コレクタ接地増幅回路（エミッタホロワ増幅回路）について

上図の等価回路はエミッタ接地における電圧帰還率 $h_\mathrm{re}$ と出力アドミタンス $h_\mathrm{oe}$ を考慮したものとなっています。 簡易等価回路と同様にして導出します。

(1) $v_\mathrm{ce}=-v_\mathrm{o}$ より、出力電圧 $v_\mathrm{o}$、入力電圧 $v_\mathrm{i}$ はそれぞれ以下の式で表されます。

したがって、電圧増幅度 $A_\mathrm{v}=v_\mathrm{o}/v_\mathrm{i}$ は次式で表されます。

(2) 出力電流は $i_\mathrm{o}=-v_\mathrm{o}/R_\mathrm{L}$ で表されることから、電流増幅度 $A_\mathrm{i}=i_\mathrm{o}/i_\mathrm{b}$ は次式で与えられます。

(3) 等価回路で $R_\mathrm{B}$ から右側を見たインピーダンス $Z_\mathrm{i}'$ は以下で求められます。

等価回路を入力端から見ると、上式の $Z_\mathrm{i}'$ に並列に $R_\mathrm{B}$ が接続されています。 したがって、入力インピーダンス $Z_\mathrm{i}$ は次式で表されます。

(4) $\gamma=i_\mathrm{i}/i_\mathrm{b}=R_\mathrm{B}/(R_\mathrm{S}+R_\mathrm{B})$ とおくと、$v_\mathrm{i}=-\gamma R_\mathrm{S}i_\mathrm{b}$ と表されます。 $v_\mathrm{i}=h_\mathrm{ie}i_\mathrm{b}+(1-h_\mathrm{re})v_\mathrm{o}$ より、$i_\mathrm{b}$ は $v_\mathrm{o}$ を用いて以下の式で表されます。

上図の点Eにおけるキルヒホッフ電流則より、出力電流は以下のように $v_\mathrm{o}$ を用いて表されます。

したがって、出力インピーダンス $Z_\mathrm{o}=v_\mathrm{o}/i_\mathrm{o}$ は次式で求められます。

題意より $R_\mathrm{S}=0$、$R_\mathrm{L}\rightarrow\infty$ として、上で導出した簡易等価回路の式を代入します。

(1)

(2)

(3)

(4)

※ (2)で求めた $Z_\mathrm{i}'$ と(4)で求めた $Z_\mathrm{o}$ を無負荷における近似式で計算すると以下のようになります。

(2)及び(4)の計算結果を高精度に近似できていることが分かります。

問題の回路は分圧方式のエミッタホロワです。

(1) 直流動作の計算問題です。 入力電圧 $v_\mathrm{i}$、出力電圧 $v_\mathrm{o}$、$v_\mathrm{CE}$ の順に計算すると、以下のようになります。

(2) 練習問題2と数値も含め全く同じ計算です。

(3) $R_1$ と $R_2$ の並列合成抵抗を $R_\mathrm{B}$ とおくと、その値は以下で与えられます。

この値を代入すると、求める $Z_\mathrm{i}$ は以下のようになります。

※ (3)の結果を練習問題2の(3)を比較すると、練習問題3のエミッタホロワの入力インピーダンスは $R_1$ の影響で10分の1以下に低下しています。

% Emitter Follower (fixed/voltage divider bias)
\documentclass[border=3mm]{standalone}
\usepackage{tikz}
\usepackage{circuitikz}
\usetikzlibrary{calc}
\ctikzset{transistors/arrow pos=end}
\usepackage{newtxtext,newtxmath}
\pagecolor{white}
\begin{document}
\begin{tikzpicture}
\draw (0,0) node[npn](Tr) {};
\draw[thick] ($(Tr)-(0.35,0)$) circle [radius=13pt] node[right=13pt] {Tr};
\draw(Tr.E)
    to[european resistor, l_=$R_\mathrm{E}$, -*] ++(0,-2)
    coordinate(tmp3) % below Re
    to[short] ++(2,0)
    coordinate(tmp6) % below Rl
    to[short] ++(2.5,0)
    to[battery2, invert, l_=$V_\mathrm{CC}$] ++(0,5.1)
    to[short] ++(-4.5,0)
    coordinate(tmp1) % above Rc
    to[short] ++(0,-2)
    to[short] (Tr.C);
% %%% [BEGIN] For fixed bias: Rb %%%
% \draw(tmp1)
%     to[short, *-] ++(-1.5,0)
%     to[european resistor, l_=$R_\mathrm{B}$] ++(0,-2.3) |- (Tr.B)
%     node[pos=0.5, coordinate] (tmp5) {};
% %%%  [END]  For fixed bias: Rb %%%
%%% [BEGIN] For voltage divider bias: R1, R2 %%%
\draw(tmp1)
    to[short, *-] ++(-1.5,0)
    to[european resistor, l_=$R_1$] ++(0,-2.3) |- (Tr.B)
    node[pos=0.5, coordinate] (tmp5) {};
\draw(tmp3)
    to[short, -*] ++(-1.5,0)
    to[european resistor, l=$R_2$] ++(0,+2.8) |- (Tr.B);
%%%  [END]  For voltage divider bias: R1, R2 %%%
\draw(tmp5) to[short, *-] (Tr.B);
\draw(Tr.E)
    to[C, l_=$C_\mathrm{o}$, *-] ++(2,0)
    coordinate(tmp7)
    to[european resistor, l=$R_\mathrm{L}$, o-o] (tmp6);
\draw(tmp5)
    to[C, l=$C_\mathrm{i}$, -o] ++(-1.5,0)
    coordinate(tmp8);
\draw(tmp3)
    to[short] ++(-3,0)
    coordinate(tmp9)
    to[short, o-] ++(-1,0)
    to[european resistor, l=$R_\mathrm{S}$] ++(0,1.6)
    to[sV=$v_\mathrm{S}$] ++(0,1) |- (tmp8) to[short, -o] (tmp8);
\draw(tmp7) to[short, o-] ++(1.3,0);
\draw[-latex] ($(tmp6)+(1.2,0.2)$) -- ($(tmp7)+(1.2,-0.2)$)
    node[pos=0.5, right] {$V_\mathrm{o}$};
\draw[-latex] ($(tmp9)+(0,0.2)$) -- ($(tmp8)+(0,-0.2)$)
    node[pos=0.5, left] {$V_\mathrm{i}$};
\end{tikzpicture}
\end{document}

% Simple Equivalent Circuit (original)
\documentclass[border=3mm]{standalone}
\usepackage{tikz}
\usepackage{circuitikz}
\usepackage{newtxtext,newtxmath}
\pagecolor{white}
\begin{document}
\begin{tikzpicture}
%%% [BEGIN] equivalent circuit of common emitter BJT %%%
\draw (0,0) to[european resistor, l_=$h_\mathrm{ie}$] ++(0,-1.75);
\draw (1.5,0) to[american current source, l=$h_\mathrm{fe}i_\mathrm{b}$] ++(0,-1.75);
\draw (0,0) to[short, f<_=$i_\mathrm{b}$] ++(-1,0);
\draw (1.5,0) to[short] ++(1.5,0) to[short] ++(0,-3.5);
\draw (-1,-3.5) to[short] (3,-3.5);
\draw (0,-1.75) to[short] ++(1.5,0);
%%%  [END]  equivalent circuit of common emitter BJT %%%
\draw (0.75,-1.75) to[european resistor, l=$R_\mathrm{E}$, *-*] ++(0,-1.75);
%%% [BEGIN] input %%%
\draw (-1,0) to[european resistor, l_=$R_\mathrm{B}$, *-*] ++(0,-3.5);
\draw (-1,0) to[short, -o] ++(-1,0) to[short, f<_=$i_\mathrm{i}$] ++(-1,0)
    to[sV_<=$v_\mathrm{S}$] ++(0,-1.5)
    to[european resistor, l_=$R_\mathrm{S}$] ++(0,-2)
    to[short] ++(1,0) to[short, o-] ++(1,0);
\draw[-latex] (-2,-3.3) -- (-2,-0.2) node[pos=0.5, left] {$v_\mathrm{i}$};
%%%  [END]  input %%%
%%% [BEGIN] output %%%
\draw (1.5,-1.75) to[short, *-o] ++(2.5,0) to[short, f<=$i_\mathrm{o}$] ++(1,0)
    to[european resistor, l=$R_\mathrm{L}$] ++(0,-1.75) to[short] ++(-1,0)
    to[short, o-*] ++(-1,0);
\draw[-latex] (4,-3.3) -- (4,-1.95) node[pos=0.5, right] {$v_\mathrm{o}$};
%%%  [END]  output %%%
\end{tikzpicture}
\end{document}

% Simple Equivalent Circuit (transformed)
\documentclass[border=3mm]{standalone}
\usepackage{tikz}
\usepackage{circuitikz}
\usepackage{newtxtext,newtxmath}
\pagecolor{white}
\begin{document}
\begin{tikzpicture}
%%% [BEGIN] equivalent circuit of common collector BJT %%%
\draw (1,-3.5) to[american current source, l=$h_\mathrm{fe}i_\mathrm{b}$, *-*] ++(0,3.5);
\draw (-2,0) to[european resistor, l=$h_\mathrm{ie}$, f>^=$i_\mathrm{b}$] ++(3,0)
    to[short, f=$(h_\mathrm{fe}+1)i_\mathrm{b}$] ++(2,0);
\draw (-2,-3.5) to[short] (3,-3.5);
%%%  [END]  equivalent circuit of common collector BJT %%%
%%% [BEGIN] input %%%
\draw (-2,0) to[european resistor, l_=$R_\mathrm{B}$, *-*] ++(0,-3.5);
\draw (-2,0) to[short, -o] ++(-1,0) to[short, f<_=$i_\mathrm{i}$] ++(-1,0)
    to[sV_<=$v_\mathrm{S}$] ++(0,-1.5)
    to[european resistor, l_=$R_\mathrm{S}$] ++(0,-2)
    to[short] ++(1,0) to[short, o-] ++(1,0);
\draw[-latex] (-3,-3.3) -- (-3,-0.2) node[pos=0.5, left] {$v_\mathrm{i}$};
%%%  [END]  input %%%
%%% [BEGIN] output %%%
\draw (3,0) node[above] {E} to[european resistor, l=$R_\mathrm{E}$, *-*] ++(0,-3.5);
\draw (3,0) to[short, -o] ++(1,0) to[short, f<=$i_\mathrm{o}$] ++(1,0)
    to[european resistor, l=$R_\mathrm{L}$] ++(0,-3.5) to[short] ++(-1,0)
    to[short, o-] ++(-1,0);
\draw[-latex] (4,-3.3) -- (4,-0.2) node[pos=0.5, right] {$v_\mathrm{o}$};
%%%  [END]  output %%%
\end{tikzpicture}
\end{document}

% Common Emitter Amplifier w/ Emitter Follower (voltage divider bias)
\documentclass[border=3mm]{standalone}
\usepackage{tikz}
\usepackage{circuitikz}
\usetikzlibrary{calc}
\ctikzset{transistors/arrow pos=end}
\usepackage{newtxtext,newtxmath}
\pagecolor{white}
\begin{document}
\begin{tikzpicture}
\draw (0,0) node[npn](Tr1) {};
\draw[thick] ($(Tr1)-(0.35,0)$) circle [radius=13pt] node[right=13pt] {$\mathrm{Tr}_1$};
\draw(Tr1.E)
    to[european resistor, l_=$R_4$] ++(0,-2) % R_\mathrm{E}
    coordinate(tmp3) % below Re
    to[short] ++(7,0)
    to[battery2, invert, l_=$V_\mathrm{CC}$] ++(0,5.6)
    to[short] ++(-7,0)
    coordinate(tmp1) % above Rc
    to[european resistor, l_=$R_3$] ++(0,-2) % R_\mathrm{C}
    to[short] (Tr1.C);
\draw(tmp1)
    to[short, *-] ++(-1.5,0)
    to[european resistor, l_=$R_1$] ++(0,-2.8) |- (Tr1.B)
    node[pos=0.5, coordinate] (tmp5) {};
\draw(tmp5) to[short, *-] (Tr1.B);
\draw(tmp3)
    to[short, *-] ++(-1.5,0)
    to[european resistor=$R_2$, *-] (tmp5);
%%% [BEGIN] Bypass capacitor %%%
\draw(Tr1.E)
    to[short, *-] ++(1.3,0)
    to[C, l_=$C_1$, -*] ++(0,-2); % C_\mathrm{E}
%%%  [END]  Bypass capacitor %%%
%%% [BEGIN] input %%%
\draw(tmp5)
    to[C, l=$C_\mathrm{i}$, -o] ++(-1.5,0)
    coordinate(tmp8);
\draw(tmp3)
    to[short] ++(-3,0)
    coordinate(tmp9)
    to[short, o-] ++(-1,0)
    to[european resistor, l=$R_\mathrm{S}$] ++(0,1.6)
    to[sV=$v_\mathrm{S}$] ++(0,1) |- (tmp8) to[short, -o] (tmp8);
\draw[-latex] ($(tmp9)+(0,0.2)$) -- ($(tmp8)+(0,-0.2)$)
    node[pos=0.5, left] {$V_\mathrm{i}$};
%%%  [END]  input %%%
%%% [BEGIN] Emitter follower %%%
\draw (Tr1.C) to[short, *-] ++(2.1,0) node[npn, anchor=B](Tr2) {};
\draw[thick] ($(Tr2)-(0.35,0)$) circle [radius=13pt] node[right=13pt] {$\mathrm{Tr}_2$};
\draw (Tr2.E) to[european resistor, l_=$R_5$, -*] (Tr2.E |- tmp3); % R_\mathrm{E}
\draw (Tr2.C) to[short, -*] (Tr2.C |- tmp1);
%%%  [END]  Emitter follower %%%
%%% [BEGIN] output %%%
\draw(Tr2.E)
    to[C, l_=$C_\mathrm{o}$, *-] ++(1.5,0)
    coordinate(tmp7)
    to[european resistor, l=$R_\mathrm{L}$, o-o] (tmp7 |- tmp3)
    coordinate(tmp6); % below Rl
\draw(tmp7) to[short, o-] ++(1.3,0);
\draw[-latex] ($(tmp6)+(1.2,0.2)$) -- ($(tmp7)+(1.2,-0.2)$)
    node[pos=0.5, right] {$V_\mathrm{o}$};
%%%  [END]  output %%%
\end{tikzpicture}
\end{document}

% Equivalent Circuit (transformed)
\documentclass[border=3mm]{standalone}
\usepackage{tikz}
\usepackage{circuitikz}
\usepackage{newtxtext,newtxmath}
\pagecolor{white}
\begin{document}
\begin{tikzpicture}
%%% [BEGIN] equivalent circuit of common collector BJT %%%
\draw (2.5,-3.5) to[american current source, l=$h_\mathrm{fe}i_\mathrm{b}$, *-*] ++(0,3.5);
\draw (4.5,-3.5) to[european resistor, l=$1/h_\mathrm{oe}$, *-*] ++(0,3.5);
\draw (-2,0) to[short, f>^=$i_\mathrm{b}$] ++(1,0) to[european resistor, l=$h_\mathrm{ie}$] ++(1.5,0)
    to[sV<=$h_\mathrm{re}v_\mathrm{ce}$, sources/symbol/rotate=auto] ++(2,0)
    to[short, f=$(h_\mathrm{fe}+1)i_\mathrm{b}$] ++(2,0) to[short, -o] ++(1,0);
\draw (-2,-3.5) to[short, -o] (5.5,-3.5);
\draw[latex-] (5.5,-3.3) -- (5.5,-0.2) node[pos=0.5, right] {$v_\mathrm{ce}$};
%%%  [END]  equivalent circuit of common collector BJT %%%
%%% [BEGIN] input %%%
\draw (-2,0) to[european resistor, l_=$R_\mathrm{B}$, *-*] ++(0,-3.5);
\draw (-2,0) to[short, -o] ++(-1,0) to[short, f<_=$i_\mathrm{i}$] ++(-1,0)
    to[sV_<=$v_\mathrm{S}$] ++(0,-1.5)
    to[european resistor, l_=$R_\mathrm{S}$] ++(0,-2)
    to[short] ++(1,0) to[short, o-] ++(1,0);
\draw[-latex] (-3,-3.3) -- (-3,-0.2) node[pos=0.5, left] {$v_\mathrm{i}$};
%%%  [END]  input %%%
%%% [BEGIN] output %%%
\draw (6.5,0) to[european resistor, l=$R_\mathrm{E}$, *-*] ++(0,-3.5);
\draw (5.5,0) node[above] {E} to[short, o-o] ++(2,0) to[short, f<=$i_\mathrm{o}$] ++(1,0)
    to[european resistor, l=$R_\mathrm{L}$] ++(0,-3.5) to[short] ++(-1,0)
    to[short, o-o] ++(-2,0) node[below] {C};
\draw[-latex] (7.5,-3.3) -- (7.5,-0.2) node[pos=0.5, right] {$v_\mathrm{o}$};
%%%  [END]  output %%%
\end{tikzpicture}
\end{document}

% Emitter Follower (unloaded, fixed/voltage divider bias)
\documentclass[border=3mm]{standalone}
\usepackage{tikz}
\usepackage{circuitikz}
\usetikzlibrary{calc}
\ctikzset{transistors/arrow pos=end}
\usepackage{newtxtext,newtxmath}
\pagecolor{white}
\begin{document}
\begin{tikzpicture}
\draw (0,0) node[npn](Tr) {};
\draw[thick] ($(Tr)-(0.35,0)$) circle [radius=13pt] node[right=13pt] {Tr};
\draw(Tr.E)
    to[european resistor, l_=$R_\mathrm{E}$, -*] ++(0,-2)
    coordinate(tmp3) % below Re
    to[short, -o] ++(2.5,0)
    coordinate(tmp6) % below Rl
    to[short] ++(1,0)
    to[battery2, invert, l_=$V_\mathrm{CC}$] ++(0,5.1)
    to[short] ++(-3.5,0)
    coordinate(tmp1) % above Rc
    to[short] ++(0,-2)
    to[short] (Tr.C);
% %%% [BEGIN] For fixed bias: Rb %%%
% \draw(tmp1)
%     to[short, *-] ++(-1.5,0)
%     to[european resistor, l_=$R_\mathrm{B}$] ++(0,-2.3) |- (Tr.B)
%     node[pos=0.5, coordinate] (tmp5) {};
% %%%  [END]  For fixed bias: Rb %%%
%%% [BEGIN] For voltage divider bias: R1, R2 %%%
\draw(tmp1)
    to[short, *-] ++(-1.5,0)
    to[european resistor, l_=$R_1$] ++(0,-2.3) |- (Tr.B)
    node[pos=0.5, coordinate] (tmp5) {};
\draw(tmp3)
    to[short, -*] ++(-1.5,0)
    to[european resistor, l=$R_2$] ++(0,+2.8) |- (Tr.B);
%%%  [END]  For voltage divider bias: R1, R2 %%%
\draw(tmp5) to[short, *-] (Tr.B);
\draw(Tr.E)
    to[C, l_=$C_\mathrm{o}$, *-o, f=$i_\mathrm{o}$] ++(2.5,0)
    coordinate(tmp7);
\draw(tmp5)
    to[C, l=$C_\mathrm{i}$, -o] ++(-1.5,0)
    coordinate(tmp8);
\draw(tmp3)
    to[short] ++(-3,0)
    coordinate(tmp9)
    to[short, o-] ++(-1,0)
    to[sV=$v_\mathrm{S}$, f=$i_\mathrm{i}$] ++(0,2.75) |- (tmp8) to[short, -o] (tmp8);
\draw[-latex] ($(tmp6)+(0,0.2)$) -- ($(tmp7)+(0,-0.2)$)
    node[pos=0.5, right] {$v_\mathrm{o}$};
\draw[-latex] ($(tmp9)+(0,0.2)$) -- ($(tmp8)+(0,-0.2)$)
    node[pos=0.5, left] {$v_\mathrm{i}$};
\end{tikzpicture}
\end{document}