Frequency Response of CE Amplifier

EEE 131 THX2/Y

Lawrence Quizon

Electrical and Electronics Engineering Institute
University of the Philippines
Diliman

2025-11-28

Coverage

  • Frequency Response of CS/CE Amplifiers
  • CE Amplifier with an Output Pole
  • CE Amplifier with an Input Pole

CE Amplifier with an Output Pole

We begin by analyzing the CE amplifier frequency response.

Assumptions:

  • Ignore BJT parasitic capacitances (for now).
  • Assume \(r_o \rightarrow \infty\).
  • An external load capacitance \(C_L\) dominates the output.

CE Amplifier with an Output Pole

KCL @ \(v_o\):

\[\frac{v_o}{R_C || \frac{1}{sC_L}} + g_m v_i = 0\]

Then, rearranging, \[A_v(s) = -g_mR_C \left( \frac{1}{1 + sR_C C_L} \right)\]

and we have the form

\[A_v(s) = A_0 \left( \frac{1}{1 + s/p_1} \right) \]

CE with Output Poles: \(A_V\)

\[A_v(s) = -g_mR_C \left( \frac{1}{1 + sR_C C_L} \right)\]

CE with Output Pole: \(R_o\)

CE with Output Pole: \(G_m\)

CE with Output Pole: \(Z_i\)

CE Transconductance \(G_m\)

Recalling that \(A_v = -G_m Z_o\), we can solve for the effective transconductance:

\[G_m = \frac{-A_v}{Z_o}\]

\[G_m = \frac{-\left[ \frac{-g_m R_C}{1 + j\omega R_C C_L} \right]}{\frac{R_C}{1 + j\omega R_C C_L}}\]

\[G_m = g_m\]

In this simplified model, the transconductance is frequency independent.

CE Input Impedance \(Z_i\)

\[Z_i = \frac{v_i}{i_i} \bigg|_{v_o=0}\]

By simple inspection of the small signal model: \[Z_i = r_\pi\]

The input impedance is purely resistive and constant across frequency (in this specific configuration).

CE Two-Port Equivalent (Output Pole Only)

We can construct a two-port network representing the amplifier so far:

\[Z_i = r_\pi\]

\[G_m = g_m\]

\[Z_o = \frac{R_C}{1 + sR_C C_L}\]

CE Amplifier with Input Pole

The CE Amplifier with an Input Pole

Now we add a source resistance \(R_S\) and a base-emitter capacitance \(C_\pi\).

We must determine how \(v_{be}\) relates to \(v_i\) considering the frequency dependent voltage divider at the input.

Deriving the Input Transfer Function

Applying voltage division at the input loop:

\[\frac{v_{be}}{v_i} = \frac{Z_\pi}{Z_\pi + R_S}\] where \(Z_\pi = r_\pi \parallel \frac{1}{sC_\pi}\).

\[\frac{v_{be}}{v_i} = \frac{\frac{r_\pi}{1 + s r_\pi C_\pi}}{\frac{r_\pi}{1 + s r_\pi C_\pi} + R_S}\]

Multiplying numerator and denominator by \((1 + s r_\pi C_\pi)\): \[\frac{v_{be}}{v_i} = \frac{r_\pi}{r_\pi + R_S + s r_\pi R_S C_\pi}\]

Deriving the Input Transfer Function

\[\frac{v_{be}}{v_i} = \frac{r_\pi}{r_\pi + R_S + s r_\pi R_S C_\pi}\]

We factor out \((r_\pi + R_S)\) from the denominator to normalize the DC term:

\[\frac{v_{be}}{v_i} = \frac{r_\pi}{r_\pi + R_S} \left( \frac{1}{1 + s C_\pi \frac{r_\pi R_S}{r_\pi + R_S}} \right)\]

This results in an input attenuation factor \(A_{i}\) and an input pole \(\omega_{p1}\): \[\frac{v_{be}}{v_i} = A_i \frac{1}{1 + j\frac{\omega}{\omega_{p1}}}\] where \(A_i = \frac{r_\pi}{r_\pi + R_S}\) and \(\omega_{p1} = \frac{1}{C_\pi (r_\pi \parallel R_S)}\).

Total Transfer Function

We combine the input section response (\(A_1\)) with the output section response (\(A_2\)).

\[A_v(s) = \underbrace{\left( \frac{v_{be}}{v_i} \right)}_{\text{Input Section}} \cdot \underbrace{\left( \frac{v_o}{v_{be}} \right)}_{\text{Output Section}}\]

\[A_v(s) = \left[ \frac{r_\pi}{r_\pi + R_S} \cdot \frac{1}{1 + j\frac{\omega}{\omega_{p1}}} \right] \cdot \left[ -g_m R_C \cdot \frac{1}{1 + j\frac{\omega}{\omega_{p2}}} \right]\]

Total Gain

\[A_v(s) = A_{mid} \frac{1}{(1 + j\frac{\omega}{\omega_{p1}})(1 + j\frac{\omega}{\omega_{p2}})}\] \[\omega_{p1} = \frac{1}{C_\pi (r_\pi \parallel R_S)}, \quad \omega_{p2} = \frac{1}{R_C C_L}\]

Magnitude Response (Two Poles)

Typically, the output pole \(\omega_{p2}\) is lower frequency than the input pole \(\omega_{p1}\).

  • Flat gain up to \(\omega_{p2}\).
  • Slope of \(-20\) dB/dec between \(\omega_{p2}\) and \(\omega_{p1}\).
  • Slope of \(-40\) dB/dec after \(\omega_{p1}\).

Phase Response

The phase response depends on the separation of the poles.

\[\angle A_v(\omega) = 180^\circ - \tan^{-1}\left(\frac{\omega}{\omega_{p1}}\right) - \tan^{-1}\left(\frac{\omega}{\omega_{p2}}\right)\]

Widely Separated If poles are far apart, the phase drops in distinct steps.

Close Proximity If poles are close, the phase shifts merge, creating a steeper descent.

Generalized Two-Port Model

Frequency Dependent Parameters

We can generalize the two-port parameters to include frequency effects:

Input Impedance \(Z_i(\omega)\) \[Z_i(\omega) = R_S + \frac{r_\pi}{1 + j\omega r_\pi C_\pi}\] Derivation involves simple series/parallel combination of \(R_S, r_\pi, C_\pi\).

Transconductance \(G_m(\omega)\) \[G_m(\omega) = \frac{g_m}{1 + j\omega (R_S \parallel r_\pi) C_\pi} \cdot \frac{r_\pi}{r_\pi + R_S}\] Effective transconductance rolls off due to the input pole.

Next Lecture

  • Miller Capacitance
    • What happens when a capacitor connects Input and Output?
  • Bypass and Coupling Capacitors
    • Effect on Low Frequency Response

End of Part 1

Frequency Response of CE Amplifier