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DEPENDENT SOURCES Objectives Notes for course EE1.1 Circuit Analysis 2004-05 TOPIC 8 – DEPENDENT SOURCES Objectives . To introduce dependent sources . To study active sub-circuits containing dependent sources . To perform nodal analysis of circuits with dependent sources 1 INTRODUCTION TO DEPENDENT SOURCES 1.1 General The elements we have introduced so far are the resistor, the capacitor, the inductor, the independent voltage source and the independent current source. These are all 2-terminal elements The power absorbed by a resistor is non-negative at all times, that is it is always positive or zero The inductor and capacitor can absorb power or deliver power at different time instants, but the average power over a period of an AC steady state signal must be zero; these elements are called lossless. Since the resistor, inductor and capacitor cannot deliver net power, they are passive elements. The independent voltage source and current source can deliver power into a suitable load, such as a resistor. The independent voltage and current source are active elements. In many situations, we separate the sources from the circuit and refer to them as excitations to the circuit. If we do this, our circuit elements are all passive. In this topic, we introduce four new elements which we describe as dependent (or controlled) sources. Like independent sources, dependent sources are either voltage sources or current sources. However, unlike independent sources, they receive a stimulus from somewhere else in the circuit and that stimulus may also be a voltage or a current, leading to four versions of the element Dependent sources are considered part of the circuit rather than the excitation and have the function of providing circuit elements which are active; they can be used to model transistors and operational amplifiers. Since there are two source terminals and also two terminals where the stimulus exists, dependent sources are in general 4-terminal elements. 1.2 An Intuitive Analogy for Dependent Sources The basic idea of the voltage-controlled voltage source can be illustrated by considering a voltage source vc connected between two nodes and a voltmeter connected between two different nodes between which there exists a voltage difference vx: Topic 8 – Dependent Sources The voltmeter measures the voltage vx and then controls the voltage source such that vc = µvx. Since the electrical equivalent of an ideal voltmeter is an open circuit, we can represent the element as follows: The basic idea of the current-controlled current source can be illustrated by considering a current source ic connected between two nodes and an ammeter connected between two different nodes between which a current ix flows: The ammeter measures the current ix and then controls the current source such that ic = βix. Since the electrical equivalent of an ideal ammeter is a short circuit, we can represent the element as follows: In these examples, the output variable vc or ic is called the controlled variable and the input variable vx or ix is called the controlling variable. The remaining two source types are obtained by combining controlled and controlling variables in different ways. For instance, the voltage-controlled current source combines open-circuit voltage measurement with a current source at the output; it is described by ic = gmvx: The current-controlled voltage source combines short-circuit current measurement with a voltage source at the output; it is described by vc = rmix: Thus the four types of dependent sources are as follows: 2 Topic 8 – Dependent Sources VCVS VCCS CCVS CCCS Dependent (or controlled) sources can be thought of intuitively as amplifiers, where the input and output variables can both be either a voltage or a current. In many cases, there is some flexibility over which type of source can be used. For example, when the load is resistive, it may be driven by a voltage source or a current source because the voltage and current are related by Ohm's law. 1.3 The Dependent Voltage Source Although dependent sources are strictly 4-terminal elements, we can show them simply as a voltage or current source and write next to it an expression for the voltage or current as a function of another voltage or current defined elsewhere in the circuit. Using this notation, the symbol for a dependent voltage source is as follows: where vx and ix are defined elsewhere in the circuit. Its outline is a diamond shape to show that it is a dependent source and to distinguish it from the round-shaped independent voltage source. Inside the diamond shape, a pair of '+' and '–' signs denote the positive reference for the voltage generated. The terminal voltage v of the dependent voltage source, when plotted as a function of its terminal current i, is exactly the same as that of an independent voltage source: This plot shows that vc is independent of i. However, whereas for an independent voltage source, the voltage is also constant with time, in the case of the dependent voltage source, the voltage depends upon either a voltage vx or a current ix somewhere else in the circuit and is therefore not, in general, constant with time. Voltage vc is referred to as the controlled voltage (or dependent voltage). When the controlling variable is a voltage, the dependent voltage source is called a voltage- controlled voltage source, or VCVS; the dependency relationship has the form: vc = µvx The parameter µ is a unit-less real constant called the voltage gain. When the controlling variable is a current, the dependent voltage source is called a current- controlled voltage source (or CCVS); the linear dependency relationship takes the form 3 Topic 8 – Dependent Sources vc = rmix The real constant rm has units of Ohms and is called the trans-resistance. The prefix "trans" is used because the dependent source "transfers" the effect of a current somewhere else in the circuit to the dependent voltage source; the term "resistance" is used because it multiplies that current by a constant having the unit of Ω to turn it into a voltage. The subscript "m" denotes "mutual" which is a synonym for "trans". The CCVS, however, is quite different from the resistor element because the current in a resistor element must be defined in the same element as that across which the voltage is defined. We could say that a resistor has self-resistance (or just resistance) and that a CCVS has mutual resistance or trans-resistance. 1.4 The Dependent Current Source The symbol for a dependent current source is as follows: It has the same diamond shape as that of the dependent voltage source, but instead of '+' and '–' signs, it has an arrow denoting the positive current reference for the controlled current ic. The v-i characteristic has the following form: It has the same terminal behaviour as that of an independent current source, but the value of the controlled current ic depends upon either a voltage vx or a current ix somewhere else in the circuit and will therefore in general vary with time. When the controlling variable is a voltage, we refer to the dependent i-source as a VCCS, for voltage-controlled current source; the dependency relationship is: ic = gmvx The real constant gm has units of Siemens and is therefore called the trans-conductance. When the controlling variable is a current,, we refer to the dependent i-source as a CCCS, for current-controlled current source; the dependency relationship is: ic = βix The real unit-less constant β is called the current gain. We develop a method for analysing a circuit containing dependent sources by considering some examples. 1.5 Dependent Source Examples Example 5.1: Find the value of current i in the circuit shown which contains a CCCS: 4 Topic 8 – Dependent Sources Solution Note that the current-controlled current source is defined by ic = 2ix, where ix is the current in the 6 Ω resistor. Note that for a dependent source, the polarities for the controlling variable and controlled variable are inter-related and neither should be changed separately. We treat the dependent source constraint relationship ic = 2ix as a label on the source. We temporarily cover this label with a small piece of tape on which we write the symbol ic; we call this procedure taping the dependent source. This leads to the following: We now treat ic as if it was an independent current source, just like the 9 A current source. The current divider rule leads to: 6 2 i = −(9 + ic ) = −6 − ic 3 + 6 3 If ic was an independent current source, we would be through; however, for a dependent source there is a further stage: We next un-tape the dependent source and express its controlled variable ic in terms of its controlling variable: ic = 2ix We then express the controlling variable in terms of our analysis variables, in this case the unknown current i. Using KCL, we have: ic = 2ix = 2(9 + i + ic ) Note that ic appears on both sides of the equation. We can now solve the two equations: ic = 2(9 + i + ic ) = 18 + 2i + 2ic ic = −18 − 2i 2 2 4 i = −6 − i = −6 + (18 + 2i) = 6 + i 3 c 3 3 −6 i = − = −18 A 1 3 We can state the general method of analysis as follows: 1) Tape all dependent sources, thus treating them temporarily as independent sources. 5 Topic 8 – Dependent Sources 2) Analyze the circuit for the unknown variable or variables you have chosen using the analysis technique of your choice. 3) Un-tape the dependent sources and express their values in terms of the unknowns you have chosen to use in step 2.
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