Exercise 8.6 A bipolar output stage

a)

An answer:
With the transistor characteristics and with the circuit in the previous figure, $v_{\mathit{OUT}}(t)=v_G(t)$.

For $0\le t\le \pi ∕\omega$, $v_G(t)\le 0$ and the NPN conducts. Hence the current from the positive supply voltage during one period of $v_G$ is:

$$\begin{array}{rcll}& i_{V\mathit{CC}}& =\frac{V_G\cdot \mathit{sin}(\mathit{\omega t})}{R_L}(0\le t\le \pi ∕\omega )& \\ & i_{V\mathit{CC}}& =0(\pi ∕\omega \le t\le 2\pi ∕\omega )& \end{array}$$

Using symmetry:

$$\begin{array}{rcll}& i_{-V\mathit{CC}}& =\frac{V_G\cdot \mathit{sin}(\mathit{\omega t})}{R_L}(\pi ∕\omega \le t\le 2\pi ∕\omega )& \\ & i_{-V\mathit{CC}}& =0(0\le t\le \pi ∕\omega )& \end{array}$$

b)

An answer:

$$\begin{array}{rcll}& P_{V\mathit{CC}}& =\frac{V_{\mathit{CC}}\cdot V_G}{2R_L\pi ∕\omega }{\int \; <!--\; nolimits\; -->}_0^{\pi ∕\omega }\mathit{sin}(\mathit{\omega t})\mathit{dt}& \\ & & =\frac{V_{\mathit{CC}}\cdot V_G}{R_L}\frac{1}{2\pi }{\int \; <!--\; nolimits\; -->}_0^{\pi }\mathit{sin}(𝜃)\mathit{d𝜃}& \\ & & =\frac{V_{\mathit{CC}}\cdot V_G}{R_L}\frac{1}{\pi }& \\ & P_{-V\mathit{CC}}& =-\frac{V_{\mathit{CC}}\cdot V_G}{2R_L\pi ∕\omega }{\int \; <!--\; nolimits\; -->}_{\pi ∕\omega }^{2\pi ∕\omega }\mathit{sin}(\mathit{\omega t})\mathit{dt}& \\ & & =-\frac{V_{\mathit{CC}}\cdot V_G}{R_L}\frac{1}{2\pi }{\int \; <!--\; nolimits\; -->}_{\pi }^{2\pi }\mathit{sin}(𝜃)\mathit{d𝜃}& \\ & & =\frac{V_{\mathit{CC}}\cdot V_G}{R_L}\frac{1}{\pi }& \end{array}$$

c)

An answer:

$$\begin{array}{rcll}& P_{\mathit{RL}}& =\frac{V_G^2}{2R_L\pi ∕\omega }{\int \; <!--\; nolimits\; -->}_0^{2\pi ∕\omega }\mathit{sin}{(\mathit{\omega t})}^2\mathit{dt}& \\ & & =\frac{V_G^2}{R_L}\frac{1}{2\pi }{\int \; <!--\; nolimits\; -->}_0^{2\pi }\mathit{sin}{(𝜃)}^2\mathit{d𝜃}& \\ & & =\frac{V_G^2}{2R_L}& \end{array}$$

d)

An answer:

$$\begin{array}{rcll}& P_{\mathit{supply}}& =\frac{2}{\pi }\cong 0.64W& \\ & P_{\mathit{RL}}& =\frac{V_G^2}{2R_L}=0.5W& \\ & \eta & \cong 0.785& \end{array}$$

e)

An answer:
This is the power dissipated by the transistors

f)