# Design of a Fixed IF Down-Conversion Double-Balanced Mixer for UHF Band Applications
## 4 Simulation and Experimental Results
### Double-Balanced Down-Conversion Mixer Noise Calculation and Simulation
#### 4.2.5 Double-Balanced Mixer Noise Figure Prediction Calculation and Simulation Validation  
Due to the double transconductance stages and switches in the double-balanced mixer, the noise contribution down-converted to the IF port is doubled. Therefore, the total noise at the mixer output can be written as from Equation (27):  
\[\overline{V_{n0,total}^{2}}=4 KTR_{L}^{\ 2}\bigg{[}\frac{2}{R_{L}}+2\bigg{(}(2r_{b2}g _{m}+1)\bigg{(}\frac{I_{\RF}}{\pi V_{LO}}\bigg{)}\bigg{)}+2\bigg{(}{g_{m}}^{2 }\bigg{(}r_{b1}+\frac{1}{2g_{m}}\bigg{)}\bigg{)}\bigg{]} \tag{28}\]  
The input referred noise of the given mixer is as follows:  
\[\overline{V_{n,total}^{2}}=\frac{1}{(gain)^{2}}\overline{V_{n0,total}^{2}} \tag{29}\]  
\[\overline{V_{n,total}^{2}}=\frac{4 KTR_{L}^{\ 2}\bigg{[}\frac{2}{R_{L}}+2\Big{(}(2r_{b2}g _{m}+1)\Big{(}\frac{I_{\RF}}{\pi V_{LO}}\Big{)}\Big{)}+2\Big{(}{g_{m}}^{2} \Big{(}r_{b1}+\frac{1}{2g_{m}}\Big{)}\Big{)}\bigg{]}}{\Big{(}\frac{2g_{m1}R_{L }}{\pi}\Big{)}^{2}} \tag{30}\]  
\[\overline{V_{n,total}^{2}}=\pi^{2}KT\bigg{[}\frac{2}{{g_{m}}^{2}R_{L}}+2\bigg{(} r_{b1}+\frac{1}{2g_{m}}\bigg{)}+\bigg{(}\frac{2}{g_{m}}^{2}(2r_{b2}g_{m}+1) \bigg{(}\frac{I_{\RF}}{\pi V_{LO}}\bigg{)}\bigg{)}\bigg{]} \tag{31}\]  
So, the single-sideband noise figure can be given as follows:  
\[NF_{SSB}=1+\frac{\overline{V_{n,total}^{2}}}{4 KTR_{S}} \tag{32}\]  
The noise equation derived in (28) and (31) for output- and input-referred noise assumes that RF stages do not feature emitter degeneration. When the emitter is degenerated, \(g_{m}\) is substituted with \(g_{m_{eff}}\) as derived in Section 3.1. In the following, we will estimate the noise figure of the double-balanced mixer illustrated in Figure 2.  
The internal base resistances of BFP840 (\(r_{b1}\) at RF stage) and BFP640 (\(r_{b2}\) at LO stage) were found in the space parameter file: the values are 16.87 ohms and 4.406 ohms, respectively. The simulated transconductance \(g_{m}\) is approximately 0.182 S. The local oscillator port has a power of \(-5\) dBm; according to the simulation, this represents about 355 mVp-p when converted to mV for the 50-ohm system.  
Upon calculating the individual terms listed in Equation (31), we obtain the data from Table 1:Substituting all of the values in Equation (31) and then in (32), the result of \(\frac{\overline{V_{n,total}^{2}}}{4 KTR_{S}}\) will be roughly 5.264251.  
\[NF_{SSB}=10 log(1+5.264251)=10\text{log}(6.264251)=7.968692\text{ dB}.\]  
The optimal condition for the gain and noise figure as a function of the LO drive is displayed in Figure 15. The predicted approximation is about 7.9686 dB, while the simulated noise figure measures about 8.31 dB when the LO power is at \(-5\) dBm.  
We swept the RF and LO frequency across the range of 400 MHz to 2.5 GHz while maintaining a constant IF of 145 MHz in order to determine the operating bandwidth of the down-conversion mixer. Figure 16 shows the single-sideband noise figure and simulated conversion gain as a function of RF frequency.  
We simplified the gain and noise figure calculation by neglecting the internal capacitances of the transconductance and switching stages, since RF and LO frequencies are much lower than \(f_{T}\). The assumption is that the \(C_{\pi}\) and \(C_{\mu}\) of each device are small enough to act as open circuits at all signal frequencies of interest. Figure 16 shows that the gain stays nearly constant (11 dB +/ - 0.5 dB) between 600 MHz and 2000 MHz.  
\begin{table}
\begin{tabular}{c c} \hline
**Terms** & **Calculated Values** \\ \hline \(g_{m\_eff}=\frac{g_{m}}{1+\beta_{m}z_{E}}\) & 0.028628 \\ \hline \(\frac{2}{g_{m}^{2}R_{L}}\) & 10.75013 \\ \hline \(2*(r_{b1}+\frac{1}{2g_{m}})\) & 68.67051 \\ \hline \(\frac{2}{g_{m}^{2}}\) & 2440.28 \\ \hline \((2r_{b2}g_{m}+1)\) & 1.252272 \\ \hline \(\frac{l_{BF}}{\pi V_{LO}}\) & 0.008954 \\ \hline \end{tabular}
\end{table}
Table 1: Calculated terms of the noise figure of the double-balanced mixer.  
Figure 15: Conversion gain and SSB noise figure as function of LO power.