-circuit is a 1 Ω resistor connected between node 1 and node 0. This approach is applicable to a circuit with any number of nodes, although a computer method of reducing the matrix would be needed for larger circuits. Having shown that an n-node circuit may be reduced to a 1-port description systematically, we are now ready to consider 2-port circuits. 3 2-PORT ADMITTANCE PARAMETERS 3.1 Definition Rather than always having to deal with all the internal variables of a 2-port circuit, it is often more convenient to deal only with the terminal voltages and currents: 4 Topic 10 – 2-port Circuits We assume that all excitations are external to the 2-port; hence the 2-port has no internal independent sources. We also assume that all dynamic elements (inductors and capacitors) are initially relaxed, i.e., have zero initial conditions, i.e. capacitor voltages and inductor currents are initially zero. Port currents are defined positive into the circuit. Under these assumptions, the admittance equations for a 2-port are expressions for the terminal currents I1 and I2 in terms of the port voltages V1 and V2, i.e.: I1 = y11V1 + y12V2 I2 = y21V1 + y22V2 where y11, y12, y21 and y22 are called admittance parameters or y-parameters. The unit for all the admittance parameters is the Siemens (S). If the circuit has no connection between port 1 and port 2, then each port can be described by Ohm's law: I1 = y11V1 I2 = y22V2 where y11 and y22 are the effective admittances at port 1 and 2, respectively. The remaining parameters y12 and y21 describe the effect of V2 on I1 and the effect of V1 on I2; these terms are necessary when there are elements which link port 1 to port 2 including where the 2-port is an active device, such as a dependent source or transistor The 2-port equations may be written more compactly in matrix notation: ⎡ I1 ⎤ ⎡y11 y12 ⎤ ⎡V1 ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣I2 ⎦ ⎣y21 y22 ⎦ ⎣V2 ⎦ Example Compute the admittance parameters of the following 2-port circuit: Solution Write the nodal admittance matrix by inspection; tape the VCCS to ic: ⎡G1 + G3 −G3 ⎤ ⎡V1 ⎤ ⎡ I1 ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎣ −G3 G2 + G3 ⎦ ⎣V2 ⎦ ⎣I2 − ic ⎦ G1, G2 and G3 are conductances of the resistors.
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