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
Opto-isolator wikipedia , lookup
Negative resistance wikipedia , lookup
Valve RF amplifier wikipedia , lookup
Audio power wikipedia , lookup
Surge protector wikipedia , lookup
Resistive opto-isolator wikipedia , lookup
Standing wave ratio wikipedia , lookup
Switched-mode power supply wikipedia , lookup
Power electronics wikipedia , lookup
Power MOSFET wikipedia , lookup
1E6 Electrical Engineering DC Circuit Analysis Lecture 7: Load Power and Internal Resistance 6.1 Introduction Even in the simplest of electric circuits the ultimate aim is usually to deliver power to a load. In the circuits we have examined to date, the load has been modelled as a pure resistance, RL. However, the load is not always a real resistor but the resistor is used to represent any load, operation or process which used up electrical energy and therefore power. Indeed, the simplified electric circuits can themselves be models for power sources, instruments or equipment which provides electrical energy to a load. If this is the case, then the factors which influence the transfer of power to the load from the source should be examined to ensure the optimum use of the energy consumed. 6.2 Maximum Power Transfer The circuit shown in Fig. 1 can be considered as a model representing a non-ideal source which has an internal resistance, RS, driving a load resistance, RL. The power dissipated in the load is given as: PL VL I L where VL where is the voltage appearing across the load and IL is the current flowing through it. power source IL RS VL E Fig. 1 RL A Power Source Driving a Resistive Load 1 The voltage across the load is essentially obtained from the potential divider action of RS and RL which gives: VL RL E RS R L The current through the load is essentially obtained from the source emf divided by the sum of the two resistors in series. E IL RS RL Then: ER L E 2R L E PL R S R L R S R L R S R L 2 It can be seen that if the voltage source were ideal with R S → 0, then the power would become: PL E2R L RL 2 E2 RL In this event, the full emf, E, is developed across the load resistance, R L, and the power delivered to the load depends only the value of this emf and the value of the load resistance. This means that all of the power drawn from the source is dissipated in the load resistance, RL. On the other hand, when the source is non-ideal and RS is finite, some power will be dissipated in this resistance and is therefore essentially ‘lost’ or ‘wasted’. The remainder of the power is delivered to the load as: PL E 2R L R S R L 2 Usually, the internal resistance of the source or the equivalent model representing the source is determined by factors beyond control and so cannot always be adjusted to suit requirements. However, the resistance of the load can often be controlled by design, at least to some degree. 2 In this case, the aim is to try to optimise the load resistance, RL, so that maximum power is transferred from the source to the load. This can be determined by finding the derivative of the expression for the load power with respect to the load resistance, RL and equating it to zero. Then if: PL E 2R L R S R L 2 The quotient rule must be used: Let u E2R L with the quotient rule so that: v R S R L 2 and PL R L v u v u R L R L v2 PL R S R L E 2 2E 2 R L R S R L 0 4 R L R S R L 2 This requires: R S R L 2 E 2 2E 2 R L R S R L 0 or: So that: R S R L 2R L R L RS Thus it can be seen that maximum power will be transferred from the source to the load when the resistance of the load is made equal to the internal resistance of the source. 3 Note that under these conditions, the actual power delivered to the load is given as: PL E2R L R S R L 2 E2R L E2 2 4R L 4R S Recall from above that when RS = 0 the power delivered to the load is: E2 PL RL This indicates that when RS ≠ 0, but RL = RS the maximum power delivered to the load is only one quarter of that obtained from an ideal source. This is indicated in Fig. 2 which shows a plot of the load power as a function of the ratio of the load and internal resistances in question. It can be seen from Fig. 2 that when the internal resistance is not zero it has a profound influence on the power delivered to the load. PL E2/RS power drawn from source power delivered to load E2/2RL E2/4RL 0 1 RL / Rs Fig. 2 Power Drawn from Source and Delivered to Load 4 When RL / RS → 0 which implies that RL → 0, maximum current will flow through the load being limited only by the internal resistance RS. However, with RL → 0 no voltage is developed across it and so no power is delivered to the load. As the load resistance, RL, is increased the current flowing through it, IL, decreases while the voltage across it, VL, increases. The maximum product of voltage and current occurs as determined above when RL = RS, shown as the peak in the curve of Fig. 2 illustrating the power delivered to the load. It is also the case, under these conditions, that the power drawn from the source is only half that obtained from the equivalent ideal source and that only half of this is delivered to the load. The other half is dissipated in the internal resistance, RS, of the non-ideal source, since it has the same value as the load resistance, RS. When the load resistance is increased further, the current flowing through it decreases even though the voltage developed across it increases. The product of voltage and current decreases, resulting in a decrease in the power delivered to the load. As the load resistance RL → ∞, the current flowing through it falls to zero so that the power delivered also falls to zero. This gives the bell-shaped curve for load power shown in Fig. 2. 6.2 Internal Resistance The above analysis highlights the importance of the effect of the internal resistance of a non-ideal voltage or current source on power transfer to the load. Given this, it is extremely useful to have a means of measuring the internal resistance of a source so that its value can be established and controlled if possible. Means used on paper in the analyses of circuits studied so far, such as using open-circuit or short-circuit load conditions, cannot always be applied as real electric circuits or systems may not tolerate such conditions without malfuction or damage. In this case a differential approach can be used to measure the internal resistance. power source IL RS VL E RL1 → RL2 Fig. 3 Measurement of Internal Source Resistance. 5 Consider again the non-ideal voltage source having an internal resistance, RS. A load resistance, RL is connected to the terminals, the value of which can be changed. Consider the scenario where two different values of load resistance, RL1 and RL2, are connected to the source in turn as indicated in Fig. 3. With the value RL1 connected: I L1 E ERL1 and VL1 I L1 RL1 RS RL1 RS RL1 With the value RL2 connected: I L2 E ERL 2 and VL 2 I L 2 RL 2 RS RL 2 RS RL 2 Taking the ratio: R RL 2 RL1 RS RL 2 VL1 ERL1 S VL 2 RS RL1 ERL 2 RL 2 RS RL1 Cross multiplying and simplifying: VL1RL 2 RS RL1 VL 2 RL1 RS RL 2 VL1RS RL 2 VL1RL1RL 2 VL 2 RS RL1 VL 2 RL1RL 2 VL1RS RL 2 VL 2 RS RL1 VL 2 RL1RL 2 VL1RL1RL 2 RS VL1RL 2 VL 2 RL1 VL 2 VL1 RL1RL 2 Then: RS VL 2 VL1 RL1 RL 2 VL1RL 2 VL 2 RL1 6 Dividing by RL1RL2: RS VL 2 VL1 VL1 VL 2 RL1 RL 2 So that: VL 2 VL1 VL RS I L1 I L 2 I L Note that if RL2 > RL1 then VL2 > VL1 and IL2 < IL1 so that RS is positive. It should also be noted that knowledge of the two values of resistance is not required, only the change in the output voltage and current which they produce. 7