Saturday , October 23 2021

Maximum power transfer theorem

Maximum power transfer theorem

 

Maximum power is transferred to the load when the load resistance equals the Thevenin resistance as seen from the load (RL  = RTh )

or

The maximum power transfer theorem states that “A load will receive maximum power from a linear bilateral dc network when its total resistive value is exactly equal to the Thevenin resistance of the network”

কোন লোডে সর্বোচ্চ পাওয়ার ট্রান্সফার হবে যখন ওই লোডের রেজিস্ট্যান্স(RL) সার্কিটের ইন্টারনাল রেজিস্টান্সের সমান হবে। সার্কিটের ইন্টারনাল রেজিস্টান্স থেভেনিন’স রেজিস্ট্যান্স (RTh)এর সমান। উল্লেখ্য লোডের রেজিস্ট্যান্স অবশ্যই পরিবর্তনশীল হতে হবে।

Question: Prove that the maximum power transfer theorem ( {p_{max}=\frac{V_{Th}^2}{4R_{Th}}} )

 

The Thevenin equivalent is useful in finding the maximum power a linear circuit can deliver to a load. We assume that we can adjust the load resistance RL . If the entire circuit is replaced by its Thevenin equivalent except for the load, as shown in Fig. 1.

Maximum power transfer theorem

Figure 1: The circuit used for maximum power transfer.

 

the power delivered to the load is

{\begin{array}{l}p=i^2R_L\\\;\;=(\frac{V_{Th}}{R_{Th}+R_L})^2\;R_L................(i)\end{array}}

To prove the maximum power transfer theorem, we differentiate p in Eq. (i) with respect to RL and set the result equal to zero. We obtain

{\begin{array}{c}\frac{dp}{dR_L}=0\\\\\Rightarrow\frac d{dR_L}(\frac{V_{Th}}{R_{Th}+R_L})^2R_L=0\\\\\Rightarrow V_{Th}^2\lbrack\frac{(R_{Th}+R_L)^2-2R_L(R_{Th}+R_L)}{(R_{Th}+R_L)^4}\rbrack=0\\\\\Rightarrow{V_{Th}^2\lbrack\frac{\displaystyle(R_{Th}^2+2R_{Th}R_L+R_L^2-2R_LR_{Th}-2R_L^2)}{\displaystyle(R_{Th}+R_L)^4}\rbrack}=0\\\\\Rightarrow\frac{R_{Th}^2-R_L^2}{(R_{Th}+R_L)^4}=0\\\\\Rightarrow\frac{\displaystyle(R_{Th}+R_L){(R_{Th}-R_L)}}{\displaystyle(R_{Th}+R_L)^4}=0\\\\\Rightarrow\frac{\displaystyle(R_{Th}-R_L)}{\displaystyle(R_{Th}+R_L)^3}=0\\\\{\Rightarrow R_{Th}-R_L}=0\\\\R_L=R_{Th}\\\\\\\end{array}}

Which showing that the maximum power transfer takes place when the load resistance RL equals the Thevenin resistance RTh

The maximum power transferred is obtained by Putting this value in equation(i) we get

{\begin{array}{c}\\p_{max}=(\frac{V_{Th}}{R_{Th}+R_{Th}})^2R_{Th}\\\\p_{max}=\frac{V_{Th}^2}{4R_{Th}}.R_{Th}\\\\p_{max}=\frac{\displaystyle V_{Th}^2}{\displaystyle4R_{Th}}\\\\\\\\\end{array}}

This equation applies only when RL=RTh. When RLRTh we compute the power delivered to the load using Eq. (1)

 

Read:Thevenin theorem

Check Also

prove that in active mode bjt can be modeled as a dependent current controlled current source

prove that in active mode bjt can be modeled as a dependent current controlled current source

Question: Prove that in active mode BJT can be modeled as a dependent current controlled …